2903:
31:
3496:
8175:
3212:
7811:
13005:
3079:
1373:
572:
4802:
2866:
4581:
269:
249:
4586:
2635:
8170:{\displaystyle m{\ddot {x}}+q\left({\frac {\partial A_{x}}{\partial t}}+{\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{x}}{\partial y}}{\dot {y}}+{\frac {\partial A_{x}}{\partial z}}{\dot {z}}\right)=-q{\frac {\partial \phi }{\partial x}}+q\left({\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{y}}{\partial x}}{\dot {y}}+{\frac {\partial A_{z}}{\partial x}}{\dot {z}}\right)}
4376:
7602:
8705:
7806:
9103:
4357:
4187:
9649:
10240:
2312:
1021:
7092:
6179:
8179:
4797:{\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,\ t)=\oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {E} (\mathbf {r} ,\ t)+\oint _{\partial \Sigma (t)}\!\!\!\!\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\ t)\,\mathrm {d} {\boldsymbol {\ell }}}
2861:{\displaystyle q_{\mathrm {G} }={\frac {q_{\mathrm {SI} }}{\sqrt {4\pi \varepsilon _{0}}}},\quad \mathbf {E} _{\mathrm {G} }={\sqrt {4\pi \varepsilon _{0}}}\,\mathbf {E} _{\mathrm {SI} },\quad \mathbf {B} _{\mathrm {G} }={\sqrt {4\pi /\mu _{0}}}\,{\mathbf {B} _{\mathrm {SI} }},\quad c={\frac {1}{\sqrt {\varepsilon _{0}\mu _{0}}}}.}
7606:
9497:
9351:
4576:{\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,t)=-\int _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot {\frac {\partial }{\partial t}}\mathbf {B} (\mathbf {r} ,t)+\oint _{\partial \Sigma (t)}\!\!\!\!\mathbf {v} \times \mathbf {B} \,\mathrm {d} {\boldsymbol {\ell }}}
8965:
4192:
4030:
6017:
5175:
8927:
4899:
Faraday's law of induction holds whether the loop of wire is rigid and stationary, or in motion or in process of deformation, and it holds whether the magnetic field is constant in time or changing. However, there are cases where
Faraday's law is either inadequate or difficult to use, and application
6985:
9502:
9962:
3142:
upon a point charge, but such electromagnetic forces are not the entire picture. Charged particles are possibly coupled to other forces, notably gravity and nuclear forces. Thus, Maxwell's equations do not stand separate from other physical laws, but are coupled to them via the charge and current
3055:
derived the modern form of the formula for the electromagnetic force which includes the contributions to the total force from both the electric and the magnetic fields. Lorentz began by abandoning the
Maxwellian descriptions of the ether and conduction. Instead, Lorentz made a distinction between
3429:), the phenomenon underlying many electrical generators. When a conductor is moved through a magnetic field, the magnetic field exerts opposite forces on electrons and nuclei in the wire, and this creates the EMF. The term "motional EMF" is applied to this phenomenon, since the EMF is due to the
2982:. From a modern perspective it is possible to identify in Maxwell's 1865 formulation of his field equations a form of the Lorentz force equation in relation to electric currents, although in the time of Maxwell it was not evident how his equations related to the forces on moving charged objects.
2962:
that same year was able to devise through experimentation the formula for the angular dependence of the force between two current elements. In all these descriptions, the force was always described in terms of the properties of the matter involved and the distances between two masses or charges
2178:
726:
1310:
The
Lorentz force is a force exerted by the electromagnetic field on the charged particle, that is, it is the rate at which linear momentum is transferred from the electromagnetic field to the particle. Associated with it is the power which is the rate at which energy is transferred from the
5611:
10880:
2622:
523:. In 1835, Gauss assumed that each segment of a DC loop contains an equal number of negative and positive point charges that move at different speeds. If Coulomb's law were completely correct, no force should act between any two short segments of such current loops. However, around 1825,
11955:
pulls a vector (in the space algebra) from the translational part, while the wedge-product creates a trivector (in the space algebra) who is dual to a vector which is the usual magnetic field vector. The relativistic velocity is given by the (time-like) changes in a time-position vector
6022:
11370:
10563:
456:
are thereby defined throughout space and time, and these are called the "electric field" and "magnetic field". The fields are defined everywhere in space and time with respect to what force a test charge would receive regardless of whether a charge is present to experience the force.
11477:
9203:
2511:
2091:
9355:
7075:
9212:
6783:
11620:
3129:
While the modern
Maxwell's equations describe how electrically charged particles and currents or moving charged particles give rise to electric and magnetic fields, the Lorentz force law completes that picture by describing the force acting on a moving point charge
6307:
2902:
7597:{\displaystyle {\begin{aligned}{\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {x}}}}=m{\ddot {x}}+q{\frac {\mathrm {d} A_{x}}{\mathrm {d} t}}&=m{\ddot {x}}+{\frac {q}{\mathrm {d} t}}\left\\&=m{\ddot {x}}+q\left\\\end{aligned}}}
2986:
was the first to attempt to derive from
Maxwell's field equations the electromagnetic forces on a moving charged object in terms of the object's properties and external fields. Interested in determining the electromagnetic behavior of the charged particles in
4895:
3861:
3050:
invented the modern vector notation and applied it to
Maxwell's field equations; he also (in 1885 and 1889) had fixed the mistakes of Thomson's derivation and arrived at the correct form of the magnetic force on a moving charged object. Finally, in 1895,
1494:
5897:
5082:
1849:
10663:
10407:
9941:
8790:
8700:{\displaystyle {\begin{aligned}F_{x}&=-q\left({\frac {\partial \phi }{\partial x}}+{\frac {\partial A_{x}}{\partial t}}\right)+q\left\\&=qE_{x}+q\\&=qE_{x}+q_{x}\\&=qE_{x}+q(\mathbf {\dot {r}} \times \mathbf {B} )_{x}\end{aligned}}}
3470:
was partially motivated by the desire to better understand this link between the two effects. In fact, the electric and magnetic fields are different facets of the same electromagnetic field, and in moving from one inertial frame to another, the
3660:
6787:
7801:{\displaystyle {\frac {\partial L}{\partial x}}=-q{\frac {\partial \phi }{\partial x}}+q\left({\frac {\partial A_{x}}{\partial x}}{\dot {x}}+{\frac {\partial A_{y}}{\partial x}}{\dot {y}}+{\frac {\partial A_{z}}{\partial x}}{\dot {z}}\right)}
3222:
When a wire carrying an electric current is placed in a magnetic field, each of the moving charges, which comprise the current, experiences the
Lorentz force, and together they can create a macroscopic force on the wire (sometimes called the
9775:
484:
fields, which would alter the electromagnetic force that it experiences. In addition, if the charge experiences acceleration, as if forced into a curved trajectory, it emits radiation that causes it to lose kinetic energy. See for example
9098:{\displaystyle {\frac {\mathrm {d} \mathbf {P} }{\mathrm {d} t}}={\frac {\partial L}{\partial \mathbf {r} }}=q{\partial \mathbf {A} \over \partial \mathbf {r} }\cdot {\dot {\mathbf {r} }}-q{\partial \phi \over \partial \mathbf {r} }}
5523:
1196:
10700:
1630:
4352:{\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q(\mathbf {r} ,\ t)=-{\frac {\mathrm {d} }{\mathrm {d} t}}\int _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot \mathbf {B} (\mathbf {r} ,\ t).}
4012:
4182:{\displaystyle \oint _{\partial \Sigma (t)}\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {E} (\mathbf {r} ,\ t)=-\ \int _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot {\frac {\mathrm {d} \mathbf {B} (\mathbf {r} ,\,t)}{\mathrm {d} t}}}
2531:
3389:
181:
712:
11891:
12396:
11060:
10412:
11823:
8779:
3154:
In real materials the
Lorentz force is inadequate to describe the collective behavior of charged particles, both in principle and as a matter of computation. The charged particles in a material medium not only respond to the
1362:
11396:
9107:
3576:
2427:
1769:
12754:
6676:
413:
3040:
8184:
12151:
Note that the order is important because between a bivector and a vector the dot product is anti-symmetric. Upon a spacetime split like one can obtain the velocity, and fields as above yielding the usual expression.
2014:
12102:
9644:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{m{\dot {\mathbf {r} }} \over {\sqrt {1-\left({\frac {\dot {\mathbf {r} }}{c}}\right)^{2}}}}=q\left(\mathbf {E} +{\dot {\mathbf {r} }}\times \mathbf {B} \right).}
3269:
11049:
6992:
6506:
10235:{\displaystyle F^{\alpha \beta }={\begin{pmatrix}0&-E_{x}/c&-E_{y}/c&-E_{z}/c\\E_{x}/c&0&-B_{z}&B_{y}\\E_{y}/c&B_{z}&0&-B_{x}\\E_{z}/c&-B_{y}&B_{x}&0\end{pmatrix}}}
6681:
731:
558:, i.e. it should not be used for higher velocities and accelerations. However, the Weber force illustrates that the Lorentz force can be traced back to central forces between numerous point-like charge carriers.
11542:
2307:{\displaystyle \mathbf {f} =\left(\rho _{f}-\nabla \cdot \mathbf {P} \right)\mathbf {E} +\left(\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\right)\times \mathbf {B} .}
1016:{\displaystyle {\begin{aligned}F_{x}&=q\left(E_{x}+v_{y}B_{z}-v_{z}B_{y}\right),\\F_{y}&=q\left(E_{y}+v_{z}B_{x}-v_{x}B_{z}\right),\\F_{z}&=q\left(E_{z}+v_{x}B_{y}-v_{y}B_{x}\right).\end{aligned}}}
6217:
6377:
4979:-field, and another portion of the circuit is held stationary, the flux linking the entire closed circuit can change due to the shift in relative position of the circuit's component parts with time (surface
6612:
7097:
5087:
11684:
6174:{\displaystyle \nabla _{\dot {\mathbf {x} }}={\hat {x}}{\dfrac {\partial }{\partial {\dot {x}}}}+{\hat {y}}{\dfrac {\partial }{\partial {\dot {y}}}}+{\hat {z}}{\dfrac {\partial }{\partial {\dot {z}}}}.}
4807:
3789:
12554:
1430:
1781:
472:, the Lorentz force is only a definition in principle because a real particle (as opposed to the hypothetical "test charge" of infinitesimally-small mass and charge) would generate its own finite
12870:
10253:
9793:
3316:
2171:
10575:
1717:
6449:
3581:
516:
can be applied. The sum of the Weber forces of all charge carriers in a closed DC loop on a single test charge produces - regardless of the shape of the current loop - the
Lorentz force.
3899:
3121:
of this point. The drift speeds may differ for various species depending on their charge states, masses, or temperatures, possibly resulting in electric currents or chemical separation.
3328:
should now be understood as the vector connecting the end points of the curved wire with direction from starting to end point of conventional current. Usually, there will also be a net
12629:
9492:{\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=-{\frac {\partial }{\partial \mathbf {r} }}\left({\sqrt {(\mathbf {P} -q\mathbf {A} )^{2}+(mc^{2})^{2}}}+q\phi \right)}
12146:
9346:{\displaystyle {\frac {\mathrm {d} \mathbf {r} }{\mathrm {d} t}}={\frac {\partial }{\partial \mathbf {p} }}\left({\sqrt {(\mathbf {P} -q\mathbf {A} )^{2}+(mc^{2})^{2}}}+q\phi \right)}
3227:). By combining the Lorentz force law above with the definition of electric current, the following equation results, in the case of a straight stationary wire in a homogeneous field:
1521:
14262:
See
Jackson, page 2. The book lists the four modern Maxwell's equations, and then states, "Also essential for consideration of charged particle motion is the Lorentz force equation,
1987:
1926:
11537:
6550:
3085:
in a homogeneous magnetic field. (A) No disturbing force (B) With an electric field, E (C) With an independent force, F (e.g. gravity) (D) In an inhomogeneous magnetic field, grad H
5712:
5436:
5297:
5889:
1028:
1576:
1303:
The magnetic force component of the Lorentz force manifests itself as the force that acts on a current-carrying wire in a magnetic field. In that context, it is also called the
3957:
2396:
12450:
3342:
124:
11914:
11719:
2524:, which is the most common. However, other conventions with the same physics (i.e. forces on e.g. an electron) are possible and used. In the conventions used with the older
12787:
11828:
1571:
1546:
12267:
11990:
5518:
5486:
5461:
2418:
2363:
2009:
1951:
1893:
1652:
11953:
11763:
11746:
2134:
13843:
8722:
11697:, so the relativistic form of the Lorentz force law can best be exhibited starting from a coordinate-independent expression for the electromagnetic and magnetic fields
2341:
14728:
See Griffiths, page 326, which states that Maxwell's equations, "together with the force law...summarize the entire theoretical content of classical electrodynamics".
12028:
9703:
6012:{\displaystyle \nabla _{\mathbf {x} }={\hat {x}}{\dfrac {\partial }{\partial x}}+{\hat {y}}{\dfrac {\partial }{\partial y}}+{\hat {z}}{\dfrac {\partial }{\partial z}}}
5170:{\displaystyle {\begin{aligned}\mathbf {E} &=-\nabla \phi -{\frac {\partial \mathbf {A} }{\partial t}}\\\mathbf {B} &=\nabla \times \mathbf {A} \end{aligned}}}
1314:
12480:
12262:
12232:
3520:
1871:
1676:
12638:
8922:{\displaystyle L=-mc^{2}{\sqrt {1-\left({\frac {\dot {\mathbf {r} }}{c}}\right)^{2}}}+q\mathbf {A} (\mathbf {r} )\cdot {\dot {\mathbf {r} }}-q\phi (\mathbf {r} )}
6621:
531:. Based on this law, Gauss concluded that the electromagnetic force between two point charges depends not only on the distance but also on the relative velocity.
363:
2994:
12890:
12202:
12182:
2111:
13562:
9675:
5032:
6980:{\displaystyle L={\frac {m}{2}}\left({\dot {x}}^{2}+{\dot {y}}^{2}+{\dot {z}}^{2}\right)+q\left({\dot {x}}A_{x}+{\dot {y}}A_{y}+{\dot {z}}A_{z}\right)-q\phi }
12033:
3230:
658:
10891:
2934:
Early attempts to quantitatively describe the electromagnetic force were made in the mid-18th century. It was proposed that the force on magnetic poles, by
2424:
density. In this way, the Lorentz force can explain the torque applied to a permanent magnet by the magnetic field. The density of the associated power is
3397:, which describes how two current-carrying wires can attract or repel each other, since each experiences a Lorentz force from the other's magnetic field.
3163:
fields but also generate these fields. Complex transport equations must be solved to determine the time and spatial response of charges, for example, the
14704:
13535:
9671:
3060:
and sought to apply the Maxwell equations at a microscopic scale. Using Heaviside's version of the Maxwell equations for a stationary ether and applying
508:
is only valid for point charges at rest. In fact, the electromagnetic force between two point charges depends not only on the distance but also on the
13547:
11694:
6562:
5606:{\displaystyle {\frac {\mathrm {d} \mathbf {A} }{\mathrm {d} t}}={\frac {\partial \mathbf {A} }{\partial t}}+(\mathbf {v} \cdot \nabla )\mathbf {A} }
10875:{\displaystyle {\frac {\mathrm {d} p^{1}}{\mathrm {d} \tau }}=qU_{\beta }F^{1\beta }=q\left(U_{0}F^{10}+U_{1}F^{11}+U_{2}F^{12}+U_{3}F^{13}\right).}
1725:
1364:
Notice that the magnetic field does not contribute to the power because the magnetic force is always perpendicular to the velocity of the particle.
11924:), which has six degrees of freedom corresponding to boosts (rotations in space-time planes) and rotations (rotations in space-space planes). The
2617:{\displaystyle \mathbf {F} =q_{\mathrm {G} }\left(\mathbf {E} _{\mathrm {G} }+{\frac {\mathbf {v} }{c}}\times \mathbf {B} _{\mathrm {G} }\right),}
11633:
1025:
In general, the electric and magnetic fields are functions of the position and time. Therefore, explicitly, the Lorentz force can be written as:
3950:
The above result can be compared with the version of Faraday's law of induction that appears in the modern Maxwell's equations, called here the
14957:
11365:{\displaystyle {\frac {\mathrm {d} p^{1}}{\mathrm {d} \tau }}=q\gamma \left=q\gamma \left(E_{x}+v_{y}B_{z}-v_{z}B_{y}\right)=q\gamma \left\,.}
10558:{\displaystyle \gamma (v)={\frac {1}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}={\frac {1}{\sqrt {1-{\frac {v_{x}^{2}+v_{y}^{2}+v_{z}^{2}}{c^{2}}}}}}}
6454:
3458:
Both of these EMFs, despite their apparently distinct origins, are described by the same equation, namely, the EMF is the rate of change of
12796:
11472:{\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} \tau }}=q\gamma \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right),}
9198:{\displaystyle \mathbf {P} -q\mathbf {A} ={\frac {m{\dot {\mathbf {r} }}}{\sqrt {1-\left({\frac {\dot {\mathbf {r} }}{c}}\right)^{2}}}}}
13798:
13567:
2506:{\displaystyle \left(\mathbf {J} _{f}+\nabla \times \mathbf {M} +{\frac {\partial \mathbf {P} }{\partial t}}\right)\cdot \mathbf {E} .}
14843:
6324:
5440:(Notice that the coordinates and the velocity components should be treated as independent variables, so the del operator acts only on
2143:
1689:
193:(proportional to the magnitude of the field and the quantity of charge), and (2) a force at right angles to both the magnetic field
13577:
3436:
In other electrical generators, the magnets move, while the conductors do not. In this case, the EMF is due to the electric force (
2175:
If we separate the total charge and total current into their free and bound parts, we get that the density of the Lorentz force is
1954:
78:
that occurs in the vicinity of electrically neutral, current-carrying conductors causing moving electrical charges to experience a
274:
Beam of electrons moving in a circle, due to the presence of a magnetic field. Purple light revealing the electron's path in this
14738:
3866:
3042:
Thomson derived the correct basic form of the formula, but, because of some miscalculations and an incomplete description of the
2528:, which are somewhat more common among some theoretical physicists as well as condensed matter experimentalists, one has instead
2086:{\displaystyle \mathbf {f} =\nabla \cdot {\boldsymbol {\sigma }}-{\dfrac {1}{c^{2}}}{\dfrac {\partial \mathbf {S} }{\partial t}}}
30:
12106:
The proper (invariant is an inadequate term because no transformation has been defined) form of the Lorentz force law is simply
3443:) term in the Lorentz Force equation. The electric field in question is created by the changing magnetic field, resulting in an
3277:
is a vector whose magnitude is the length of the wire, and whose direction is along the wire, aligned with the direction of the
13402:
7070:{\displaystyle {\frac {\mathrm {d} }{\mathrm {d} t}}{\frac {\partial L}{\partial {\dot {x}}}}={\frac {\partial L}{\partial x}}}
3200:
12936:
In its manifestation as the Laplace force on an electric current in a conductor, this force occurs in many devices including:
6778:{\displaystyle L=T-V={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot \mathbf {\dot {r}} -q\phi }
3448:
1292:. According to some definitions, the term "Lorentz force" refers specifically to the formula for the magnetic force, with the
15098:
14880:
14337:
14205:
14180:
14053:
14023:
13417:
13412:
13039:
12486:
11615:{\displaystyle {\frac {\mathrm {d} \mathbf {p} }{\mathrm {d} t}}=q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right).}
8784:
The potential energy depends on the velocity of the particle, so the force is velocity dependent, so it is not conservative.
2894:. In practice, the subscripts "G" and "SI" are omitted, and the used convention (and unit) must be determined from context.
11494:
15123:
13427:
9651:
This formula is the Lorentz force, representing the rate at which the EM field adds relativistic momentum to the particle.
6302:{\displaystyle L={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} +q\mathbf {A} \cdot \mathbf {\dot {r}} -q\phi }
5616:
5308:
5199:
15115:
13848:
5491:
3335:
If, in addition, the magnetic field is inhomogeneous, the net force on a stationary rigid wire carrying a steady current
3294:
4901:
1300:
follow this nomenclature: In what follows, the term "Lorentz force" will refer to the expression for the total force.
15077:
15058:
15039:
15017:
14933:
14905:
14853:
14823:
14790:
14713:
14705:
A History of the Theories of Aether and Electricity: From the Age of Descartes to the Close of the Nineteenth Century
14683:
14614:
14542:
14509:
14478:
14445:
14414:
14387:
14133:
13029:
2975:
2520:
The above-mentioned formulae use the conventions for the definition of the electric and magnetic field used with the
13297:
4890:{\displaystyle \mathbf {F} =q\,\mathbf {E} (\mathbf {r} ,\,t)+q\,\mathbf {v} \times \mathbf {B} (\mathbf {r} ,\,t).}
3856:{\displaystyle {\mathcal {E}}=\oint _{\partial \Sigma (t)}\!\!\mathrm {d} {\boldsymbol {\ell }}\cdot \mathbf {F} /q}
2991:, Thomson published a paper in 1881 wherein he gave the force on the particles due to an external magnetic field as
12790:
6412:
15153:
13212:
4987:
time-dependent). In all three cases, Faraday's law of induction then predicts the EMF generated by the change in
2946:. However, in both cases the experimental proof was neither complete nor conclusive. It was not until 1784 when
14362:
13791:
13557:
13034:
12562:
2125:. Rather than the amount of charge and its velocity in electric and magnetic fields, this equation relates the
1489:{\displaystyle \mathrm {d} \mathbf {F} =\mathrm {d} q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}
231:
arrived at a complete derivation in 1895, identifying the contribution of the electric force a few years after
17:
12112:
11997:
1844:{\displaystyle \mathbf {F} =\int \left(\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} \right)\mathrm {d} V.}
1499:
723:
is the vector cross product (all boldface quantities are vectors). In terms of Cartesian components, we have:
13572:
13277:
12961:
3750:
3738:
3504:
1970:
1909:
217:
118:
6515:
2632:. Although this equation looks slightly different, it is equivalent, since one has the following relations:
1296:
electromagnetic force (including the electric force) given some other (nonstandard) name. This article will
13437:
13177:
13044:
12161:
10402:{\displaystyle U_{\beta }=\left(U_{0},U_{1},U_{2},U_{3}\right)=\gamma \left(c,-v_{x},-v_{y},-v_{z}\right),}
3467:
2954:, was able to definitively show through experiment that this was true. Soon after the discovery in 1820 by
13167:
9936:{\displaystyle p^{\alpha }=\left(p_{0},p_{1},p_{2},p_{3}\right)=\left(\gamma mc,p_{x},p_{y},p_{z}\right),}
6189:
4907:
If the magnetic field is fixed in time and the conducting loop moves through the field, the magnetic flux
1548:. If both sides of this equation are divided by the volume of this small piece of the charge distribution
208:
Variations on this basic formula describe the magnetic force on a current-carrying wire (sometimes called
15143:
13730:
13605:
13502:
13477:
13397:
5732:
3172:
2947:
3064:(see below), Lorentz arrived at the correct and complete form of the force law that now bears his name.
13230:
10658:{\displaystyle F'^{\mu \nu }={\Lambda ^{\mu }}_{\alpha }{\Lambda ^{\nu }}_{\beta }F^{\alpha \beta }\,,}
3168:
539:
3655:{\displaystyle \Phi _{B}=\int _{\Sigma (t)}\mathrm {d} \mathbf {A} \cdot \mathbf {B} (\mathbf {r} ,t)}
2372:
419:
15148:
15121:
Interactive Java applet on the magnetic deflection of a particle beam in a homogeneous magnetic field
13853:
13784:
13745:
13272:
13262:
13202:
13197:
13137:
12930:
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5302:
5058:
4021:
3944:
555:
494:
11895:
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3196:
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5501:
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2133:
per unit time per unit distance) in the fields to the force exerted on a charge distribution. See
1992:
1934:
1876:
1635:
528:
14818:(Second ed.). Oxford: Butterworth-Heinemann. p. §63 (§49 pp. 205–207 in 1960 edition).
13878:
13595:
13122:
13112:
13107:
11931:
11724:
9770:{\displaystyle {\frac {\mathrm {d} p^{\alpha }}{\mathrm {d} \tau }}=qF^{\alpha \beta }U_{\beta }}
5071:
5040:
3472:
2959:
524:
13720:
13690:
6210:
in an electromagnetic field equivalently describes the dynamics of the particle in terms of its
512:. For small relative velocities and very small accelerations, instead of the Coulomb force, the
293:
In many textbook treatments of classical electromagnetism, the Lorentz force law is used as the
14141:
13838:
13542:
13312:
13087:
12893:
11757:
10675:
9957:
8955:
5036:
4973:. As a third example, if a portion of the circuit is swept through a uniform, time-independent
3452:
3319:
2319:
1896:
547:
15088:
14780:
3287:
If the wire is not straight, the force on it can be computed by applying this formula to each
3195:. An entire physical apparatus for dealing with these matters has developed. See for example,
3143:
densities. The response of a point charge to the Lorentz law is one aspect; the generation of
2906:
Lorentz' theory of electrons. Formulas for the Lorentz force (I, ponderomotive force) and the
14501:
14495:
14437:
14431:
14125:
14043:
14013:
13640:
13327:
13317:
13267:
13257:
11921:
9206:
4362:
3110:
1904:
513:
67:
14810:
14667:
14631:
14598:
14526:
14462:
14117:
12455:
12237:
12207:
1191:{\displaystyle \mathbf {F} \left(\mathbf {r} (t),{\dot {\mathbf {r} }}(t),t,q\right)=q\left}
14560:"XXXIII. On the electric and magnetic effects produced by the motion of electrified bodies"
13765:
13665:
13630:
13382:
13247:
13147:
13132:
13067:
12966:
12912:
6380:
6195:
5067:
3278:
3184:
3176:
3061:
3043:
2366:
1856:
1661:
1625:{\displaystyle \mathbf {f} =\rho \left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}
520:
309:. To be specific, the Lorentz force is understood to be the following empirical statement:
14632:"On the Electromagnetic Effects due to the Motion of Electrification through a Dielectric"
13004:
4007:{\displaystyle \nabla \times \mathbf {E} =-{\frac {\partial \mathbf {B} }{\partial t}}\,.}
205:
of the charge (proportional to the magnitude of the field, the charge, and the velocity).
8:
14839:
14668:
14599:
14527:
14463:
13863:
13725:
13705:
13700:
13507:
13492:
13377:
13347:
13242:
13172:
12971:
3514:
3422:
3134:
in the presence of electromagnetic fields. The Lorentz force law describes the effect of
3057:
2979:
2935:
2891:
2878:
1424:
1383:
519:
The interpretation of magnetism by means of a modified Coulomb law was first proposed by
224:
213:
35:
3745:, the Lorentz Force can be deduced. The reverse is also true, the Lorentz force and the
3384:{\displaystyle \mathbf {F} =I\int \mathrm {d} {\boldsymbol {\ell }}\times \mathbf {B} .}
176:{\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right).}
15029:
14951:
14380:
The New Cambridge Modern History Volume 8: The American and French Revolutions, 1763–93
14327:
13600:
13340:
13142:
13102:
12875:
12187:
12167:
9694:
6407:
5190:
5020:
3495:
3164:
2943:
2096:
1961:
1237:(in detail, if the fingers of the right hand are extended to point in the direction of
707:{\displaystyle \mathbf {F} =q\left(\mathbf {E} +\mathbf {v} \times \mathbf {B} \right)}
14746:
14654:
Versuch einer Theorie der electrischen und optischen Erscheinungen in bewegten Körpern
14332:. reprint. with corr. (3rd ed.). Upper Saddle River, New Jersey : Prentice Hall.
13422:
11886:{\displaystyle i\mathbf {B} =\left({\mathcal {F}}\wedge \gamma _{0}\right)\gamma _{0}}
15094:
15073:
15054:
15035:
15013:
14939:
14929:
14921:
14901:
14876:
14849:
14819:
14786:
14709:
14679:
14610:
14579:
14538:
14505:
14474:
14441:
14410:
14383:
14358:
14333:
14221:
14201:
14176:
14129:
14049:
14019:
13868:
13660:
12976:
12917:
12391:{\displaystyle m{\frac {du_{c}}{ds}}-m{\frac {1}{2}}g_{ab,c}u^{a}u^{b}=eF_{cb}u^{b},}
11749:
8940:
3746:
3742:
3192:
3188:
2907:
509:
490:
14118:
11818:{\displaystyle \mathbf {E} =\left({\mathcal {F}}\cdot \gamma _{0}\right)\gamma _{0}}
14571:
14308:
13893:
13873:
13823:
13818:
13760:
13675:
13635:
13625:
13512:
13467:
13450:
13367:
13302:
13072:
12996:
11753:
9681:
8774:{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {\dot {r}} \times \mathbf {B} )}
6556:
5495:
5016:
3106:
3047:
2915:
582:
551:
505:
284:
232:
51:
14984:
2966:
The modern concept of electric and magnetic fields first arose in the theories of
15127:
15005:
14870:
14699:
14404:
14092:
13833:
13695:
13620:
13615:
13482:
13357:
13322:
13182:
13082:
10572:
The fields are transformed to a frame moving with constant relative velocity by:
8959:
3940:
3094:
3052:
2967:
2951:
2939:
1929:
1900:
1775:
1683:
1396:
1357:{\displaystyle \mathbf {v} \cdot \mathbf {F} =q\,\mathbf {v} \cdot \mathbf {E} .}
1234:
624:
228:
15120:
13948:
13735:
4937:
will change. Alternatively, if the loop changes orientation with respect to the
3764:
be the moving wire, moving together without rotation and with constant velocity
1221:
field, but will curve perpendicularly to both the instantaneous velocity vector
527:
demonstrated experimentally that this is not the case. Ampère also formulated a
14980:
14564:
The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science
13986:
13933:
13883:
13858:
13655:
13650:
13472:
13362:
13287:
13237:
13187:
13160:
13117:
13092:
13062:
13055:
12941:
11624:
This is precisely the Lorentz force law, however, it is important to note that
10566:
6615:
3510:
3180:
3090:
3073:
2971:
2919:
2629:
2525:
2140:
The density of power associated with the Lorentz force in a material medium is
2114:
1679:
1411:
1387:
605:
596:
486:
109:
100:
39:
14575:
3571:{\displaystyle {\mathcal {E}}=-{\frac {\mathrm {d} \Phi _{B}}{\mathrm {d} t}}}
1764:{\displaystyle \mathbf {f} =\rho \mathbf {E} +\mathbf {J} \times \mathbf {B} }
15137:
14583:
13770:
13755:
13740:
13680:
13392:
13292:
13207:
13192:
13097:
11054:
10247:
9787:
8944:
6214:, rather than the force exerted on it. The classical expression is given by:
4922:-field varies with position, and the loop moves to a location with different
3909:
3723:
3663:
3459:
3288:
3211:
2983:
2924:
La théorie electromagnétique de Maxwell et son application aux corps mouvants
2421:
535:
14943:
14559:
12749:{\displaystyle m{\frac {du_{c}}{ds}}-m\Gamma _{abc}u^{a}u^{b}=eF_{cb}u^{b},}
11393:
directions) yields similar results, so collecting the 3 equations into one:
8939:, minus the potential energy contribution, plus an extra contribution which
6671:{\displaystyle T={\frac {m}{2}}\mathbf {\dot {r}} \cdot \mathbf {\dot {r}} }
408:{\displaystyle \mathbf {F} =q(\mathbf {E} +\mathbf {v} \times \mathbf {B} )}
13943:
13929:
13750:
13645:
13610:
13552:
13487:
13407:
13372:
13252:
13127:
12951:
5012:
3035:{\displaystyle \mathbf {F} ={\frac {q}{2}}\mathbf {v} \times \mathbf {B} .}
2122:
2118:
497:) and indirectly (by affecting the motion of nearby charges and currents).
442:
418:
This is valid, even for particles approaching the speed of light (that is,
275:
63:
3711:(magnitude is the area of an infinitesimal patch of surface, direction is
3078:
1372:
14195:
14170:
13813:
13670:
13522:
13352:
13014:
12956:
11925:
9947:
4999:
Note that the Maxwell Faraday's equation implies that the Electric Field
3700:
2988:
2126:
571:
323:
42:. Positive and negative charge trajectories curve in opposite directions.
6383:, the equation for the Lorentz force given above can be obtained again.
4951:
differential element will change because of the different angle between
3046:, included an incorrect scale-factor of a half in front of the formula.
14249:. Königliche Gesellschaft der Wissenschaften zu Göttingen. p. 617.
14140:. These authors use the Lorentz force in tensor form as definer of the
13888:
13387:
12097:{\displaystyle \mathbf {v} =cv\wedge \gamma _{0}/(v\cdot \gamma _{0}).}
3712:
3264:{\displaystyle \mathbf {F} =I{\boldsymbol {\ell }}\times \mathbf {B} ,}
2911:
187:
is a combination of (1) a force in the direction of the electric field
75:
11044:{\displaystyle {\frac {\mathrm {d} p^{1}}{\mathrm {d} \tau }}=q\left.}
6501:{\displaystyle q\mathbf {A} (\mathbf {r} ,t)\cdot \mathbf {\dot {r}} }
14999:
The numbered references refer in part to the list immediately below.
13710:
13685:
13497:
13019:
12925:
12908:
12793:(of the torsion-free metric connection in general relativity), or as
8936:
8932:
3067:
1523:
is the force on a small piece of the charge distribution with charge
14875:(Fourth ed.). NY/Oxford: Oxford University Press. p. 391.
10884:
Substituting the components of the covariant electromagnetic tensor
8947:
a charged particle gets when it is moving along a vector potential.
3778:
be the internal surface of the wire. The EMF around the closed path
13462:
13457:
13077:
11917:
8951:
Derivation of Lorentz force from relativistic Lagrangian (SI units)
3741:(that is valid for a moving wire, for instance in a motor) and the
3463:
3098:
543:
268:
5726:, we can put the equation into the convenient Euler–Lagrange form
14222:"The Feynman Lectures on Physics Vol. II Ch. 1: Electromagnetism"
13981:
13432:
12946:
6379:
can be thought as a velocity-dependent potential function. Using
4916:
linking the loop can change in several ways. For example, if the
238:
47:
27:
Force acting on charged particles in electric and magnetic fields
14115:
6387:
Derivation of Lorentz force from classical Lagrangian (SI units)
6372:{\displaystyle V=q(\phi -\mathbf {A} \cdot \mathbf {\dot {r}} )}
3215:
Right-hand rule for a current-carrying wire in a magnetic field
248:
13993:. National Institute of Standards and Technology. 12 April 2010
13977:
13517:
13024:
6607:{\displaystyle V=q\phi -q\mathbf {A} \cdot \mathbf {\dot {r}} }
3490:
3329:
3113:
of a relatively fast circular motion around a point called the
500:
493:. These effects occur through both a direct effect (called the
14074:
Flash of the Cathode Rays: A History of J J Thomson's Electron
2011:, another way to write the Lorentz force (per unit volume) is
326:
at a given point and time is a certain function of its charge
14497:
Hidden Attraction : The History And Mystery Of Magnetism
14433:
Hidden Attraction : The History And Mystery Of Magnetism
235:
correctly identified the contribution of the magnetic force.
216:
in a wire loop moving through a magnetic field (an aspect of
59:
12030:(which shows our choice for the metric) and the velocity is
5011:
varies in time, and is not expressible as the gradient of a
4361:
The two are equivalent if the wire is not moving. Using the
254:
Trajectory of a particle with a positive or negative charge
14900:. European Physics Series (2nd ed.). McGraw Hill. UK.
11688:
11679:{\displaystyle \mathbf {p} =\gamma (v)m_{0}\mathbf {v} \,.}
4804:
since this is valid for any wire position it implies that,
3462:
through the wire. (This is Faraday's law of induction, see
14809:
Landau, L. D.; Lifshitz, E. M.; Pitaevskiĭ, L. P. (1984).
223:
Historians suggest that the law is implicit in a paper by
15070:
Physics for scientists and engineers, with modern physics
14357:. Chicago, IL: Fitzroy Dearborn Publishers. p. 538.
9655:
6183:
3102:
2958:
that a magnetic needle is acted on by a voltaic current,
2938:
and others in 1760, and electrically charged objects, by
1249:, then the extended thumb will point in the direction of
1211:
A positively charged particle will be accelerated in the
278:
is created by the electrons colliding with gas molecules.
14808:
14409:. Norwalk, Connecticut: Burndy Library. pp. 30–31.
13844:
Formulation of Maxwell's equations in special relativity
14321:
14319:
14317:
12549:{\displaystyle g_{ab,c}=\partial g_{ab}/\partial x^{c}}
5026:
4016:
The Maxwell–Faraday equation also can be written in an
2521:
1686:
corresponding to the motion of the charge continuum is
649:
15093:. Cambridge, ; New York : Cambridge University Press.
14382:. Cambridge: Cambridge University Press. p. 130.
14196:
I.S. Grant; W.R. Phillips; Manchester Physics (1990).
14171:
I.S. Grant; W.R. Phillips; Manchester Physics (1990).
12155:
9987:
9676:
Mathematical descriptions of the electromagnetic field
9660:
5033:
Mathematical descriptions of the electromagnetic field
4900:
of the underlying Lorentz force law is necessary. See
2963:
rather than in terms of electric and magnetic fields.
1903:, this form of the equation can be used to derive the
262:, which is directed perpendicularly out of the screen.
15034:(3rd ed.). Upper Saddle River, : Prentice-Hall.
14848:. Mineola, New York: Dover Publications. p. 56.
12904:
The Lorentz force occurs in many devices, including:
12878:
12799:
12762:
12641:
12565:
12489:
12458:
12404:
12270:
12240:
12210:
12190:
12170:
12115:
12036:
12000:
11963:
11934:
11920:(an oriented plane segment, just like a vector is an
11898:
11831:
11766:
11727:
11703:
11636:
11545:
11497:
11399:
11063:
10894:
10703:
10578:
10415:
10256:
9965:
9796:
9706:
9505:
9358:
9215:
9110:
8968:
8793:
8725:
8182:
7814:
7609:
7095:
6995:
6790:
6684:
6624:
6565:
6518:
6457:
6415:
6327:
6220:
6143:
6102:
6061:
6025:
5993:
5961:
5929:
5900:
5735:
5619:
5526:
5504:
5472:
5447:
5311:
5202:
5085:
4810:
4589:
4379:
4195:
4033:
3960:
3869:
3792:
3584:
3523:
3345:
3297:
3233:
3089:
In many cases of practical interest, the motion in a
2997:
2974:, later to be given full mathematical description by
2638:
2534:
2515:
2430:
2404:
2375:
2349:
2322:
2181:
2146:
2099:
2060:
2041:
2017:
1995:
1973:
1937:
1912:
1879:
1859:
1784:
1728:
1692:
1664:
1638:
1579:
1554:
1529:
1502:
1433:
1317:
1311:
electromagnetic field to the particle. That power is
1304:
1031:
729:
661:
366:
209:
127:
15004:
14314:
14200:(2nd ed.). John Wiley & Sons. p. 123.
14175:(2nd ed.). John Wiley & Sons. p. 122.
11752:(or the geometric algebra of space-time), a type of
3939:
have a sign ambiguity; to get the correct sign, the
3418:) component of the Lorentz force is responsible for
3124:
342:, which can be parameterized by exactly two vectors
14838:
13991:
NIST reference on constants, units, and uncertainty
12865:{\displaystyle m{\frac {Du_{c}}{ds}}=eF_{cb}u^{b},}
9672:
Covariant formulation of classical electromagnetism
3206:
2135:
Covariant formulation of classical electromagnetism
183:It says that the electromagnetic force on a charge
12884:
12864:
12781:
12748:
12623:
12548:
12474:
12444:
12390:
12256:
12226:
12196:
12176:
12140:
12096:
12022:
11984:
11947:
11908:
11885:
11817:
11740:
11713:
11678:
11614:
11531:
11471:
11364:
11043:
10874:
10657:
10557:
10401:
10234:
9935:
9769:
9643:
9491:
9345:
9197:
9097:
8921:
8773:
8699:
8169:
7800:
7596:
7069:
6979:
6777:
6670:
6606:
6544:
6500:
6443:
6371:
6301:
6173:
6011:
5883:
5706:
5605:
5512:
5494:in the equation above). Using the chain rule, the
5480:
5455:
5430:
5291:
5169:
4889:
4796:
4575:
4351:
4181:
4006:
3893:
3855:
3654:
3570:
3383:
3322:. This results in the same formal expression, but
3310:
3263:
3068:Trajectories of particles due to the Lorentz force
3034:
2860:
2616:
2505:
2412:
2390:
2357:
2335:
2306:
2165:
2105:
2085:
2003:
1981:
1945:
1920:
1887:
1865:
1843:
1763:
1711:
1670:
1646:
1624:
1565:
1540:
1515:
1488:
1356:
1190:
1015:
706:
407:
175:
15008:; Leighton, Robert B.; Sands, Matthew L. (2006).
14629:
14285:, which gives the force acting on a point charge
10681:
4749:
4748:
4747:
4746:
4548:
4547:
4546:
4545:
3826:
3825:
3311:{\displaystyle \mathrm {d} {\boldsymbol {\ell }}}
1367:
1243:and are then curled to point in the direction of
15135:
15068:Serway, Raymond A.; Jewett, John W. Jr. (2004).
12164:the equation of motion for a particle with mass
6321:are the potential fields as above. The quantity
3513:, Faraday's law of induction states the induced
1404:corresponds to the motion of the charge element
1204:is the position vector of the charged particle,
14785:. Sudbury MA: Jones and Bartlett. p. 395.
14765:
14763:
2166:{\displaystyle \mathbf {J} \cdot \mathbf {E} .}
1427:in motion, the Lorentz force equation becomes:
1208:is time, and the overdot is a time derivative.
220:), and the force on a moving charged particle.
14812:Electrodynamics of continuous media; Volume 8
14605:. Oxford, : Oxford University Press. pp.
14533:. Oxford, : Oxford University Press. pp.
14469:. Oxford, : Oxford University Press. pp.
14436:. New York: Oxford University Press. pp.
12896:in general relativity (metric, torsion-free).
9499:both are equivalent to the noncanonical form:
1719:so the continuous analogue to the equation is
1712:{\displaystyle \mathbf {J} =\rho \mathbf {v} }
239:Lorentz force law as the definition of E and B
14674:. Oxford, : Oxford University Press. p.
14500:. New York: Oxford University Press. p.
14116:J.A. Wheeler; C. Misner; K.S. Thorne (1973).
13792:
6444:{\displaystyle q\mathbf {A} (\mathbf {r} ,t)}
15067:
14868:
14760:
14708:. Longmans, Green and Co. pp. 420–423.
5005:is non conservative when the Magnetic Field
3491:Lorentz force and Faraday's law of induction
501:Physical interpretation of the Lorentz force
58:is the combination of electric and magnetic
34:Lorentz force acting on fast-moving charged
14845:Introduction to electromagnetic engineering
14778:
14722:
14289:in the presence of electromagnetic fields."
14258:
14256:
11479:and since differentials in coordinate time
7089:). So calculating the partial derivatives:
3894:{\displaystyle \mathbf {E} =\mathbf {F} /q}
3680:is a surface bounded by the closed contour
3479:-field can change in whole or in part to a
14956:: CS1 maint: location missing publisher (
14832:
14301:
14292:
13799:
13785:
13003:
6512:field, the particle's potential energy is
4027:So we have, the Maxwell Faraday equation:
15086:
15027:
14804:
14802:
14698:
14493:
14429:
14325:
13949:"Geomagnetism Frequently Asked Questions"
12624:{\displaystyle ds^{2}=g_{ab}dx^{a}dx^{b}}
11672:
11519:
11358:
10651:
8719:directions. Hence the force equation is:
4877:
4852:
4839:
4822:
4783:
4562:
4159:
4000:
3726:. Note that this is valid for not only a
2796:
2732:
1899:, and manipulating using the theorems of
1337:
15010:The Feynman lectures on physics (3 vol.)
14665:
14596:
14524:
14460:
14253:
14247:Carl Friedrich Gauss Werke. Fünfter Band
14100:
12141:{\displaystyle F=q{\mathcal {F}}\cdot v}
11695:dependent on the velocity of an observer
11689:Lorentz force in spacetime algebra (STA)
4583:and using the Maxwell Faraday equation,
3499:Lorentz force -image on a wall in Leiden
3494:
3339:is given by integration along the wire,
3210:
3077:
2901:
1928:, in turn this can be combined with the
1516:{\displaystyle \mathrm {d} \mathbf {F} }
1371:
570:
258:under the influence of a magnetic field
29:
15048:
14926:The variational principles of mechanics
14920:
14670:Electrodynamics from Ampère to Einstein
14601:Electrodynamics from Ampère to Einstein
14529:Electrodynamics from Ampère to Einstein
14465:Electrodynamics from Ampère to Einstein
14377:
14087:
14085:
14083:
14037:
14035:
13918:
13916:
13914:
13548:Electromagnetism and special relativity
12204:, moving in a space with metric tensor
6396:field, a particle moving with velocity
4790:
4689:
4618:
4569:
4408:
4224:
4062:
3833:
3366:
3304:
3246:
2033:
1982:{\displaystyle {\boldsymbol {\sigma }}}
1975:
1921:{\displaystyle {\boldsymbol {\sigma }}}
1914:
554:apply. Weber electrodynamics is only a
14:
15136:
14895:
14799:
14406:A History of Electricity and Magnetism
14067:
14065:
11532:{\displaystyle dt=\gamma (v)\,d\tau ,}
9656:Relativistic form of the Lorentz force
6545:{\displaystyle q\phi (\mathbf {r} ,t)}
6184:Lorentz force and analytical mechanics
5613:so that the above expression becomes:
14402:
14352:
14244:
14106:See, for example, Jackson, pp. 777–8.
14041:
14011:
13964:
13568:Maxwell equations in curved spacetime
11693:The electric and magnetic fields are
3943:is used, as explained in the article
3730:wire – but also for a
3318:, then adding up all these forces by
14979:
14772:
14080:
14071:
14032:
13987:"International system of units (SI)"
13980:per metre (A/m) in SI units, and in
13911:
12635:The equation can also be written as
5707:{\displaystyle \mathbf {F} =q\left.}
5431:{\displaystyle \mathbf {F} =q\left,}
5292:{\displaystyle \mathbf {F} =q\left.}
5027:Lorentz force in terms of potentials
3151:by currents and charges is another.
2398:is the density of free current; and
1955:electromagnetic stress–energy tensor
1682:(charge per unit volume). Next, the
1419:and varies throughout the continuum.
636:, due to an external electric field
589:) in motion (instantaneous velocity
297:of the electric and magnetic fields
15053:(3rd ed.). New York, : Wiley.
14557:
14062:
13849:Moving magnet and conductor problem
12156:Lorentz force in general relativity
11721:, and an arbitrary time-direction,
11491:are related by the Lorentz factor,
9661:Covariant form of the Lorentz force
8954:The equations of motion derived by
5884:{\displaystyle \mathbf {F} =q\left}
3201:Green's function (many-body theory)
566:
24:
15072:. Belmont, : Thomson Brooks/Cole.
14124:. W.H. Freeman & Co. pp.
13951:. National Geophysical Data Center
12764:
12679:
12533:
12512:
12127:
11901:
11850:
11782:
11706:
11562:
11550:
11416:
11404:
11085:
11068:
11053:Using the components of covariant
10916:
10899:
10725:
10708:
10622:
10603:
9728:
9711:
9515:
9509:
9397:
9393:
9375:
9363:
9251:
9247:
9232:
9220:
9084:
9076:
9039:
9029:
9009:
9001:
8985:
8973:
8607:
8533:
8494:
8428:
8413:
8398:
8383:
8346:
8331:
8316:
8301:
8256:
8241:
8226:
8218:
8141:
8126:
8099:
8084:
8057:
8042:
8019:
8011:
7973:
7958:
7931:
7916:
7889:
7874:
7859:
7844:
7772:
7757:
7730:
7715:
7688:
7673:
7650:
7642:
7621:
7613:
7564:
7549:
7522:
7507:
7480:
7465:
7450:
7435:
7376:
7361:
7340:
7325:
7304:
7289:
7268:
7253:
7235:
7193:
7176:
7130:
7122:
7109:
7103:
7058:
7050:
7026:
7018:
7005:
6999:
6149:
6145:
6108:
6104:
6067:
6063:
6027:
5999:
5995:
5967:
5963:
5935:
5931:
5902:
5824:
5813:
5807:
5756:
5686:
5674:
5639:
5592:
5569:
5559:
5543:
5531:
5407:
5365:
5353:
5343:
5331:
5267:
5244:
5234:
5222:
5152:
5127:
5117:
5105:
4785:
4732:
4729:
4684:
4669:
4666:
4613:
4598:
4595:
4564:
4531:
4528:
4489:
4485:
4471:
4456:
4403:
4388:
4385:
4309:
4294:
4279:
4273:
4219:
4204:
4201:
4169:
4139:
4123:
4108:
4057:
4042:
4039:
3991:
3981:
3961:
3828:
3811:
3808:
3795:
3618:
3603:
3586:
3558:
3546:
3541:
3526:
3361:
3299:
2809:
2806:
2762:
2744:
2741:
2703:
2665:
2662:
2645:
2600:
2568:
2549:
2516:Equations with Gaussian quantities
2478:
2468:
2451:
2279:
2269:
2252:
2208:
2073:
2063:
2026:
1831:
1556:
1531:
1504:
1448:
1435:
1376:Lorentz force (per unit 3-volume)
25:
15170:
15109:
14558:M.A, J. J. Thomson (1881-04-01).
14355:Encyclopedia of the Enlightenment
9687:, the Lorentz force for a charge
3125:Significance of the Lorentz force
542:. This demonstrates not only the
89:states that a particle of charge
14630:Heaviside, Oliver (April 1889).
14309:website of the Lorentz Institute
12482:is taken along the trajectory),
12038:
11836:
11768:
11668:
11638:
11630:is the relativistic expression,
11600:
11592:
11584:
11555:
11457:
11449:
11441:
11409:
11338:
11330:
9629:
9615:
9604:
9564:
9535:
9430:
9419:
9401:
9368:
9284:
9273:
9255:
9225:
9168:
9140:
9123:
9112:
9088:
9057:
9043:
9033:
9013:
8978:
8912:
8889:
8875:
8867:
8834:
8764:
8754:
8751:
8741:
8727:
8679:
8669:
8666:
8614:
8595:
8592:
8540:
8501:
6760:
6757:
6747:
6734:
6731:
6719:
6716:
6662:
6659:
6647:
6644:
6598:
6595:
6585:
6529:
6492:
6489:
6470:
6462:
6428:
6420:
6360:
6357:
6347:
6284:
6281:
6271:
6258:
6255:
6243:
6240:
6034:
5907:
5869:
5855:
5831:
5794:
5780:
5761:
5737:
5679:
5660:
5652:
5621:
5599:
5585:
5563:
5536:
5506:
5490:thus, there is no need of using
5474:
5449:
5416:
5400:
5382:
5374:
5347:
5313:
5274:
5257:
5238:
5204:
5159:
5141:
5121:
5091:
4902:inapplicability of Faraday's law
4870:
4862:
4854:
4832:
4824:
4812:
4767:
4759:
4751:
4705:
4697:
4642:
4626:
4558:
4550:
4507:
4499:
4476:
4432:
4416:
4330:
4322:
4314:
4248:
4232:
4152:
4144:
4128:
4078:
4070:
3985:
3968:
3879:
3871:
3841:
3639:
3631:
3623:
3374:
3347:
3254:
3235:
3207:Force on a current-carrying wire
3025:
3017:
2999:
2800:
2756:
2735:
2697:
2594:
2580:
2562:
2536:
2496:
2472:
2458:
2438:
2406:
2391:{\displaystyle \mathbf {J} _{f}}
2378:
2351:
2297:
2273:
2259:
2239:
2225:
2215:
2183:
2156:
2148:
2067:
2019:
1997:
1939:
1881:
1821:
1813:
1805:
1786:
1757:
1749:
1741:
1730:
1705:
1694:
1640:
1613:
1605:
1597:
1581:
1509:
1477:
1469:
1461:
1440:
1347:
1339:
1327:
1319:
1170:
1162:
1139:
1119:
1111:
1063:
1043:
1033:
695:
687:
679:
663:
552:conservation of angular momentum
398:
390:
382:
368:
267:
247:
161:
153:
145:
129:
15031:Introduction to electrodynamics
14973:
14964:
14914:
14889:
14862:
14731:
14692:
14659:
14646:
14623:
14590:
14551:
14518:
14487:
14454:
14423:
14396:
14371:
14346:
14329:Introduction to electrodynamics
14238:
14226:www.feynmanlectures.caltech.edu
14214:
14189:
12899:
12445:{\displaystyle u^{a}=dx^{a}/ds}
9665:
8935:of the path of the particle in
8931:The action is the relativistic
8787:The relativistic Lagrangian is
6198:for a charged particle of mass
5303:identity for the triple product
2819:
2753:
2694:
2343:is the density of free charge;
287:experiencing the Lorentz force.
14245:Gauss, Carl Friedrich (1867).
14164:
14109:
14005:
12962:Magnetoplasmadynamic thrusters
12088:
12069:
11909:{\displaystyle {\mathcal {F}}}
11748:. This can be settled through
11714:{\displaystyle {\mathcal {F}}}
11654:
11648:
11516:
11510:
11213:
11200:
11197:
11181:
11175:
11159:
11156:
11140:
11030:
11017:
11001:
10985:
10682:Translation to vector notation
10425:
10419:
9464:
9447:
9435:
9415:
9318:
9301:
9289:
9269:
9207:Hamilton's equations of motion
8916:
8908:
8879:
8871:
8768:
8737:
8684:
8660:
8622:
8618:
8604:
8586:
8554:
8545:
8530:
8506:
8491:
8476:
6539:
6525:
6480:
6466:
6438:
6424:
6366:
6337:
6136:
6095:
6054:
5986:
5954:
5922:
5873:
5842:
5798:
5767:
5664:
5642:
5595:
5581:
5278:
5264:
5057:fields can be replaced by the
4881:
4866:
4843:
4828:
4780:
4763:
4741:
4735:
4718:
4701:
4678:
4672:
4655:
4638:
4607:
4601:
4540:
4534:
4517:
4503:
4465:
4459:
4442:
4428:
4397:
4391:
4343:
4326:
4303:
4297:
4261:
4244:
4213:
4207:
4163:
4148:
4117:
4111:
4091:
4074:
4051:
4045:
3912:vector element of the contour
3820:
3814:
3649:
3635:
3612:
3606:
2121:· denotes the divergence of a
1778:over the charge distribution:
1368:Continuous charge distribution
1180:
1166:
1155:
1149:
1129:
1115:
1079:
1073:
1053:
1047:
402:
378:
13:
1:
15116:Lorentz force (demonstration)
14994:
14928:(Fourth ed.). New York.
14814:Course of Theoretical Physics
14494:Verschuur, Gerrit L. (1993).
14430:Verschuur, Gerrit L. (1993).
14012:Huray, Paul G. (2009-11-16).
13573:Relativistic electromagnetism
12782:{\displaystyle \Gamma _{abc}}
10250:of the particle, defined as:
6451:, so its potential energy is
1566:{\displaystyle \mathrm {d} V}
1541:{\displaystyle \mathrm {d} q}
15049:Jackson, John David (1999).
15028:Griffiths, David J. (1999).
15012:. Pearson / Addison-Wesley.
14872:Elements of electromagnetics
14326:Griffiths, David J. (1999).
13905:
12162:general theory of relativity
11985:{\displaystyle v={\dot {x}}}
10697:-component) of the force is
5513:{\displaystyle \mathbf {A} }
5492:Feynman's subscript notation
5481:{\displaystyle \mathbf {v} }
5456:{\displaystyle \mathbf {A} }
3722:of the EMF is determined by
3468:special theory of relativity
2413:{\displaystyle \mathbf {M} }
2358:{\displaystyle \mathbf {P} }
2004:{\displaystyle \mathbf {S} }
1946:{\displaystyle \mathbf {S} }
1888:{\displaystyle \mathbf {J} }
1658:(force per unit volume) and
1647:{\displaystyle \mathbf {f} }
630:with instantaneous velocity
7:
15087:Srednicki, Mark A. (2007).
14769:See Griffiths, pages 301–3.
14743:www.physicsexperiment.co.uk
14378:Goodwin, Elliot H. (1965).
12983:
11948:{\displaystyle \gamma _{0}}
11741:{\displaystyle \gamma _{0}}
3393:One application of this is
2970:, particularly his idea of
2948:Charles-Augustin de Coulomb
652:definition of quantities):
561:
10:
15175:
14666:Darrigol, Olivier (2000).
14597:Darrigol, Olivier (2000).
14525:Darrigol, Olivier (2000).
14461:Darrigol, Olivier (2000).
14403:Meyer, Herbert W. (1972).
14048:. Wiley-IEEE. p. 22.
13298:Liénard–Wiechert potential
12234:and electromagnetic field
9669:
7808:equating and simplifying:
6187:
5305:this can be rewritten as,
5030:
3901:is the electric field and
3749:can be used to derive the
3739:Faraday's law of induction
3509:Given a loop of wire in a
3505:Faraday's law of induction
3502:
3071:
2897:
1215:linear orientation as the
314:The electromagnetic force
218:Faraday's law of induction
74:, on the other hand, is a
15051:Classical electrodynamics
15006:Feynman, Richard Phillips
14652:Lorentz, Hendrik Antoon,
14576:10.1080/14786448108627008
14018:. John Wiley & Sons.
13563:Mathematical descriptions
13273:Electromagnetic radiation
13263:Electromagnetic induction
13203:Magnetic vector potential
13198:Magnetic scalar potential
12931:Lorentz force velocimetry
11381:(force components in the
6989:Lagrange's equations are
5059:magnetic vector potential
5015:, and not subject to the
3447:EMF, as described by the
2930:is the velocity of light.
2336:{\displaystyle \rho _{f}}
556:quasistatic approximation
14970:Jackson, J.D. Chapter 11
14298:See Griffiths, page 204.
14076:. CRC Press. p. 10.
13899:Electromagnetic buoyancy
12911:and other circular path
12023:{\displaystyle v^{2}=1,}
3952:Maxwell–Faraday equation
3715:to that surface patch).
3451:(one of the four modern
3449:Maxwell–Faraday equation
3109:) can be treated as the
623:acting on a particle of
544:conservation of momentum
495:radiation reaction force
358:, in the functional form
117:experiences a force (in
14896:Kibble, T.W.B. (1973).
14042:Huray, Paul G. (2010).
13879:Helmholtz decomposition
13113:Electrostatic induction
13108:Electrostatic discharge
5185:is the divergence, and
5072:electrostatic potential
5041:Helmholtz decomposition
3672:is the magnetic field,
3473:solenoidal vector field
3173:Navier–Stokes equations
3083:Charged particle drifts
1774:The total force is the
1233:field according to the
613:vary in space and time.
93:moving with a velocity
14782:Electromagnetic theory
14636:Philosophical Magazine
14353:Delon, Michel (2001).
14142:electromagnetic tensor
13976:-field is measured in
13946:(symbol: G). See e.g.
13543:Electromagnetic tensor
12894:covariant differential
12886:
12866:
12783:
12750:
12625:
12550:
12476:
12475:{\displaystyle dx^{a}}
12446:
12392:
12258:
12257:{\displaystyle F_{ab}}
12228:
12227:{\displaystyle g_{ab}}
12198:
12178:
12142:
12098:
12024:
11986:
11949:
11910:
11887:
11819:
11758:pseudo-Euclidean space
11742:
11715:
11680:
11616:
11533:
11473:
11366:
11045:
10876:
10676:Lorentz transformation
10659:
10559:
10403:
10236:
9958:electromagnetic tensor
9937:
9771:
9645:
9493:
9347:
9199:
9099:
8923:
8775:
8707:and similarly for the
8701:
8171:
7802:
7598:
7071:
6981:
6779:
6678:hence the Lagrangian:
6672:
6608:
6546:
6502:
6445:
6373:
6303:
6175:
6013:
5885:
5708:
5607:
5514:
5482:
5457:
5432:
5293:
5171:
4891:
4798:
4577:
4353:
4183:
4008:
3895:
3857:
3656:
3572:
3517:(EMF) in the wire is:
3500:
3400:
3385:
3312:
3265:
3219:
3169:Fokker–Planck equation
3117:and a relatively slow
3086:
3036:
2931:
2862:
2618:
2507:
2414:
2392:
2359:
2337:
2308:
2167:
2107:
2087:
2005:
1983:
1947:
1922:
1889:
1867:
1845:
1765:
1713:
1672:
1648:
1626:
1567:
1542:
1517:
1490:
1420:
1358:
1192:
1017:
708:
614:
548:conservation of energy
416:
409:
177:
68:electromagnetic fields
43:
14869:M N O Sadiku (2007).
14739:"Physics Experiments"
14307:For example, see the
14149:, in turn the fields
14072:Dahl, Per F. (1997).
13854:Abraham–Lorentz force
13536:Covariant formulation
13328:Synchrotron radiation
13268:Electromagnetic pulse
13258:Electromagnetic field
12967:Electrical generators
12913:particle accelerators
12887:
12867:
12784:
12751:
12626:
12551:
12477:
12447:
12393:
12259:
12229:
12199:
12179:
12143:
12099:
12025:
11987:
11950:
11922:oriented line segment
11911:
11888:
11820:
11743:
11716:
11681:
11617:
11534:
11474:
11367:
11046:
10877:
10660:
10560:
10404:
10237:
9938:
9772:
9646:
9494:
9348:
9200:
9100:
8924:
8776:
8702:
8172:
7803:
7599:
7072:
6982:
6780:
6673:
6609:
6547:
6503:
6446:
6374:
6304:
6176:
6014:
5886:
5709:
5608:
5515:
5483:
5458:
5433:
5294:
5172:
4892:
4799:
4578:
4363:Leibniz integral rule
4354:
4189:and the Faraday Law,
4184:
4022:Kelvin–Stokes theorem
4009:
3945:Kelvin–Stokes theorem
3896:
3858:
3657:
3573:
3498:
3386:
3313:
3266:
3214:
3097:particle (such as an
3081:
3037:
2956:Hans Christian Ørsted
2905:
2863:
2619:
2508:
2415:
2393:
2360:
2338:
2309:
2168:
2108:
2088:
2006:
1984:
1948:
1923:
1905:Maxwell stress tensor
1890:
1868:
1866:{\displaystyle \rho }
1846:
1766:
1714:
1673:
1671:{\displaystyle \rho }
1649:
1627:
1568:
1543:
1518:
1491:
1375:
1359:
1193:
1018:
709:
574:
534:The Weber force is a
410:
311:
227:, published in 1865.
178:
33:
15090:Quantum field theory
14985:"SpaceTime Calculus"
14779:Tai L. Chow (2006).
13578:Stress–energy tensor
13503:Reluctance (complex)
13248:Displacement current
12972:Homopolar generators
12876:
12797:
12760:
12639:
12563:
12487:
12456:
12402:
12268:
12238:
12208:
12188:
12168:
12113:
12034:
11998:
11961:
11932:
11896:
11829:
11764:
11725:
11701:
11634:
11543:
11495:
11397:
11374:The calculation for
11061:
10892:
10701:
10576:
10413:
10254:
9963:
9794:
9704:
9503:
9356:
9213:
9108:
8966:
8941:quantum mechanically
8791:
8723:
8180:
7812:
7607:
7093:
6993:
6788:
6682:
6622:
6563:
6516:
6455:
6413:
6381:Lagrange's equations
6325:
6218:
6023:
5898:
5733:
5617:
5524:
5502:
5470:
5445:
5309:
5200:
5083:
4808:
4587:
4377:
4193:
4031:
3958:
3867:
3790:
3699:is an infinitesimal
3582:
3521:
3405:The magnetic force (
3343:
3295:
3279:conventional current
3231:
3197:Green–Kubo relations
3185:electrohydrodynamics
3177:magnetohydrodynamics
3095:electrically charged
3062:Lagrangian mechanics
3044:displacement current
2995:
2636:
2532:
2428:
2402:
2373:
2367:polarization density
2347:
2320:
2179:
2144:
2097:
2015:
1993:
1971:
1935:
1910:
1877:
1857:
1782:
1726:
1690:
1662:
1636:
1577:
1552:
1527:
1500:
1431:
1315:
1029:
727:
659:
521:Carl Friedrich Gauss
364:
125:
15154:Maxwell's equations
14898:Classical Mechanics
14840:Roger F. Harrington
14045:Maxwell's Equations
14015:Maxwell's Equations
13984:(Oe) in cgs units.
13864:Cyclotron radiation
13839:Maxwell's equations
13493:Magnetomotive force
13378:Electromotive force
13348:Alternating current
13283:Jefimenko equations
13243:Cyclotron radiation
10538:
10520:
10502:
8962:for the notation):
5037:Maxwell's equations
3515:electromotive force
3453:Maxwell's equations
3423:electromotive force
3175:. For example, see
3058:luminiferous aether
2980:James Clerk Maxwell
2942:in 1762, obeyed an
2936:Johann Tobias Mayer
2892:vacuum permeability
2879:vacuum permittivity
1897:Maxwell's equations
1425:charge distribution
1395:) in motion. The 3-
1384:charge distribution
642:and magnetic field
460:As a definition of
225:James Clerk Maxwell
214:electromotive force
15144:Physical phenomena
15126:2011-08-13 at the
14922:Lanczos, Cornelius
14097:, JHU Press, 2002.
13934:Gaussian-cgs units
13829:Ampère's force law
13341:Electrical network
13178:Gauss magnetic law
13143:Static electricity
13103:Electric potential
12977:Linear alternators
12918:Mass spectrometers
12882:
12862:
12791:Christoffel symbol
12779:
12746:
12621:
12546:
12472:
12442:
12388:
12254:
12224:
12194:
12174:
12138:
12094:
12020:
11982:
11945:
11906:
11883:
11815:
11750:Space-Time Algebra
11738:
11711:
11676:
11612:
11529:
11469:
11362:
11041:
10872:
10655:
10555:
10524:
10506:
10488:
10399:
10232:
10226:
9956:the contravariant
9933:
9767:
9693:can be written in
9641:
9489:
9343:
9195:
9095:
8952:
8919:
8771:
8697:
8695:
8167:
7798:
7594:
7592:
7067:
6977:
6775:
6668:
6604:
6542:
6498:
6441:
6408:potential momentum
6388:
6369:
6299:
6171:
6166:
6125:
6084:
6009:
6007:
5975:
5943:
5881:
5704:
5603:
5510:
5478:
5453:
5428:
5289:
5196:The force becomes
5167:
5165:
4887:
4794:
4573:
4349:
4179:
4004:
3891:
3853:
3666:through the loop,
3652:
3568:
3501:
3395:Ampère's force law
3381:
3308:
3261:
3220:
3165:Boltzmann equation
3087:
3032:
2960:André-Marie Ampère
2944:inverse-square law
2932:
2858:
2614:
2526:CGS-Gaussian units
2503:
2410:
2388:
2355:
2333:
2304:
2163:
2137:for more details.
2103:
2083:
2081:
2057:
2001:
1979:
1962:general relativity
1943:
1918:
1885:
1863:
1841:
1761:
1709:
1668:
1644:
1622:
1563:
1538:
1513:
1486:
1421:
1354:
1188:
1013:
1011:
704:
615:
546:but also that the
540:Newton's third law
538:and complies with
525:André-Marie Ampère
405:
173:
50:, specifically in
44:
15130:by Wolfgang Bauer
15100:978-0-521-86449-7
14882:978-0-19-530048-2
14339:978-0-13-805326-0
14207:978-0-471-92712-9
14182:978-0-471-92712-9
14055:978-0-470-54276-7
14025:978-0-470-54276-7
13869:Magnetoresistance
13809:
13808:
13508:Reluctance (real)
13478:Gyrator–capacitor
13423:Resonant cavities
13313:Maxwell equations
12885:{\displaystyle D}
12828:
12670:
12315:
12299:
12197:{\displaystyle e}
12177:{\displaystyle m}
11979:
11570:
11424:
11131:
11093:
10966:
10924:
10733:
10553:
10552:
10550:
10468:
10467:
10465:
10246:is the covariant
9950:of the particle,
9736:
9622:
9590:
9588:
9576:
9571:
9542:
9523:
9473:
9406:
9383:
9327:
9260:
9240:
9193:
9192:
9180:
9175:
9147:
9093:
9064:
9048:
9018:
8993:
8950:
8896:
8858:
8846:
8841:
8757:
8672:
8598:
8527:
8488:
8435:
8405:
8372:
8353:
8323:
8290:
8263:
8233:
8159:
8148:
8117:
8106:
8075:
8064:
8026:
7991:
7980:
7949:
7938:
7907:
7896:
7866:
7827:
7790:
7779:
7748:
7737:
7706:
7695:
7657:
7628:
7582:
7571:
7540:
7529:
7498:
7487:
7457:
7418:
7383:
7347:
7311:
7275:
7243:
7222:
7201:
7163:
7146:
7142:
7117:
7065:
7042:
7038:
7013:
6950:
6925:
6900:
6866:
6844:
6822:
6805:
6763:
6737:
6722:
6711:
6665:
6650:
6639:
6601:
6495:
6386:
6363:
6287:
6261:
6246:
6235:
6165:
6161:
6139:
6124:
6120:
6098:
6083:
6079:
6057:
6041:
6006:
5989:
5974:
5957:
5942:
5925:
5862:
5838:
5821:
5787:
5694:
5576:
5551:
5360:
5251:
5181:is the gradient,
5134:
4776:
4714:
4651:
4496:
4339:
4287:
4257:
4177:
4102:
4087:
3998:
3747:Maxwell Equations
3743:Maxwell Equations
3566:
3193:stellar evolution
3189:superconductivity
3014:
2908:Maxwell equations
2853:
2852:
2794:
2730:
2689:
2688:
2587:
2485:
2286:
2106:{\displaystyle c}
2080:
2056:
1573:, the result is:
1423:For a continuous
1271:, while the term
1146:
1070:
510:relative velocity
491:synchrotron light
285:Charged particles
199:and the velocity
87:Lorentz force law
56:Lorentz force law
16:(Redirected from
15166:
15149:Electromagnetism
15104:
15083:
15064:
15045:
15023:
14989:
14988:
14977:
14971:
14968:
14962:
14961:
14955:
14947:
14924:(January 1986).
14918:
14912:
14911:
14893:
14887:
14886:
14866:
14860:
14859:
14836:
14830:
14829:
14806:
14797:
14796:
14776:
14770:
14767:
14758:
14757:
14755:
14754:
14745:. Archived from
14735:
14729:
14726:
14720:
14719:
14700:Whittaker, E. T.
14696:
14690:
14689:
14673:
14663:
14657:
14650:
14644:
14643:
14627:
14621:
14620:
14604:
14594:
14588:
14587:
14555:
14549:
14548:
14532:
14522:
14516:
14515:
14491:
14485:
14484:
14468:
14458:
14452:
14451:
14427:
14421:
14420:
14400:
14394:
14393:
14375:
14369:
14368:
14350:
14344:
14343:
14323:
14312:
14305:
14299:
14296:
14290:
14284:
14260:
14251:
14250:
14242:
14236:
14235:
14233:
14232:
14218:
14212:
14211:
14198:Electromagnetism
14193:
14187:
14186:
14173:Electromagnetism
14168:
14162:
14160:
14154:
14148:
14139:
14123:
14113:
14107:
14104:
14098:
14094:Oliver Heaviside
14089:
14078:
14077:
14069:
14060:
14059:
14039:
14030:
14029:
14009:
14003:
14002:
14000:
13998:
13975:
13968:
13962:
13960:
13958:
13956:
13941:
13932:(symbol: T). In
13927:
13920:
13874:Scalar potential
13824:Gravitomagnetism
13819:Electromagnetism
13801:
13794:
13787:
13468:Electric machine
13451:Magnetic circuit
13413:Parallel circuit
13403:Network analysis
13368:Electric current
13303:London equations
13148:Triboelectricity
13138:Potential energy
13007:
12997:Electromagnetism
12988:
12987:
12922:Velocity Filters
12891:
12889:
12888:
12883:
12871:
12869:
12868:
12863:
12858:
12857:
12848:
12847:
12829:
12827:
12819:
12818:
12817:
12804:
12788:
12786:
12785:
12780:
12778:
12777:
12755:
12753:
12752:
12747:
12742:
12741:
12732:
12731:
12713:
12712:
12703:
12702:
12693:
12692:
12671:
12669:
12661:
12660:
12659:
12646:
12632:
12630:
12628:
12627:
12622:
12620:
12619:
12607:
12606:
12594:
12593:
12578:
12577:
12557:
12555:
12553:
12552:
12547:
12545:
12544:
12532:
12527:
12526:
12508:
12507:
12481:
12479:
12478:
12473:
12471:
12470:
12451:
12449:
12448:
12443:
12435:
12430:
12429:
12414:
12413:
12397:
12395:
12394:
12389:
12384:
12383:
12374:
12373:
12355:
12354:
12345:
12344:
12335:
12334:
12316:
12308:
12300:
12298:
12290:
12289:
12288:
12275:
12263:
12261:
12260:
12255:
12253:
12252:
12233:
12231:
12230:
12225:
12223:
12222:
12203:
12201:
12200:
12195:
12183:
12181:
12180:
12175:
12147:
12145:
12144:
12139:
12131:
12130:
12103:
12101:
12100:
12095:
12087:
12086:
12068:
12063:
12062:
12041:
12029:
12027:
12026:
12021:
12010:
12009:
11993:
11991:
11989:
11988:
11983:
11981:
11980:
11972:
11954:
11952:
11951:
11946:
11944:
11943:
11928:with the vector
11916:is a space-time
11915:
11913:
11912:
11907:
11905:
11904:
11892:
11890:
11889:
11884:
11882:
11881:
11872:
11868:
11867:
11866:
11854:
11853:
11839:
11824:
11822:
11821:
11816:
11814:
11813:
11804:
11800:
11799:
11798:
11786:
11785:
11771:
11754:Clifford algebra
11747:
11745:
11744:
11739:
11737:
11736:
11720:
11718:
11717:
11712:
11710:
11709:
11685:
11683:
11682:
11677:
11671:
11666:
11665:
11641:
11629:
11621:
11619:
11618:
11613:
11608:
11604:
11603:
11595:
11587:
11571:
11569:
11565:
11559:
11558:
11553:
11547:
11539:so we arrive at
11538:
11536:
11535:
11530:
11490:
11485:and proper time
11484:
11478:
11476:
11475:
11470:
11465:
11461:
11460:
11452:
11444:
11425:
11423:
11419:
11413:
11412:
11407:
11401:
11392:
11386:
11380:
11371:
11369:
11368:
11363:
11357:
11353:
11352:
11351:
11346:
11342:
11341:
11333:
11319:
11318:
11295:
11291:
11290:
11289:
11280:
11279:
11267:
11266:
11257:
11256:
11244:
11243:
11220:
11216:
11212:
11211:
11196:
11195:
11174:
11173:
11155:
11154:
11136:
11132:
11127:
11126:
11117:
11094:
11092:
11088:
11082:
11081:
11080:
11071:
11065:
11050:
11048:
11047:
11042:
11037:
11033:
11029:
11028:
11016:
11015:
11000:
10999:
10984:
10983:
10971:
10967:
10962:
10961:
10952:
10946:
10945:
10925:
10923:
10919:
10913:
10912:
10911:
10902:
10896:
10881:
10879:
10878:
10873:
10868:
10864:
10863:
10862:
10853:
10852:
10840:
10839:
10830:
10829:
10817:
10816:
10807:
10806:
10794:
10793:
10784:
10783:
10763:
10762:
10750:
10749:
10734:
10732:
10728:
10722:
10721:
10720:
10711:
10705:
10692:
10673:
10664:
10662:
10661:
10656:
10650:
10649:
10637:
10636:
10631:
10630:
10629:
10618:
10617:
10612:
10611:
10610:
10596:
10595:
10594:
10564:
10562:
10561:
10556:
10554:
10551:
10549:
10548:
10539:
10537:
10532:
10519:
10514:
10501:
10496:
10486:
10478:
10474:
10469:
10466:
10464:
10463:
10454:
10453:
10444:
10436:
10432:
10408:
10406:
10405:
10400:
10395:
10391:
10390:
10389:
10374:
10373:
10358:
10357:
10328:
10324:
10323:
10322:
10310:
10309:
10297:
10296:
10284:
10283:
10266:
10265:
10245:
10241:
10239:
10238:
10233:
10231:
10230:
10218:
10217:
10206:
10205:
10188:
10183:
10182:
10169:
10168:
10149:
10148:
10134:
10129:
10128:
10115:
10114:
10103:
10102:
10080:
10075:
10074:
10058:
10053:
10052:
10035:
10030:
10029:
10012:
10007:
10006:
9978:
9977:
9955:
9945:
9942:
9940:
9939:
9934:
9929:
9925:
9924:
9923:
9911:
9910:
9898:
9897:
9868:
9864:
9863:
9862:
9850:
9849:
9837:
9836:
9824:
9823:
9806:
9805:
9785:
9776:
9774:
9773:
9768:
9766:
9765:
9756:
9755:
9737:
9735:
9731:
9725:
9724:
9723:
9714:
9708:
9692:
9686:
9682:metric signature
9650:
9648:
9647:
9642:
9637:
9633:
9632:
9624:
9623:
9618:
9613:
9607:
9591:
9589:
9587:
9586:
9581:
9577:
9572:
9567:
9562:
9560:
9547:
9545:
9544:
9543:
9538:
9533:
9526:
9524:
9522:
9518:
9512:
9507:
9498:
9496:
9495:
9490:
9488:
9484:
9474:
9472:
9471:
9462:
9461:
9443:
9442:
9433:
9422:
9414:
9407:
9405:
9404:
9392:
9384:
9382:
9378:
9372:
9371:
9366:
9360:
9352:
9350:
9349:
9344:
9342:
9338:
9328:
9326:
9325:
9316:
9315:
9297:
9296:
9287:
9276:
9268:
9261:
9259:
9258:
9246:
9241:
9239:
9235:
9229:
9228:
9223:
9217:
9205:are the same as
9204:
9202:
9201:
9196:
9194:
9191:
9190:
9185:
9181:
9176:
9171:
9166:
9164:
9151:
9150:
9149:
9148:
9143:
9138:
9131:
9126:
9115:
9104:
9102:
9101:
9096:
9094:
9092:
9091:
9082:
9074:
9066:
9065:
9060:
9055:
9049:
9047:
9046:
9037:
9036:
9027:
9019:
9017:
9016:
9007:
8999:
8994:
8992:
8988:
8982:
8981:
8976:
8970:
8958:the action (see
8928:
8926:
8925:
8920:
8915:
8898:
8897:
8892:
8887:
8878:
8870:
8859:
8857:
8856:
8851:
8847:
8842:
8837:
8832:
8830:
8817:
8815:
8814:
8780:
8778:
8777:
8772:
8767:
8759:
8758:
8750:
8744:
8730:
8718:
8712:
8706:
8704:
8703:
8698:
8696:
8692:
8691:
8682:
8674:
8673:
8665:
8653:
8652:
8634:
8630:
8629:
8617:
8600:
8599:
8591:
8579:
8578:
8560:
8553:
8552:
8543:
8529:
8528:
8520:
8514:
8513:
8504:
8490:
8489:
8481:
8469:
8468:
8450:
8446:
8442:
8441:
8437:
8436:
8434:
8426:
8425:
8424:
8411:
8406:
8404:
8396:
8395:
8394:
8381:
8374:
8373:
8365:
8359:
8355:
8354:
8352:
8344:
8343:
8342:
8329:
8324:
8322:
8314:
8313:
8312:
8299:
8292:
8291:
8283:
8269:
8265:
8264:
8262:
8254:
8253:
8252:
8239:
8234:
8232:
8224:
8216:
8196:
8195:
8176:
8174:
8173:
8168:
8166:
8162:
8161:
8160:
8152:
8149:
8147:
8139:
8138:
8137:
8124:
8119:
8118:
8110:
8107:
8105:
8097:
8096:
8095:
8082:
8077:
8076:
8068:
8065:
8063:
8055:
8054:
8053:
8040:
8027:
8025:
8017:
8009:
7998:
7994:
7993:
7992:
7984:
7981:
7979:
7971:
7970:
7969:
7956:
7951:
7950:
7942:
7939:
7937:
7929:
7928:
7927:
7914:
7909:
7908:
7900:
7897:
7895:
7887:
7886:
7885:
7872:
7867:
7865:
7857:
7856:
7855:
7842:
7829:
7828:
7820:
7807:
7805:
7804:
7799:
7797:
7793:
7792:
7791:
7783:
7780:
7778:
7770:
7769:
7768:
7755:
7750:
7749:
7741:
7738:
7736:
7728:
7727:
7726:
7713:
7708:
7707:
7699:
7696:
7694:
7686:
7685:
7684:
7671:
7658:
7656:
7648:
7640:
7629:
7627:
7619:
7611:
7603:
7601:
7600:
7595:
7593:
7589:
7585:
7584:
7583:
7575:
7572:
7570:
7562:
7561:
7560:
7547:
7542:
7541:
7533:
7530:
7528:
7520:
7519:
7518:
7505:
7500:
7499:
7491:
7488:
7486:
7478:
7477:
7476:
7463:
7458:
7456:
7448:
7447:
7446:
7433:
7420:
7419:
7411:
7399:
7395:
7391:
7384:
7382:
7374:
7373:
7372:
7359:
7348:
7346:
7338:
7337:
7336:
7323:
7312:
7310:
7302:
7301:
7300:
7287:
7276:
7274:
7266:
7265:
7264:
7251:
7244:
7242:
7238:
7229:
7224:
7223:
7215:
7202:
7200:
7196:
7190:
7189:
7188:
7179:
7173:
7165:
7164:
7156:
7147:
7145:
7144:
7143:
7135:
7128:
7120:
7118:
7116:
7112:
7106:
7101:
7088:
7082:
7076:
7074:
7073:
7068:
7066:
7064:
7056:
7048:
7043:
7041:
7040:
7039:
7031:
7024:
7016:
7014:
7012:
7008:
7002:
6997:
6986:
6984:
6983:
6978:
6967:
6963:
6962:
6961:
6952:
6951:
6943:
6937:
6936:
6927:
6926:
6918:
6912:
6911:
6902:
6901:
6893:
6879:
6875:
6874:
6873:
6868:
6867:
6859:
6852:
6851:
6846:
6845:
6837:
6830:
6829:
6824:
6823:
6815:
6806:
6798:
6784:
6782:
6781:
6776:
6765:
6764:
6756:
6750:
6739:
6738:
6730:
6724:
6723:
6715:
6712:
6704:
6677:
6675:
6674:
6669:
6667:
6666:
6658:
6652:
6651:
6643:
6640:
6632:
6613:
6611:
6610:
6605:
6603:
6602:
6594:
6588:
6557:potential energy
6551:
6549:
6548:
6543:
6532:
6507:
6505:
6504:
6499:
6497:
6496:
6488:
6473:
6465:
6450:
6448:
6447:
6442:
6431:
6423:
6405:
6395:
6378:
6376:
6375:
6370:
6365:
6364:
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6289:
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6280:
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6236:
6228:
6209:
6203:
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6154:
6144:
6141:
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6132:
6126:
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6018:
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5990:
5982:
5976:
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5528:
5519:
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5511:
5509:
5496:total derivative
5489:
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5485:
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5479:
5477:
5464:
5462:
5460:
5459:
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5410:
5403:
5390:
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5350:
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5316:
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5260:
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5232:
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5144:
5135:
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5125:
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5115:
5094:
5078:
5065:
5056:
5050:
5017:gradient theorem
5010:
5004:
4995:
4986:
4978:
4972:
4964:, also changing
4963:
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3307:
3302:
3291:segment of wire
3283:
3276:
3270:
3268:
3267:
3262:
3257:
3249:
3238:
3048:Oliver Heaviside
3041:
3039:
3038:
3033:
3028:
3020:
3015:
3007:
3002:
2926:, 1892, p. 451.
2916:electrical field
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1952:
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1927:
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1495:
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1485:
1481:
1480:
1472:
1464:
1451:
1443:
1438:
1418:
1409:
1403:
1394:
1382:on a continuous
1381:
1363:
1361:
1360:
1355:
1350:
1342:
1330:
1322:
1287:
1266:
1254:
1248:
1242:
1232:
1226:
1220:
1207:
1203:
1197:
1195:
1194:
1189:
1187:
1183:
1173:
1165:
1148:
1147:
1142:
1137:
1122:
1114:
1098:
1094:
1072:
1071:
1066:
1061:
1046:
1036:
1022:
1020:
1019:
1014:
1012:
1005:
1001:
1000:
999:
990:
989:
977:
976:
967:
966:
954:
953:
929:
928:
912:
908:
907:
906:
897:
896:
884:
883:
874:
873:
861:
860:
836:
835:
819:
815:
814:
813:
804:
803:
791:
790:
781:
780:
768:
767:
743:
742:
722:
713:
711:
710:
705:
703:
699:
698:
690:
682:
666:
647:
641:
635:
629:
622:
610:
601:
594:
588:
583:charged particle
580:
567:Charged particle
483:
477:
471:
465:
455:
449:
440:
435:
427:
414:
412:
411:
406:
401:
393:
385:
371:
356:
348:
340:
332:
320:
308:
302:
271:
251:
233:Oliver Heaviside
204:
198:
192:
186:
182:
180:
179:
174:
169:
165:
164:
156:
148:
132:
116:
107:
98:
92:
52:electromagnetism
21:
15174:
15173:
15169:
15168:
15167:
15165:
15164:
15163:
15159:Hendrik Lorentz
15134:
15133:
15128:Wayback Machine
15112:
15107:
15101:
15080:
15061:
15042:
15020:
14997:
14992:
14981:Hestenes, David
14978:
14974:
14969:
14965:
14949:
14948:
14936:
14919:
14915:
14908:
14894:
14890:
14883:
14867:
14863:
14856:
14837:
14833:
14826:
14807:
14800:
14793:
14777:
14773:
14768:
14761:
14752:
14750:
14737:
14736:
14732:
14727:
14723:
14716:
14697:
14693:
14686:
14664:
14660:
14651:
14647:
14628:
14624:
14617:
14595:
14591:
14570:(68): 229–249.
14556:
14552:
14545:
14537:–131, 139–144.
14523:
14519:
14512:
14492:
14488:
14481:
14459:
14455:
14448:
14428:
14424:
14417:
14401:
14397:
14390:
14376:
14372:
14365:
14351:
14347:
14340:
14324:
14315:
14306:
14302:
14297:
14293:
14280:
14263:
14261:
14254:
14243:
14239:
14230:
14228:
14220:
14219:
14215:
14208:
14194:
14190:
14183:
14169:
14165:
14156:
14150:
14144:
14136:
14114:
14110:
14105:
14101:
14091:Paul J. Nahin,
14090:
14081:
14070:
14063:
14056:
14040:
14033:
14026:
14010:
14006:
13996:
13994:
13985:
13971:
13969:
13965:
13954:
13952:
13947:
13942:is measured in
13937:
13928:is measured in
13923:
13921:
13912:
13908:
13903:
13834:Hendrik Lorentz
13805:
13776:
13775:
13591:
13583:
13582:
13538:
13528:
13527:
13483:Induction motor
13453:
13443:
13442:
13358:Current density
13343:
13333:
13332:
13323:Poynting vector
13233:
13231:Electrodynamics
13223:
13222:
13218:Right-hand rule
13183:Magnetic dipole
13173:Biot–Savart law
13163:
13153:
13152:
13088:Electric dipole
13083:Electric charge
13058:
12986:
12981:
12942:Electric motors
12902:
12877:
12874:
12873:
12853:
12849:
12840:
12836:
12820:
12813:
12809:
12805:
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12678:
12662:
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12645:
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12573:
12569:
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12559:
12540:
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12528:
12519:
12515:
12494:
12490:
12488:
12485:
12484:
12483:
12466:
12462:
12457:
12454:
12453:
12431:
12425:
12421:
12409:
12405:
12403:
12400:
12399:
12379:
12375:
12366:
12362:
12350:
12346:
12340:
12336:
12321:
12317:
12307:
12291:
12284:
12280:
12276:
12274:
12269:
12266:
12265:
12245:
12241:
12239:
12236:
12235:
12215:
12211:
12209:
12206:
12205:
12189:
12186:
12185:
12169:
12166:
12165:
12158:
12149:
12126:
12125:
12114:
12111:
12110:
12082:
12078:
12064:
12058:
12054:
12037:
12035:
12032:
12031:
12005:
12001:
11999:
11996:
11995:
11971:
11970:
11962:
11959:
11958:
11957:
11939:
11935:
11933:
11930:
11929:
11900:
11899:
11897:
11894:
11893:
11877:
11873:
11862:
11858:
11849:
11848:
11847:
11843:
11835:
11830:
11827:
11826:
11809:
11805:
11794:
11790:
11781:
11780:
11779:
11775:
11767:
11765:
11762:
11761:
11732:
11728:
11726:
11723:
11722:
11705:
11704:
11702:
11699:
11698:
11691:
11667:
11661:
11657:
11637:
11635:
11632:
11631:
11625:
11599:
11591:
11583:
11582:
11578:
11561:
11560:
11554:
11549:
11548:
11546:
11544:
11541:
11540:
11496:
11493:
11492:
11486:
11480:
11456:
11448:
11440:
11439:
11435:
11415:
11414:
11408:
11403:
11402:
11400:
11398:
11395:
11394:
11388:
11382:
11375:
11347:
11337:
11329:
11328:
11324:
11323:
11314:
11310:
11309:
11305:
11285:
11281:
11275:
11271:
11262:
11258:
11252:
11248:
11239:
11235:
11234:
11230:
11207:
11203:
11191:
11187:
11169:
11165:
11150:
11146:
11122:
11118:
11116:
11112:
11108:
11104:
11084:
11083:
11076:
11072:
11067:
11066:
11064:
11062:
11059:
11058:
11024:
11020:
11011:
11007:
10995:
10991:
10979:
10975:
10957:
10953:
10951:
10947:
10941:
10937:
10936:
10932:
10915:
10914:
10907:
10903:
10898:
10897:
10895:
10893:
10890:
10889:
10858:
10854:
10848:
10844:
10835:
10831:
10825:
10821:
10812:
10808:
10802:
10798:
10789:
10785:
10779:
10775:
10774:
10770:
10755:
10751:
10745:
10741:
10724:
10723:
10716:
10712:
10707:
10706:
10704:
10702:
10699:
10698:
10687:
10684:
10672:
10666:
10642:
10638:
10632:
10625:
10621:
10620:
10619:
10613:
10606:
10602:
10601:
10600:
10587:
10583:
10579:
10577:
10574:
10573:
10544:
10540:
10533:
10528:
10515:
10510:
10497:
10492:
10487:
10485:
10473:
10459:
10455:
10449:
10445:
10443:
10431:
10414:
10411:
10410:
10385:
10381:
10369:
10365:
10353:
10349:
10339:
10335:
10318:
10314:
10305:
10301:
10292:
10288:
10279:
10275:
10274:
10270:
10261:
10257:
10255:
10252:
10251:
10243:
10225:
10224:
10219:
10213:
10209:
10207:
10201:
10197:
10192:
10184:
10178:
10174:
10171:
10170:
10164:
10160:
10155:
10150:
10144:
10140:
10138:
10130:
10124:
10120:
10117:
10116:
10110:
10106:
10104:
10098:
10094:
10089:
10084:
10076:
10070:
10066:
10063:
10062:
10054:
10048:
10044:
10039:
10031:
10025:
10021:
10016:
10008:
10002:
9998:
9993:
9983:
9982:
9970:
9966:
9964:
9961:
9960:
9951:
9943:
9919:
9915:
9906:
9902:
9893:
9889:
9876:
9872:
9858:
9854:
9845:
9841:
9832:
9828:
9819:
9815:
9814:
9810:
9801:
9797:
9795:
9792:
9791:
9781:
9778:
9761:
9757:
9748:
9744:
9727:
9726:
9719:
9715:
9710:
9709:
9707:
9705:
9702:
9701:
9688:
9685:(1, −1, −1, −1)
9684:
9678:
9670:Main articles:
9668:
9663:
9658:
9653:
9628:
9614:
9612:
9611:
9603:
9602:
9598:
9582:
9563:
9561:
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9554:
9546:
9534:
9532:
9531:
9527:
9525:
9514:
9513:
9508:
9506:
9504:
9501:
9500:
9467:
9463:
9457:
9453:
9438:
9434:
9429:
9418:
9413:
9412:
9408:
9400:
9396:
9391:
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9353:
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9311:
9307:
9292:
9288:
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9245:
9231:
9230:
9224:
9219:
9218:
9216:
9214:
9211:
9210:
9186:
9167:
9165:
9163:
9159:
9158:
9139:
9137:
9136:
9132:
9130:
9122:
9111:
9109:
9106:
9105:
9087:
9083:
9075:
9073:
9056:
9054:
9053:
9042:
9038:
9032:
9028:
9026:
9012:
9008:
9000:
8998:
8984:
8983:
8977:
8972:
8971:
8969:
8967:
8964:
8963:
8960:matrix calculus
8911:
8888:
8886:
8885:
8874:
8866:
8852:
8833:
8831:
8829:
8825:
8824:
8816:
8810:
8806:
8792:
8789:
8788:
8782:
8763:
8749:
8748:
8740:
8726:
8724:
8721:
8720:
8714:
8708:
8694:
8693:
8687:
8683:
8678:
8664:
8663:
8648:
8644:
8632:
8631:
8625:
8621:
8613:
8590:
8589:
8574:
8570:
8558:
8557:
8548:
8544:
8539:
8519:
8518:
8509:
8505:
8500:
8480:
8479:
8464:
8460:
8448:
8447:
8427:
8420:
8416:
8412:
8410:
8397:
8390:
8386:
8382:
8380:
8379:
8375:
8364:
8363:
8345:
8338:
8334:
8330:
8328:
8315:
8308:
8304:
8300:
8298:
8297:
8293:
8282:
8281:
8280:
8276:
8255:
8248:
8244:
8240:
8238:
8225:
8217:
8215:
8214:
8210:
8197:
8191:
8187:
8183:
8181:
8178:
8177:
8151:
8150:
8140:
8133:
8129:
8125:
8123:
8109:
8108:
8098:
8091:
8087:
8083:
8081:
8067:
8066:
8056:
8049:
8045:
8041:
8039:
8038:
8034:
8018:
8010:
8008:
7983:
7982:
7972:
7965:
7961:
7957:
7955:
7941:
7940:
7930:
7923:
7919:
7915:
7913:
7899:
7898:
7888:
7881:
7877:
7873:
7871:
7858:
7851:
7847:
7843:
7841:
7840:
7836:
7819:
7818:
7813:
7810:
7809:
7782:
7781:
7771:
7764:
7760:
7756:
7754:
7740:
7739:
7729:
7722:
7718:
7714:
7712:
7698:
7697:
7687:
7680:
7676:
7672:
7670:
7669:
7665:
7649:
7641:
7639:
7620:
7612:
7610:
7608:
7605:
7604:
7591:
7590:
7574:
7573:
7563:
7556:
7552:
7548:
7546:
7532:
7531:
7521:
7514:
7510:
7506:
7504:
7490:
7489:
7479:
7472:
7468:
7464:
7462:
7449:
7442:
7438:
7434:
7432:
7431:
7427:
7410:
7409:
7397:
7396:
7375:
7368:
7364:
7360:
7358:
7339:
7332:
7328:
7324:
7322:
7303:
7296:
7292:
7288:
7286:
7267:
7260:
7256:
7252:
7250:
7249:
7245:
7234:
7233:
7228:
7214:
7213:
7203:
7192:
7191:
7184:
7180:
7175:
7174:
7172:
7155:
7154:
7134:
7133:
7129:
7121:
7119:
7108:
7107:
7102:
7100:
7096:
7094:
7091:
7090:
7084:
7078:
7057:
7049:
7047:
7030:
7029:
7025:
7017:
7015:
7004:
7003:
6998:
6996:
6994:
6991:
6990:
6957:
6953:
6942:
6941:
6932:
6928:
6917:
6916:
6907:
6903:
6892:
6891:
6890:
6886:
6869:
6858:
6857:
6856:
6847:
6836:
6835:
6834:
6825:
6814:
6813:
6812:
6811:
6807:
6797:
6789:
6786:
6785:
6755:
6754:
6746:
6729:
6728:
6714:
6713:
6703:
6683:
6680:
6679:
6657:
6656:
6642:
6641:
6631:
6623:
6620:
6619:
6593:
6592:
6584:
6564:
6561:
6560:
6528:
6517:
6514:
6513:
6487:
6486:
6469:
6461:
6456:
6453:
6452:
6427:
6419:
6414:
6411:
6410:
6397:
6391:
6355:
6354:
6346:
6326:
6323:
6322:
6316:
6310:
6279:
6278:
6270:
6253:
6252:
6238:
6237:
6227:
6219:
6216:
6215:
6205:
6199:
6192:
6186:
6153:
6152:
6148:
6142:
6131:
6130:
6112:
6111:
6107:
6101:
6090:
6089:
6071:
6070:
6066:
6060:
6049:
6048:
6033:
6031:
6030:
6026:
6024:
6021:
6020:
5998:
5992:
5981:
5980:
5966:
5960:
5949:
5948:
5934:
5928:
5917:
5916:
5906:
5905:
5901:
5899:
5896:
5895:
5892:
5868:
5854:
5852:
5851:
5830:
5828:
5827:
5823:
5812:
5811:
5806:
5804:
5793:
5779:
5777:
5776:
5760:
5759:
5755:
5751:
5747:
5736:
5734:
5731:
5730:
5717:
5685:
5684:
5678:
5673:
5672:
5670:
5659:
5651:
5635:
5631:
5620:
5618:
5615:
5614:
5598:
5584:
5568:
5562:
5558:
5556:
5542:
5541:
5535:
5530:
5529:
5527:
5525:
5522:
5521:
5505:
5503:
5500:
5499:
5473:
5471:
5468:
5467:
5466:
5448:
5446:
5443:
5442:
5441:
5415:
5399:
5398:
5394:
5381:
5373:
5372:
5368:
5352:
5346:
5342:
5340:
5327:
5323:
5312:
5310:
5307:
5306:
5273:
5256:
5243:
5237:
5233:
5231:
5218:
5214:
5203:
5201:
5198:
5197:
5186:
5182:
5178:
5164:
5163:
5158:
5145:
5140:
5137:
5136:
5126:
5120:
5116:
5114:
5095:
5090:
5086:
5084:
5081:
5080:
5074:
5061:
5052:
5046:
5043:
5029:
5006:
5000:
4994:
4988:
4980:
4974:
4971:
4965:
4958:
4952:
4942:
4935:
4929:
4923:
4917:
4914:
4908:
4869:
4861:
4853:
4831:
4823:
4811:
4809:
4806:
4805:
4789:
4784:
4766:
4758:
4750:
4728:
4724:
4704:
4696:
4688:
4683:
4665:
4661:
4641:
4630:
4625:
4617:
4612:
4594:
4590:
4588:
4585:
4584:
4568:
4563:
4557:
4549:
4527:
4523:
4506:
4498:
4488:
4483:
4475:
4470:
4455:
4451:
4431:
4420:
4415:
4407:
4402:
4384:
4380:
4378:
4375:
4374:
4366:
4329:
4321:
4313:
4308:
4293:
4289:
4278:
4277:
4272:
4270:
4247:
4236:
4231:
4223:
4218:
4200:
4196:
4194:
4191:
4190:
4168:
4167:
4151:
4143:
4138:
4137:
4135:
4127:
4122:
4107:
4103:
4077:
4069:
4061:
4056:
4038:
4034:
4032:
4029:
4028:
3990:
3984:
3980:
3978:
3967:
3959:
3956:
3955:
3941:right-hand rule
3933:
3924:
3913:
3902:
3883:
3878:
3870:
3868:
3865:
3864:
3845:
3840:
3832:
3827:
3807:
3803:
3794:
3793:
3791:
3788:
3787:
3779:
3771:
3765:
3757:
3704:
3693:
3689:
3681:
3673:
3667:
3638:
3630:
3622:
3617:
3602:
3598:
3589:
3585:
3583:
3580:
3579:
3557:
3556:
3549:
3545:
3540:
3539:
3537:
3525:
3524:
3522:
3519:
3518:
3507:
3493:
3475:portion of the
3406:
3403:
3373:
3365:
3360:
3346:
3344:
3341:
3340:
3336:
3323:
3303:
3298:
3296:
3293:
3292:
3281:
3272:
3253:
3245:
3234:
3232:
3229:
3228:
3209:
3127:
3076:
3070:
3056:matter and the
3053:Hendrik Lorentz
3024:
3016:
3006:
2998:
2996:
2993:
2992:
2968:Michael Faraday
2952:torsion balance
2940:Henry Cavendish
2918:E (II) and the
2900:
2888:
2882:
2875:
2869:
2846:
2842:
2836:
2832:
2826:
2805:
2804:
2799:
2798:
2797:
2788:
2784:
2779:
2771:
2761:
2760:
2755:
2754:
2740:
2739:
2734:
2733:
2724:
2720:
2712:
2702:
2701:
2696:
2695:
2682:
2678:
2661:
2660:
2656:
2654:
2644:
2643:
2639:
2637:
2634:
2633:
2599:
2598:
2593:
2592:
2579:
2577:
2567:
2566:
2561:
2560:
2559:
2555:
2548:
2547:
2543:
2535:
2533:
2530:
2529:
2518:
2495:
2477:
2471:
2467:
2465:
2457:
2442:
2437:
2436:
2435:
2431:
2429:
2426:
2425:
2405:
2403:
2400:
2399:
2382:
2377:
2376:
2374:
2371:
2370:
2350:
2348:
2345:
2344:
2327:
2323:
2321:
2318:
2317:
2296:
2278:
2272:
2268:
2266:
2258:
2243:
2238:
2237:
2236:
2232:
2224:
2214:
2199:
2195:
2194:
2190:
2182:
2180:
2177:
2176:
2155:
2147:
2145:
2142:
2141:
2098:
2095:
2094:
2072:
2066:
2062:
2059:
2050:
2046:
2040:
2032:
2018:
2016:
2013:
2012:
1996:
1994:
1991:
1990:
1974:
1972:
1969:
1968:
1938:
1936:
1933:
1932:
1930:Poynting vector
1913:
1911:
1908:
1907:
1901:vector calculus
1880:
1878:
1875:
1874:
1858:
1855:
1854:
1853:By eliminating
1830:
1820:
1812:
1804:
1800:
1796:
1785:
1783:
1780:
1779:
1776:volume integral
1772:
1756:
1748:
1740:
1729:
1727:
1724:
1723:
1704:
1693:
1691:
1688:
1687:
1684:current density
1663:
1660:
1659:
1639:
1637:
1634:
1633:
1612:
1604:
1596:
1595:
1591:
1580:
1578:
1575:
1574:
1555:
1553:
1550:
1549:
1530:
1528:
1525:
1524:
1508:
1503:
1501:
1498:
1497:
1476:
1468:
1460:
1459:
1455:
1447:
1439:
1434:
1432:
1429:
1428:
1414:
1405:
1399:
1397:current density
1390:
1377:
1370:
1346:
1338:
1326:
1318:
1316:
1313:
1312:
1282:
1272:
1259:
1250:
1244:
1238:
1235:right-hand rule
1228:
1222:
1216:
1205:
1199:
1169:
1161:
1138:
1136:
1135:
1118:
1110:
1109:
1105:
1062:
1060:
1059:
1042:
1041:
1037:
1032:
1030:
1027:
1026:
1010:
1009:
995:
991:
985:
981:
972:
968:
962:
958:
949:
945:
944:
940:
930:
924:
920:
917:
916:
902:
898:
892:
888:
879:
875:
869:
865:
856:
852:
851:
847:
837:
831:
827:
824:
823:
809:
805:
799:
795:
786:
782:
776:
772:
763:
759:
758:
754:
744:
738:
734:
730:
728:
725:
724:
721:
718:
715:
694:
686:
678:
677:
673:
662:
660:
657:
656:
648:, is given by (
643:
637:
631:
627:
625:electric charge
618:
606:
597:
590:
586:
576:
569:
564:
503:
479:
473:
467:
461:
451:
445:
431:
429:
423:
397:
389:
381:
367:
365:
362:
361:
352:
344:
336:
328:
316:
304:
298:
291:
290:
289:
288:
281:
280:
279:
272:
264:
263:
252:
241:
229:Hendrik Lorentz
200:
194:
188:
184:
160:
152:
144:
143:
139:
128:
126:
123:
122:
112:
103:
94:
90:
76:physical effect
28:
23:
22:
15:
12:
11:
5:
15172:
15162:
15161:
15156:
15151:
15146:
15132:
15131:
15118:
15111:
15110:External links
15108:
15106:
15105:
15099:
15084:
15078:
15065:
15059:
15046:
15040:
15025:
15018:
15001:
14996:
14993:
14991:
14990:
14972:
14963:
14934:
14913:
14906:
14888:
14881:
14861:
14854:
14831:
14824:
14798:
14791:
14771:
14759:
14730:
14721:
14714:
14691:
14684:
14658:
14645:
14622:
14615:
14589:
14550:
14543:
14517:
14510:
14486:
14479:
14453:
14446:
14422:
14415:
14395:
14388:
14370:
14363:
14345:
14338:
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14300:
14291:
14278:
14252:
14237:
14213:
14206:
14188:
14181:
14163:
14134:
14108:
14099:
14079:
14061:
14054:
14031:
14024:
14004:
13963:
13909:
13907:
13904:
13902:
13901:
13896:
13891:
13886:
13884:Guiding center
13881:
13876:
13871:
13866:
13861:
13859:Larmor formula
13856:
13851:
13846:
13841:
13836:
13831:
13826:
13821:
13816:
13810:
13807:
13806:
13804:
13803:
13796:
13789:
13781:
13778:
13777:
13774:
13773:
13768:
13763:
13758:
13753:
13748:
13743:
13738:
13733:
13728:
13723:
13718:
13713:
13708:
13703:
13698:
13693:
13688:
13683:
13678:
13673:
13668:
13663:
13658:
13653:
13648:
13643:
13638:
13633:
13628:
13623:
13618:
13613:
13608:
13603:
13598:
13592:
13589:
13588:
13585:
13584:
13581:
13580:
13575:
13570:
13565:
13560:
13558:Four-potential
13555:
13550:
13545:
13539:
13534:
13533:
13530:
13529:
13526:
13525:
13520:
13515:
13510:
13505:
13500:
13495:
13490:
13485:
13480:
13475:
13473:Electric motor
13470:
13465:
13460:
13454:
13449:
13448:
13445:
13444:
13441:
13440:
13435:
13430:
13428:Series circuit
13425:
13420:
13415:
13410:
13405:
13400:
13398:Kirchhoff laws
13395:
13390:
13385:
13380:
13375:
13370:
13365:
13363:Direct current
13360:
13355:
13350:
13344:
13339:
13338:
13335:
13334:
13331:
13330:
13325:
13320:
13318:Maxwell tensor
13315:
13310:
13305:
13300:
13295:
13290:
13288:Larmor formula
13285:
13280:
13275:
13270:
13265:
13260:
13255:
13250:
13245:
13240:
13238:Bremsstrahlung
13234:
13229:
13228:
13225:
13224:
13221:
13220:
13215:
13210:
13205:
13200:
13195:
13190:
13188:Magnetic field
13185:
13180:
13175:
13170:
13164:
13161:Magnetostatics
13159:
13158:
13155:
13154:
13151:
13150:
13145:
13140:
13135:
13130:
13125:
13120:
13115:
13110:
13105:
13100:
13095:
13093:Electric field
13090:
13085:
13080:
13075:
13070:
13065:
13063:Charge density
13059:
13056:Electrostatics
13054:
13053:
13050:
13049:
13048:
13047:
13042:
13037:
13032:
13027:
13022:
13017:
13009:
13008:
13000:
12999:
12993:
12992:
12991:Articles about
12985:
12982:
12980:
12979:
12974:
12969:
12964:
12959:
12954:
12949:
12944:
12938:
12934:
12933:
12928:
12923:
12920:
12915:
12901:
12898:
12881:
12861:
12856:
12852:
12846:
12843:
12839:
12835:
12832:
12826:
12823:
12816:
12812:
12808:
12802:
12776:
12773:
12770:
12766:
12745:
12740:
12736:
12730:
12727:
12723:
12719:
12716:
12711:
12707:
12701:
12697:
12691:
12688:
12685:
12681:
12677:
12674:
12668:
12665:
12658:
12654:
12650:
12644:
12618:
12614:
12610:
12605:
12601:
12597:
12592:
12589:
12585:
12581:
12576:
12572:
12568:
12543:
12539:
12535:
12531:
12525:
12522:
12518:
12514:
12511:
12506:
12503:
12500:
12497:
12493:
12469:
12465:
12461:
12441:
12438:
12434:
12428:
12424:
12420:
12417:
12412:
12408:
12387:
12382:
12378:
12372:
12369:
12365:
12361:
12358:
12353:
12349:
12343:
12339:
12333:
12330:
12327:
12324:
12320:
12314:
12311:
12306:
12303:
12297:
12294:
12287:
12283:
12279:
12273:
12264:, is given as
12251:
12248:
12244:
12221:
12218:
12214:
12193:
12173:
12157:
12154:
12137:
12134:
12129:
12124:
12121:
12118:
12108:
12093:
12090:
12085:
12081:
12077:
12074:
12071:
12067:
12061:
12057:
12053:
12050:
12047:
12044:
12040:
12019:
12016:
12013:
12008:
12004:
11978:
11975:
11969:
11966:
11942:
11938:
11903:
11880:
11876:
11871:
11865:
11861:
11857:
11852:
11846:
11842:
11838:
11834:
11812:
11808:
11803:
11797:
11793:
11789:
11784:
11778:
11774:
11770:
11735:
11731:
11708:
11690:
11687:
11675:
11670:
11664:
11660:
11656:
11653:
11650:
11647:
11644:
11640:
11611:
11607:
11602:
11598:
11594:
11590:
11586:
11581:
11577:
11574:
11568:
11564:
11557:
11552:
11528:
11525:
11522:
11518:
11515:
11512:
11509:
11506:
11503:
11500:
11468:
11464:
11459:
11455:
11451:
11447:
11443:
11438:
11434:
11431:
11428:
11422:
11418:
11411:
11406:
11361:
11356:
11350:
11345:
11340:
11336:
11332:
11327:
11322:
11317:
11313:
11308:
11304:
11301:
11298:
11294:
11288:
11284:
11278:
11274:
11270:
11265:
11261:
11255:
11251:
11247:
11242:
11238:
11233:
11229:
11226:
11223:
11219:
11215:
11210:
11206:
11202:
11199:
11194:
11190:
11186:
11183:
11180:
11177:
11172:
11168:
11164:
11161:
11158:
11153:
11149:
11145:
11142:
11139:
11135:
11130:
11125:
11121:
11115:
11111:
11107:
11103:
11100:
11097:
11091:
11087:
11079:
11075:
11070:
11040:
11036:
11032:
11027:
11023:
11019:
11014:
11010:
11006:
11003:
10998:
10994:
10990:
10987:
10982:
10978:
10974:
10970:
10965:
10960:
10956:
10950:
10944:
10940:
10935:
10931:
10928:
10922:
10918:
10910:
10906:
10901:
10871:
10867:
10861:
10857:
10851:
10847:
10843:
10838:
10834:
10828:
10824:
10820:
10815:
10811:
10805:
10801:
10797:
10792:
10788:
10782:
10778:
10773:
10769:
10766:
10761:
10758:
10754:
10748:
10744:
10740:
10737:
10731:
10727:
10719:
10715:
10710:
10683:
10680:
10668:
10654:
10648:
10645:
10641:
10635:
10628:
10624:
10616:
10609:
10605:
10599:
10593:
10590:
10586:
10582:
10567:Lorentz factor
10547:
10543:
10536:
10531:
10527:
10523:
10518:
10513:
10509:
10505:
10500:
10495:
10491:
10484:
10481:
10477:
10472:
10462:
10458:
10452:
10448:
10442:
10439:
10435:
10430:
10427:
10424:
10421:
10418:
10398:
10394:
10388:
10384:
10380:
10377:
10372:
10368:
10364:
10361:
10356:
10352:
10348:
10345:
10342:
10338:
10334:
10331:
10327:
10321:
10317:
10313:
10308:
10304:
10300:
10295:
10291:
10287:
10282:
10278:
10273:
10269:
10264:
10260:
10229:
10223:
10220:
10216:
10212:
10208:
10204:
10200:
10196:
10193:
10191:
10187:
10181:
10177:
10173:
10172:
10167:
10163:
10159:
10156:
10154:
10151:
10147:
10143:
10139:
10137:
10133:
10127:
10123:
10119:
10118:
10113:
10109:
10105:
10101:
10097:
10093:
10090:
10088:
10085:
10083:
10079:
10073:
10069:
10065:
10064:
10061:
10057:
10051:
10047:
10043:
10040:
10038:
10034:
10028:
10024:
10020:
10017:
10015:
10011:
10005:
10001:
9997:
9994:
9992:
9989:
9988:
9986:
9981:
9976:
9973:
9969:
9932:
9928:
9922:
9918:
9914:
9909:
9905:
9901:
9896:
9892:
9888:
9885:
9882:
9879:
9875:
9871:
9867:
9861:
9857:
9853:
9848:
9844:
9840:
9835:
9831:
9827:
9822:
9818:
9813:
9809:
9804:
9800:
9764:
9760:
9754:
9751:
9747:
9743:
9740:
9734:
9730:
9722:
9718:
9713:
9699:
9695:covariant form
9667:
9664:
9662:
9659:
9657:
9654:
9640:
9636:
9631:
9627:
9621:
9617:
9610:
9606:
9601:
9597:
9594:
9585:
9580:
9575:
9570:
9566:
9558:
9553:
9550:
9541:
9537:
9530:
9521:
9517:
9511:
9487:
9483:
9480:
9477:
9470:
9466:
9460:
9456:
9452:
9449:
9446:
9441:
9437:
9432:
9428:
9425:
9421:
9417:
9411:
9403:
9399:
9395:
9390:
9387:
9381:
9377:
9370:
9365:
9341:
9337:
9334:
9331:
9324:
9320:
9314:
9310:
9306:
9303:
9300:
9295:
9291:
9286:
9282:
9279:
9275:
9271:
9265:
9257:
9253:
9249:
9244:
9238:
9234:
9227:
9222:
9189:
9184:
9179:
9174:
9170:
9162:
9157:
9154:
9146:
9142:
9135:
9129:
9125:
9121:
9118:
9114:
9090:
9086:
9081:
9078:
9072:
9069:
9063:
9059:
9052:
9045:
9041:
9035:
9031:
9025:
9022:
9015:
9011:
9006:
9003:
8997:
8991:
8987:
8980:
8975:
8949:
8918:
8914:
8910:
8907:
8904:
8901:
8895:
8891:
8884:
8881:
8877:
8873:
8869:
8865:
8862:
8855:
8850:
8845:
8840:
8836:
8828:
8823:
8820:
8813:
8809:
8805:
8802:
8799:
8796:
8770:
8766:
8762:
8756:
8753:
8747:
8743:
8739:
8736:
8733:
8729:
8690:
8686:
8681:
8677:
8671:
8668:
8662:
8659:
8656:
8651:
8647:
8643:
8640:
8637:
8635:
8633:
8628:
8624:
8620:
8616:
8612:
8609:
8606:
8603:
8597:
8594:
8588:
8585:
8582:
8577:
8573:
8569:
8566:
8563:
8561:
8559:
8556:
8551:
8547:
8542:
8538:
8535:
8532:
8526:
8523:
8517:
8512:
8508:
8503:
8499:
8496:
8493:
8487:
8484:
8478:
8475:
8472:
8467:
8463:
8459:
8456:
8453:
8451:
8449:
8445:
8440:
8433:
8430:
8423:
8419:
8415:
8409:
8403:
8400:
8393:
8389:
8385:
8378:
8371:
8368:
8362:
8358:
8351:
8348:
8341:
8337:
8333:
8327:
8321:
8318:
8311:
8307:
8303:
8296:
8289:
8286:
8279:
8275:
8272:
8268:
8261:
8258:
8251:
8247:
8243:
8237:
8231:
8228:
8223:
8220:
8213:
8209:
8206:
8203:
8200:
8198:
8194:
8190:
8186:
8185:
8165:
8158:
8155:
8146:
8143:
8136:
8132:
8128:
8122:
8116:
8113:
8104:
8101:
8094:
8090:
8086:
8080:
8074:
8071:
8062:
8059:
8052:
8048:
8044:
8037:
8033:
8030:
8024:
8021:
8016:
8013:
8007:
8004:
8001:
7997:
7990:
7987:
7978:
7975:
7968:
7964:
7960:
7954:
7948:
7945:
7936:
7933:
7926:
7922:
7918:
7912:
7906:
7903:
7894:
7891:
7884:
7880:
7876:
7870:
7864:
7861:
7854:
7850:
7846:
7839:
7835:
7832:
7826:
7823:
7817:
7796:
7789:
7786:
7777:
7774:
7767:
7763:
7759:
7753:
7747:
7744:
7735:
7732:
7725:
7721:
7717:
7711:
7705:
7702:
7693:
7690:
7683:
7679:
7675:
7668:
7664:
7661:
7655:
7652:
7647:
7644:
7638:
7635:
7632:
7626:
7623:
7618:
7615:
7588:
7581:
7578:
7569:
7566:
7559:
7555:
7551:
7545:
7539:
7536:
7527:
7524:
7517:
7513:
7509:
7503:
7497:
7494:
7485:
7482:
7475:
7471:
7467:
7461:
7455:
7452:
7445:
7441:
7437:
7430:
7426:
7423:
7417:
7414:
7408:
7405:
7402:
7400:
7398:
7394:
7390:
7387:
7381:
7378:
7371:
7367:
7363:
7357:
7354:
7351:
7345:
7342:
7335:
7331:
7327:
7321:
7318:
7315:
7309:
7306:
7299:
7295:
7291:
7285:
7282:
7279:
7273:
7270:
7263:
7259:
7255:
7248:
7241:
7237:
7232:
7227:
7221:
7218:
7212:
7209:
7206:
7204:
7199:
7195:
7187:
7183:
7178:
7171:
7168:
7162:
7159:
7153:
7150:
7141:
7138:
7132:
7127:
7124:
7115:
7111:
7105:
7099:
7098:
7063:
7060:
7055:
7052:
7046:
7037:
7034:
7028:
7023:
7020:
7011:
7007:
7001:
6976:
6973:
6970:
6966:
6960:
6956:
6949:
6946:
6940:
6935:
6931:
6924:
6921:
6915:
6910:
6906:
6899:
6896:
6889:
6885:
6882:
6878:
6872:
6865:
6862:
6855:
6850:
6843:
6840:
6833:
6828:
6821:
6818:
6810:
6804:
6801:
6796:
6793:
6774:
6771:
6768:
6762:
6759:
6753:
6749:
6745:
6742:
6736:
6733:
6727:
6721:
6718:
6710:
6707:
6702:
6699:
6696:
6693:
6690:
6687:
6664:
6661:
6655:
6649:
6646:
6638:
6635:
6630:
6627:
6616:kinetic energy
6600:
6597:
6591:
6587:
6583:
6580:
6577:
6574:
6571:
6568:
6541:
6538:
6535:
6531:
6527:
6524:
6521:
6494:
6491:
6485:
6482:
6479:
6476:
6472:
6468:
6464:
6460:
6440:
6437:
6434:
6430:
6426:
6422:
6418:
6385:
6368:
6362:
6359:
6353:
6349:
6345:
6342:
6339:
6336:
6333:
6330:
6298:
6295:
6292:
6286:
6283:
6277:
6273:
6269:
6266:
6260:
6257:
6251:
6245:
6242:
6234:
6231:
6226:
6223:
6185:
6182:
6170:
6160:
6157:
6151:
6147:
6138:
6135:
6129:
6119:
6116:
6110:
6106:
6097:
6094:
6088:
6078:
6075:
6069:
6065:
6056:
6053:
6047:
6040:
6036:
6029:
6004:
6001:
5997:
5988:
5985:
5979:
5972:
5969:
5965:
5956:
5953:
5947:
5940:
5937:
5933:
5924:
5921:
5915:
5909:
5904:
5879:
5875:
5871:
5867:
5861:
5857:
5850:
5847:
5844:
5837:
5833:
5826:
5819:
5815:
5809:
5803:
5800:
5796:
5792:
5786:
5782:
5775:
5772:
5769:
5763:
5758:
5754:
5750:
5746:
5743:
5739:
5728:
5703:
5699:
5692:
5688:
5681:
5676:
5669:
5666:
5662:
5658:
5654:
5650:
5647:
5644:
5641:
5638:
5634:
5630:
5627:
5623:
5601:
5597:
5594:
5591:
5587:
5583:
5580:
5574:
5571:
5565:
5561:
5555:
5549:
5545:
5538:
5533:
5508:
5476:
5451:
5427:
5423:
5418:
5413:
5409:
5406:
5402:
5397:
5393:
5389:
5384:
5380:
5376:
5371:
5367:
5364:
5358:
5355:
5349:
5345:
5339:
5336:
5333:
5330:
5326:
5322:
5319:
5315:
5288:
5284:
5280:
5276:
5272:
5269:
5266:
5263:
5259:
5255:
5249:
5246:
5240:
5236:
5230:
5227:
5224:
5221:
5217:
5213:
5210:
5206:
5161:
5157:
5154:
5151:
5148:
5146:
5143:
5139:
5138:
5132:
5129:
5123:
5119:
5113:
5110:
5107:
5104:
5101:
5098:
5096:
5093:
5089:
5088:
5028:
5025:
4990:
4967:
4931:
4910:
4886:
4883:
4880:
4876:
4872:
4868:
4864:
4860:
4856:
4851:
4848:
4845:
4842:
4838:
4834:
4830:
4826:
4821:
4818:
4814:
4792:
4787:
4782:
4779:
4773:
4769:
4765:
4761:
4757:
4753:
4743:
4740:
4737:
4734:
4731:
4727:
4723:
4720:
4717:
4711:
4707:
4703:
4699:
4695:
4691:
4686:
4680:
4677:
4674:
4671:
4668:
4664:
4660:
4657:
4654:
4648:
4644:
4640:
4637:
4633:
4628:
4624:
4620:
4615:
4609:
4606:
4603:
4600:
4597:
4593:
4571:
4566:
4560:
4556:
4552:
4542:
4539:
4536:
4533:
4530:
4526:
4522:
4519:
4516:
4513:
4509:
4505:
4501:
4494:
4491:
4487:
4482:
4478:
4473:
4467:
4464:
4461:
4458:
4454:
4450:
4447:
4444:
4441:
4438:
4434:
4430:
4427:
4423:
4418:
4414:
4410:
4405:
4399:
4396:
4393:
4390:
4387:
4383:
4373:, results in,
4348:
4345:
4342:
4336:
4332:
4328:
4324:
4320:
4316:
4311:
4305:
4302:
4299:
4296:
4292:
4285:
4281:
4275:
4269:
4266:
4263:
4260:
4254:
4250:
4246:
4243:
4239:
4234:
4230:
4226:
4221:
4215:
4212:
4209:
4206:
4203:
4199:
4175:
4171:
4165:
4162:
4158:
4154:
4150:
4146:
4141:
4134:
4130:
4125:
4119:
4116:
4113:
4110:
4106:
4099:
4096:
4093:
4090:
4084:
4080:
4076:
4072:
4068:
4064:
4059:
4053:
4050:
4047:
4044:
4041:
4037:
4003:
3996:
3993:
3987:
3983:
3977:
3974:
3970:
3966:
3963:
3890:
3886:
3881:
3877:
3873:
3852:
3848:
3843:
3839:
3835:
3830:
3822:
3819:
3816:
3813:
3810:
3806:
3802:
3797:
3651:
3648:
3645:
3641:
3637:
3633:
3629:
3625:
3620:
3614:
3611:
3608:
3605:
3601:
3597:
3592:
3588:
3564:
3560:
3552:
3548:
3543:
3536:
3533:
3528:
3511:magnetic field
3503:Main article:
3492:
3489:
3466:.) Einstein's
3402:
3399:
3380:
3376:
3372:
3368:
3363:
3359:
3356:
3353:
3349:
3306:
3301:
3260:
3256:
3252:
3248:
3244:
3241:
3237:
3208:
3205:
3181:fluid dynamics
3126:
3123:
3115:guiding center
3091:magnetic field
3074:Guiding center
3072:Main article:
3069:
3066:
3031:
3027:
3023:
3019:
3013:
3010:
3005:
3001:
2972:lines of force
2920:magnetic field
2899:
2896:
2886:
2873:
2857:
2849:
2845:
2839:
2835:
2830:
2825:
2822:
2818:
2811:
2808:
2802:
2791:
2787:
2782:
2778:
2775:
2770:
2764:
2758:
2752:
2746:
2743:
2737:
2727:
2723:
2719:
2716:
2711:
2705:
2699:
2693:
2685:
2681:
2677:
2674:
2667:
2664:
2659:
2653:
2647:
2642:
2630:speed of light
2613:
2609:
2602:
2596:
2591:
2586:
2582:
2576:
2570:
2564:
2558:
2551:
2546:
2542:
2538:
2517:
2514:
2502:
2498:
2494:
2490:
2483:
2480:
2474:
2470:
2464:
2460:
2456:
2453:
2450:
2445:
2440:
2434:
2408:
2385:
2380:
2353:
2330:
2326:
2303:
2299:
2295:
2291:
2284:
2281:
2275:
2271:
2265:
2261:
2257:
2254:
2251:
2246:
2241:
2235:
2231:
2227:
2222:
2217:
2213:
2210:
2207:
2202:
2198:
2193:
2189:
2185:
2162:
2158:
2154:
2150:
2115:speed of light
2102:
2078:
2075:
2069:
2065:
2053:
2049:
2045:
2039:
2035:
2031:
2028:
2025:
2021:
1999:
1977:
1953:to obtain the
1941:
1916:
1883:
1862:
1840:
1837:
1833:
1828:
1823:
1819:
1815:
1811:
1807:
1803:
1799:
1795:
1792:
1788:
1759:
1755:
1751:
1747:
1743:
1739:
1736:
1732:
1721:
1707:
1703:
1700:
1696:
1680:charge density
1667:
1642:
1620:
1615:
1611:
1607:
1603:
1599:
1594:
1590:
1587:
1583:
1562:
1558:
1537:
1533:
1511:
1506:
1484:
1479:
1475:
1471:
1467:
1463:
1458:
1454:
1450:
1446:
1442:
1437:
1412:volume element
1388:charge density
1369:
1366:
1353:
1349:
1345:
1341:
1336:
1333:
1329:
1325:
1321:
1290:magnetic force
1288:is called the
1280:
1269:electric force
1267:is called the
1186:
1182:
1179:
1176:
1172:
1168:
1164:
1160:
1157:
1154:
1151:
1145:
1141:
1134:
1131:
1128:
1125:
1121:
1117:
1113:
1108:
1104:
1101:
1097:
1093:
1090:
1087:
1084:
1081:
1078:
1075:
1069:
1065:
1058:
1055:
1052:
1049:
1045:
1040:
1035:
1008:
1004:
998:
994:
988:
984:
980:
975:
971:
965:
961:
957:
952:
948:
943:
939:
936:
933:
931:
927:
923:
919:
918:
915:
911:
905:
901:
895:
891:
887:
882:
878:
872:
868:
864:
859:
855:
850:
846:
843:
840:
838:
834:
830:
826:
825:
822:
818:
812:
808:
802:
798:
794:
789:
785:
779:
775:
771:
766:
762:
757:
753:
750:
747:
745:
741:
737:
733:
732:
719:
702:
697:
693:
689:
685:
681:
676:
672:
669:
665:
654:
575:Lorentz force
568:
565:
563:
560:
502:
499:
487:Bremsstrahlung
441:). So the two
404:
400:
396:
392:
388:
384:
380:
377:
374:
370:
283:
282:
273:
266:
265:
253:
246:
245:
244:
243:
242:
240:
237:
172:
168:
163:
159:
155:
151:
147:
142:
138:
135:
131:
110:magnetic field
101:electric field
80:magnetic force
40:bubble chamber
26:
18:Magnetic force
9:
6:
4:
3:
2:
15171:
15160:
15157:
15155:
15152:
15150:
15147:
15145:
15142:
15141:
15139:
15129:
15125:
15122:
15119:
15117:
15114:
15113:
15102:
15096:
15092:
15091:
15085:
15081:
15079:0-534-40846-X
15075:
15071:
15066:
15062:
15060:0-471-30932-X
15056:
15052:
15047:
15043:
15041:0-13-805326-X
15037:
15033:
15032:
15026:
15021:
15019:0-8053-9047-2
15015:
15011:
15007:
15003:
15002:
15000:
14986:
14982:
14976:
14967:
14959:
14953:
14945:
14941:
14937:
14935:0-486-65067-7
14931:
14927:
14923:
14917:
14909:
14907:0-07-084018-0
14903:
14899:
14892:
14884:
14878:
14874:
14873:
14865:
14857:
14855:0-486-43241-6
14851:
14847:
14846:
14841:
14835:
14827:
14825:0-7506-2634-8
14821:
14817:
14816:
14813:
14805:
14803:
14794:
14792:0-7637-3827-1
14788:
14784:
14783:
14775:
14766:
14764:
14749:on 2018-07-08
14748:
14744:
14740:
14734:
14725:
14717:
14715:1-143-01208-9
14711:
14707:
14706:
14701:
14695:
14687:
14685:0-19-850593-0
14681:
14677:
14672:
14671:
14662:
14655:
14649:
14641:
14637:
14633:
14626:
14618:
14616:0-19-850593-0
14612:
14608:
14603:
14602:
14593:
14585:
14581:
14577:
14573:
14569:
14565:
14561:
14554:
14546:
14544:0-19-850593-0
14540:
14536:
14531:
14530:
14521:
14513:
14511:0-19-506488-7
14507:
14503:
14499:
14498:
14490:
14482:
14480:0-19-850593-0
14476:
14472:
14467:
14466:
14457:
14449:
14447:0-19-506488-7
14443:
14439:
14435:
14434:
14426:
14418:
14416:0-262-13070-X
14412:
14408:
14407:
14399:
14391:
14389:9780521045469
14385:
14381:
14374:
14366:
14360:
14356:
14349:
14341:
14335:
14331:
14330:
14322:
14320:
14318:
14311:or Griffiths.
14310:
14304:
14295:
14288:
14282:
14274:
14270:
14266:
14259:
14257:
14248:
14241:
14227:
14223:
14217:
14209:
14203:
14199:
14192:
14184:
14178:
14174:
14167:
14159:
14153:
14147:
14143:
14137:
14135:0-7167-0344-0
14131:
14127:
14122:
14121:
14112:
14103:
14096:
14095:
14088:
14086:
14084:
14075:
14068:
14066:
14057:
14051:
14047:
14046:
14038:
14036:
14027:
14021:
14017:
14016:
14008:
13992:
13988:
13983:
13979:
13974:
13967:
13950:
13945:
13940:
13935:
13931:
13926:
13922:In SI units,
13919:
13917:
13915:
13910:
13900:
13897:
13895:
13894:Coulomb's law
13892:
13890:
13887:
13885:
13882:
13880:
13877:
13875:
13872:
13870:
13867:
13865:
13862:
13860:
13857:
13855:
13852:
13850:
13847:
13845:
13842:
13840:
13837:
13835:
13832:
13830:
13827:
13825:
13822:
13820:
13817:
13815:
13812:
13811:
13802:
13797:
13795:
13790:
13788:
13783:
13782:
13780:
13779:
13772:
13769:
13767:
13764:
13762:
13759:
13757:
13754:
13752:
13749:
13747:
13744:
13742:
13739:
13737:
13734:
13732:
13729:
13727:
13724:
13722:
13719:
13717:
13714:
13712:
13709:
13707:
13704:
13702:
13699:
13697:
13694:
13692:
13689:
13687:
13684:
13682:
13679:
13677:
13674:
13672:
13669:
13667:
13664:
13662:
13659:
13657:
13654:
13652:
13649:
13647:
13644:
13642:
13639:
13637:
13634:
13632:
13629:
13627:
13624:
13622:
13619:
13617:
13614:
13612:
13609:
13607:
13604:
13602:
13599:
13597:
13594:
13593:
13587:
13586:
13579:
13576:
13574:
13571:
13569:
13566:
13564:
13561:
13559:
13556:
13554:
13551:
13549:
13546:
13544:
13541:
13540:
13537:
13532:
13531:
13524:
13521:
13519:
13516:
13514:
13511:
13509:
13506:
13504:
13501:
13499:
13496:
13494:
13491:
13489:
13486:
13484:
13481:
13479:
13476:
13474:
13471:
13469:
13466:
13464:
13461:
13459:
13456:
13455:
13452:
13447:
13446:
13439:
13436:
13434:
13431:
13429:
13426:
13424:
13421:
13419:
13416:
13414:
13411:
13409:
13406:
13404:
13401:
13399:
13396:
13394:
13393:Joule heating
13391:
13389:
13386:
13384:
13381:
13379:
13376:
13374:
13371:
13369:
13366:
13364:
13361:
13359:
13356:
13354:
13351:
13349:
13346:
13345:
13342:
13337:
13336:
13329:
13326:
13324:
13321:
13319:
13316:
13314:
13311:
13309:
13308:Lorentz force
13306:
13304:
13301:
13299:
13296:
13294:
13291:
13289:
13286:
13284:
13281:
13279:
13276:
13274:
13271:
13269:
13266:
13264:
13261:
13259:
13256:
13254:
13251:
13249:
13246:
13244:
13241:
13239:
13236:
13235:
13232:
13227:
13226:
13219:
13216:
13214:
13211:
13209:
13208:Magnetization
13206:
13204:
13201:
13199:
13196:
13194:
13193:Magnetic flux
13191:
13189:
13186:
13184:
13181:
13179:
13176:
13174:
13171:
13169:
13166:
13165:
13162:
13157:
13156:
13149:
13146:
13144:
13141:
13139:
13136:
13134:
13131:
13129:
13126:
13124:
13121:
13119:
13116:
13114:
13111:
13109:
13106:
13104:
13101:
13099:
13098:Electric flux
13096:
13094:
13091:
13089:
13086:
13084:
13081:
13079:
13076:
13074:
13071:
13069:
13066:
13064:
13061:
13060:
13057:
13052:
13051:
13046:
13043:
13041:
13038:
13036:
13035:Computational
13033:
13031:
13028:
13026:
13023:
13021:
13018:
13016:
13013:
13012:
13011:
13010:
13006:
13002:
13001:
12998:
12995:
12994:
12990:
12989:
12978:
12975:
12973:
12970:
12968:
12965:
12963:
12960:
12958:
12955:
12953:
12952:Linear motors
12950:
12948:
12945:
12943:
12940:
12939:
12937:
12932:
12929:
12927:
12924:
12921:
12919:
12916:
12914:
12910:
12907:
12906:
12905:
12897:
12895:
12879:
12859:
12854:
12850:
12844:
12841:
12837:
12833:
12830:
12824:
12821:
12814:
12810:
12806:
12800:
12792:
12774:
12771:
12768:
12743:
12738:
12734:
12728:
12725:
12721:
12717:
12714:
12709:
12705:
12699:
12695:
12689:
12686:
12683:
12675:
12672:
12666:
12663:
12656:
12652:
12648:
12642:
12633:
12616:
12612:
12608:
12603:
12599:
12595:
12590:
12587:
12583:
12579:
12574:
12570:
12566:
12541:
12537:
12529:
12523:
12520:
12516:
12509:
12504:
12501:
12498:
12495:
12491:
12467:
12463:
12459:
12439:
12436:
12432:
12426:
12422:
12418:
12415:
12410:
12406:
12385:
12380:
12376:
12370:
12367:
12363:
12359:
12356:
12351:
12347:
12341:
12337:
12331:
12328:
12325:
12322:
12318:
12312:
12309:
12304:
12301:
12295:
12292:
12285:
12281:
12277:
12271:
12249:
12246:
12242:
12219:
12216:
12212:
12191:
12171:
12163:
12153:
12148:
12135:
12132:
12122:
12119:
12116:
12107:
12104:
12091:
12083:
12079:
12075:
12072:
12065:
12059:
12055:
12051:
12048:
12045:
12042:
12017:
12014:
12011:
12006:
12002:
11976:
11973:
11967:
11964:
11940:
11936:
11927:
11923:
11919:
11878:
11874:
11869:
11863:
11859:
11855:
11844:
11840:
11832:
11810:
11806:
11801:
11795:
11791:
11787:
11776:
11772:
11759:
11756:defined on a
11755:
11751:
11733:
11729:
11696:
11686:
11673:
11662:
11658:
11651:
11645:
11642:
11628:
11622:
11609:
11605:
11596:
11588:
11579:
11575:
11572:
11566:
11526:
11523:
11520:
11513:
11507:
11504:
11501:
11498:
11489:
11483:
11466:
11462:
11453:
11445:
11436:
11432:
11429:
11426:
11420:
11391:
11385:
11378:
11372:
11359:
11354:
11348:
11343:
11334:
11325:
11320:
11315:
11311:
11306:
11302:
11299:
11296:
11292:
11286:
11282:
11276:
11272:
11268:
11263:
11259:
11253:
11249:
11245:
11240:
11236:
11231:
11227:
11224:
11221:
11217:
11208:
11204:
11192:
11188:
11184:
11178:
11170:
11166:
11162:
11151:
11147:
11143:
11137:
11133:
11128:
11123:
11119:
11113:
11109:
11105:
11101:
11098:
11095:
11089:
11077:
11073:
11056:
11055:four-velocity
11051:
11038:
11034:
11025:
11021:
11012:
11008:
11004:
10996:
10992:
10988:
10980:
10976:
10972:
10968:
10963:
10958:
10954:
10948:
10942:
10938:
10933:
10929:
10926:
10920:
10908:
10904:
10887:
10882:
10869:
10865:
10859:
10855:
10849:
10845:
10841:
10836:
10832:
10826:
10822:
10818:
10813:
10809:
10803:
10799:
10795:
10790:
10786:
10780:
10776:
10771:
10767:
10764:
10759:
10756:
10752:
10746:
10742:
10738:
10735:
10729:
10717:
10713:
10696:
10690:
10679:
10677:
10671:
10652:
10646:
10643:
10639:
10633:
10626:
10614:
10607:
10597:
10591:
10588:
10584:
10580:
10570:
10568:
10545:
10541:
10534:
10529:
10525:
10521:
10516:
10511:
10507:
10503:
10498:
10493:
10489:
10482:
10479:
10475:
10470:
10460:
10456:
10450:
10446:
10440:
10437:
10433:
10428:
10422:
10416:
10396:
10392:
10386:
10382:
10378:
10375:
10370:
10366:
10362:
10359:
10354:
10350:
10346:
10343:
10340:
10336:
10332:
10329:
10325:
10319:
10315:
10311:
10306:
10302:
10298:
10293:
10289:
10285:
10280:
10276:
10271:
10267:
10262:
10258:
10249:
10227:
10221:
10214:
10210:
10202:
10198:
10194:
10189:
10185:
10179:
10175:
10165:
10161:
10157:
10152:
10145:
10141:
10135:
10131:
10125:
10121:
10111:
10107:
10099:
10095:
10091:
10086:
10081:
10077:
10071:
10067:
10059:
10055:
10049:
10045:
10041:
10036:
10032:
10026:
10022:
10018:
10013:
10009:
10003:
9999:
9995:
9990:
9984:
9979:
9974:
9971:
9967:
9959:
9954:
9949:
9930:
9926:
9920:
9916:
9912:
9907:
9903:
9899:
9894:
9890:
9886:
9883:
9880:
9877:
9873:
9869:
9865:
9859:
9855:
9851:
9846:
9842:
9838:
9833:
9829:
9825:
9820:
9816:
9811:
9807:
9802:
9798:
9790:, defined as
9789:
9788:four-momentum
9784:
9777:
9762:
9758:
9752:
9749:
9745:
9741:
9738:
9732:
9720:
9716:
9698:
9696:
9691:
9683:
9677:
9673:
9652:
9638:
9634:
9625:
9619:
9608:
9599:
9595:
9592:
9583:
9578:
9573:
9568:
9556:
9551:
9548:
9539:
9528:
9519:
9485:
9481:
9478:
9475:
9468:
9458:
9454:
9450:
9444:
9439:
9426:
9423:
9409:
9388:
9385:
9379:
9339:
9335:
9332:
9329:
9322:
9312:
9308:
9304:
9298:
9293:
9280:
9277:
9263:
9242:
9236:
9208:
9187:
9182:
9177:
9172:
9160:
9155:
9152:
9144:
9133:
9127:
9119:
9116:
9079:
9070:
9067:
9061:
9050:
9023:
9020:
9004:
8995:
8989:
8961:
8957:
8948:
8946:
8942:
8938:
8934:
8929:
8905:
8902:
8899:
8893:
8882:
8863:
8860:
8853:
8848:
8843:
8838:
8826:
8821:
8818:
8811:
8807:
8803:
8800:
8797:
8794:
8785:
8781:
8760:
8745:
8734:
8731:
8717:
8711:
8688:
8675:
8657:
8654:
8649:
8645:
8641:
8638:
8636:
8626:
8610:
8601:
8583:
8580:
8575:
8571:
8567:
8564:
8562:
8549:
8536:
8524:
8521:
8515:
8510:
8497:
8485:
8482:
8473:
8470:
8465:
8461:
8457:
8454:
8452:
8443:
8438:
8431:
8421:
8417:
8407:
8401:
8391:
8387:
8376:
8369:
8366:
8360:
8356:
8349:
8339:
8335:
8325:
8319:
8309:
8305:
8294:
8287:
8284:
8277:
8273:
8270:
8266:
8259:
8249:
8245:
8235:
8229:
8221:
8211:
8207:
8204:
8201:
8199:
8192:
8188:
8163:
8156:
8153:
8144:
8134:
8130:
8120:
8114:
8111:
8102:
8092:
8088:
8078:
8072:
8069:
8060:
8050:
8046:
8035:
8031:
8028:
8022:
8014:
8005:
8002:
7999:
7995:
7988:
7985:
7976:
7966:
7962:
7952:
7946:
7943:
7934:
7924:
7920:
7910:
7904:
7901:
7892:
7882:
7878:
7868:
7862:
7852:
7848:
7837:
7833:
7830:
7824:
7821:
7815:
7794:
7787:
7784:
7775:
7765:
7761:
7751:
7745:
7742:
7733:
7723:
7719:
7709:
7703:
7700:
7691:
7681:
7677:
7666:
7662:
7659:
7653:
7645:
7636:
7633:
7630:
7624:
7616:
7586:
7579:
7576:
7567:
7557:
7553:
7543:
7537:
7534:
7525:
7515:
7511:
7501:
7495:
7492:
7483:
7473:
7469:
7459:
7453:
7443:
7439:
7428:
7424:
7421:
7415:
7412:
7406:
7403:
7401:
7392:
7388:
7385:
7379:
7369:
7365:
7355:
7352:
7349:
7343:
7333:
7329:
7319:
7316:
7313:
7307:
7297:
7293:
7283:
7280:
7277:
7271:
7261:
7257:
7246:
7239:
7230:
7225:
7219:
7216:
7210:
7207:
7205:
7197:
7185:
7181:
7169:
7166:
7160:
7157:
7151:
7148:
7139:
7136:
7125:
7113:
7087:
7081:
7061:
7053:
7044:
7035:
7032:
7021:
7009:
6987:
6974:
6971:
6968:
6964:
6958:
6954:
6947:
6944:
6938:
6933:
6929:
6922:
6919:
6913:
6908:
6904:
6897:
6894:
6887:
6883:
6880:
6876:
6870:
6863:
6860:
6853:
6848:
6841:
6838:
6831:
6826:
6819:
6816:
6808:
6802:
6799:
6794:
6791:
6772:
6769:
6766:
6751:
6743:
6740:
6725:
6708:
6705:
6700:
6697:
6694:
6691:
6688:
6685:
6653:
6636:
6633:
6628:
6625:
6617:
6589:
6581:
6578:
6575:
6572:
6569:
6566:
6558:
6553:
6536:
6533:
6522:
6519:
6511:
6483:
6477:
6474:
6458:
6435:
6432:
6416:
6409:
6404:
6400:
6394:
6384:
6382:
6351:
6343:
6340:
6334:
6331:
6328:
6319:
6313:
6296:
6293:
6290:
6275:
6267:
6264:
6249:
6232:
6229:
6224:
6221:
6213:
6208:
6202:
6197:
6191:
6181:
6168:
6158:
6155:
6133:
6127:
6117:
6114:
6092:
6086:
6076:
6073:
6051:
6045:
6038:
6002:
5983:
5977:
5970:
5951:
5945:
5938:
5919:
5913:
5891:
5877:
5865:
5859:
5848:
5845:
5835:
5817:
5801:
5790:
5784:
5773:
5770:
5752:
5748:
5744:
5741:
5727:
5724:
5720:
5714:
5701:
5697:
5690:
5667:
5656:
5648:
5645:
5636:
5632:
5628:
5625:
5589:
5578:
5572:
5553:
5547:
5497:
5493:
5438:
5425:
5421:
5411:
5404:
5395:
5391:
5387:
5378:
5369:
5362:
5356:
5337:
5334:
5328:
5324:
5320:
5317:
5304:
5299:
5286:
5282:
5270:
5261:
5253:
5247:
5228:
5225:
5219:
5215:
5211:
5208:
5194:
5192:
5155:
5149:
5147:
5130:
5111:
5108:
5102:
5099:
5097:
5077:
5073:
5069:
5064:
5060:
5055:
5049:
5042:
5038:
5034:
5024:
5023:is not zero.
5022:
5018:
5014:
5009:
5003:
4997:
4993:
4984:
4977:
4970:
4962:
4955:
4949:
4945:
4940:
4934:
4926:
4920:
4913:
4905:
4903:
4897:
4884:
4878:
4874:
4858:
4849:
4846:
4840:
4836:
4819:
4816:
4777:
4771:
4755:
4738:
4725:
4721:
4715:
4709:
4693:
4675:
4662:
4658:
4652:
4646:
4635:
4631:
4622:
4604:
4591:
4554:
4537:
4524:
4520:
4514:
4511:
4492:
4480:
4462:
4452:
4448:
4445:
4439:
4436:
4425:
4421:
4412:
4394:
4381:
4370:
4364:
4359:
4346:
4340:
4334:
4318:
4300:
4290:
4283:
4267:
4264:
4258:
4252:
4241:
4237:
4228:
4210:
4197:
4173:
4160:
4156:
4132:
4114:
4104:
4097:
4094:
4088:
4082:
4066:
4048:
4035:
4025:
4023:
4019:
4018:integral form
4014:
4001:
3994:
3975:
3972:
3964:
3953:
3948:
3946:
3942:
3937:
3930:
3929:
3921:
3917:
3911:
3910:infinitesimal
3906:
3888:
3884:
3875:
3850:
3846:
3837:
3817:
3804:
3800:
3786:is given by:
3783:
3775:
3768:
3761:
3754:
3752:
3748:
3744:
3740:
3735:
3733:
3729:
3725:
3721:
3716:
3714:
3708:
3702:
3697:
3685:
3677:
3670:
3665:
3664:magnetic flux
3646:
3643:
3627:
3609:
3599:
3595:
3590:
3562:
3550:
3534:
3531:
3516:
3512:
3506:
3497:
3488:
3486:
3482:
3478:
3474:
3469:
3465:
3461:
3460:magnetic flux
3456:
3454:
3450:
3446:
3442:
3439:
3434:
3433:of the wire.
3432:
3428:
3424:
3421:
3416:
3412:
3409:
3398:
3396:
3391:
3378:
3370:
3357:
3354:
3351:
3333:
3331:
3326:
3321:
3290:
3289:infinitesimal
3285:
3280:
3275:
3258:
3250:
3242:
3239:
3226:
3225:Laplace force
3218:
3213:
3204:
3202:
3198:
3194:
3190:
3186:
3182:
3178:
3174:
3170:
3166:
3162:
3158:
3152:
3150:
3146:
3141:
3137:
3133:
3122:
3120:
3116:
3112:
3111:superposition
3108:
3104:
3100:
3096:
3092:
3084:
3080:
3075:
3065:
3063:
3059:
3054:
3049:
3045:
3029:
3021:
3011:
3008:
3003:
2990:
2985:
2984:J. J. Thomson
2981:
2977:
2973:
2969:
2964:
2961:
2957:
2953:
2949:
2945:
2941:
2937:
2929:
2925:
2921:
2917:
2913:
2909:
2904:
2895:
2893:
2885:
2880:
2872:
2855:
2847:
2843:
2837:
2833:
2828:
2823:
2820:
2816:
2789:
2785:
2780:
2776:
2773:
2768:
2750:
2725:
2721:
2717:
2714:
2709:
2691:
2683:
2679:
2675:
2672:
2657:
2651:
2640:
2631:
2627:
2611:
2607:
2589:
2584:
2574:
2556:
2544:
2540:
2527:
2523:
2513:
2500:
2492:
2488:
2481:
2462:
2454:
2448:
2443:
2432:
2423:
2422:magnetization
2383:
2368:
2328:
2324:
2314:
2301:
2293:
2289:
2282:
2263:
2255:
2249:
2244:
2233:
2229:
2220:
2211:
2205:
2200:
2196:
2191:
2187:
2173:
2160:
2152:
2138:
2136:
2132:
2128:
2124:
2120:
2116:
2100:
2076:
2051:
2047:
2043:
2037:
2029:
2023:
1965:
1963:
1959:
1956:
1931:
1906:
1902:
1898:
1860:
1851:
1838:
1835:
1826:
1817:
1809:
1801:
1797:
1793:
1790:
1777:
1771:
1753:
1745:
1737:
1734:
1720:
1701:
1698:
1685:
1681:
1665:
1657:
1656:force density
1618:
1609:
1601:
1592:
1588:
1585:
1560:
1535:
1482:
1473:
1465:
1456:
1452:
1444:
1426:
1417:
1413:
1408:
1402:
1398:
1393:
1389:
1385:
1380:
1374:
1365:
1351:
1343:
1334:
1331:
1323:
1308:
1306:
1305:Laplace force
1301:
1299:
1295:
1291:
1285:
1279:
1275:
1270:
1265:
1262:
1256:
1253:
1247:
1241:
1236:
1231:
1225:
1219:
1214:
1209:
1202:
1184:
1177:
1174:
1158:
1152:
1143:
1132:
1126:
1123:
1106:
1102:
1099:
1095:
1091:
1088:
1085:
1082:
1076:
1067:
1056:
1050:
1038:
1023:
1006:
1002:
996:
992:
986:
982:
978:
973:
969:
963:
959:
955:
950:
946:
941:
937:
934:
932:
925:
921:
913:
909:
903:
899:
893:
889:
885:
880:
876:
870:
866:
862:
857:
853:
848:
844:
841:
839:
832:
828:
820:
816:
810:
806:
800:
796:
792:
787:
783:
777:
773:
769:
764:
760:
755:
751:
748:
746:
739:
735:
714:
700:
691:
683:
674:
670:
667:
653:
651:
646:
640:
634:
626:
621:
612:
609:
603:
600:
593:
584:
579:
573:
559:
557:
553:
549:
545:
541:
537:
536:central force
532:
530:
526:
522:
517:
515:
511:
507:
506:Coulomb's law
498:
496:
492:
488:
482:
476:
470:
464:
458:
454:
448:
444:
443:vector fields
439:
434:
426:
421:
415:
394:
386:
375:
372:
359:
355:
351:
347:
343:
339:
335:
334:and velocity
331:
327:
325:
319:
315:
310:
307:
301:
296:
286:
277:
270:
261:
257:
250:
236:
234:
230:
226:
221:
219:
215:
211:
210:Laplace force
206:
203:
197:
191:
170:
166:
157:
149:
140:
136:
133:
120:
115:
111:
106:
102:
97:
88:
83:
81:
77:
73:
72:Lorentz force
69:
65:
61:
57:
53:
49:
41:
37:
32:
19:
15089:
15069:
15050:
15030:
15009:
14998:
14975:
14966:
14925:
14916:
14897:
14891:
14871:
14864:
14844:
14834:
14815:
14811:
14781:
14774:
14751:. Retrieved
14747:the original
14742:
14733:
14724:
14703:
14694:
14669:
14661:
14653:
14648:
14639:
14635:
14625:
14600:
14592:
14567:
14563:
14553:
14528:
14520:
14496:
14489:
14464:
14456:
14432:
14425:
14405:
14398:
14379:
14373:
14354:
14348:
14328:
14303:
14294:
14286:
14276:
14272:
14268:
14264:
14246:
14240:
14229:. Retrieved
14225:
14216:
14197:
14191:
14172:
14166:
14157:
14151:
14145:
14119:
14111:
14102:
14093:
14073:
14044:
14014:
14007:
13995:. Retrieved
13990:
13972:
13966:
13953:. Retrieved
13938:
13924:
13553:Four-current
13488:Linear motor
13373:Electrolysis
13307:
13253:Eddy current
13213:Permeability
13133:Polarization
13128:Permittivity
12957:Loudspeakers
12935:
12903:
12900:Applications
12634:
12159:
12150:
12109:
12105:
11692:
11626:
11623:
11487:
11481:
11389:
11383:
11376:
11373:
11052:
10885:
10883:
10694:
10688:
10685:
10669:
10571:
9952:
9782:
9779:
9700:
9689:
9679:
9666:Field tensor
8953:
8943:is an extra
8930:
8786:
8783:
8715:
8709:
7085:
7079:
6988:
6554:
6509:
6402:
6398:
6392:
6389:
6317:
6311:
6211:
6206:
6200:
6193:
5893:
5729:
5722:
5718:
5715:
5439:
5300:
5195:
5075:
5062:
5053:
5047:
5044:
5013:scalar field
5007:
5001:
4998:
4991:
4982:
4975:
4968:
4960:
4953:
4947:
4943:
4941:-field, the
4938:
4932:
4924:
4918:
4911:
4906:
4898:
4368:
4360:
4026:
4017:
4015:
3951:
3949:
3935:
3927:
3926:
3922:
3915:
3904:
3781:
3773:
3766:
3759:
3755:
3736:
3731:
3727:
3719:
3717:
3706:
3695:
3683:
3675:
3668:
3508:
3484:
3480:
3476:
3457:
3444:
3440:
3437:
3435:
3430:
3427:motional EMF
3426:
3419:
3414:
3410:
3407:
3404:
3392:
3334:
3324:
3286:
3273:
3224:
3221:
3216:
3160:
3156:
3153:
3148:
3144:
3139:
3135:
3131:
3128:
3118:
3114:
3088:
3082:
2989:cathode rays
2965:
2933:
2927:
2923:
2883:
2870:
2625:
2519:
2315:
2174:
2139:
2130:
2123:tensor field
1967:In terms of
1966:
1957:
1852:
1773:
1722:
1655:
1422:
1415:
1406:
1400:
1391:
1378:
1309:
1302:
1297:
1293:
1289:
1283:
1277:
1273:
1268:
1263:
1260:
1257:
1251:
1245:
1239:
1229:
1223:
1217:
1212:
1210:
1200:
1024:
716:
655:
644:
638:
632:
619:
616:
607:
598:
591:
577:
533:
518:
504:
480:
474:
468:
462:
459:
452:
446:
437:
432:
424:
417:
357:
353:
349:
345:
341:
337:
333:
329:
321:
317:
313:
312:
305:
299:
294:
292:
276:Teltron tube
259:
255:
222:
207:
201:
195:
189:
113:
104:
95:
86:
84:
79:
71:
64:point charge
55:
45:
15024:: volume 2.
14609:, 429–430.
14120:Gravitation
13814:Hall effect
13523:Transformer
13353:Capacitance
13278:Faraday law
13073:Coulomb law
13015:Electricity
12184:and charge
11926:dot product
10693:component (
9948:proper time
8956:extremizing
6204:and charge
3751:Faraday Law
3703:element of
3701:vector area
3320:integration
2976:Lord Kelvin
2127:energy flux
585:(of charge
514:Weber force
324:test charge
15138:Categories
14995:References
14753:2018-08-14
14364:157958246X
14231:2022-07-06
13955:21 October
13889:Field line
13590:Scientists
13438:Waveguides
13418:Resistance
13388:Inductance
13168:Ampère law
12926:Magnetrons
12909:Cyclotrons
10248:4-velocity
9680:Using the
7077:(same for
6555:The total
6196:Lagrangian
6188:See also:
5031:See also:
5019:since its
4020:using the
3728:stationary
3724:Lenz's law
3713:orthogonal
3688:, at time
3485:vice versa
3483:-field or
2950:, using a
2912:divergence
617:The force
295:definition
14952:cite book
14584:1941-5982
13906:Footnotes
13746:Steinmetz
13676:Kirchhoff
13661:Jefimenko
13656:Hopkinson
13641:Helmholtz
13636:Heaviside
13498:Permeance
13383:Impedance
13123:Insulator
13118:Gauss law
13068:Conductor
13045:Phenomena
13040:Textbooks
13020:Magnetism
12765:Γ
12680:Γ
12673:−
12534:∂
12513:∂
12302:−
12133:⋅
12080:γ
12076:⋅
12056:γ
12052:∧
11977:˙
11937:γ
11875:γ
11860:γ
11856:∧
11807:γ
11792:γ
11788:⋅
11730:γ
11646:γ
11597:×
11524:τ
11508:γ
11454:×
11433:γ
11421:τ
11335:×
11303:γ
11269:−
11228:γ
11185:−
11163:−
11144:−
11102:γ
11090:τ
10989:−
10921:τ
10760:β
10747:β
10730:τ
10647:β
10644:α
10634:β
10627:ν
10623:Λ
10615:α
10608:μ
10604:Λ
10592:ν
10589:μ
10483:−
10441:−
10417:γ
10409:in which
10379:−
10363:−
10347:−
10333:γ
10263:β
10195:−
10158:−
10092:−
10042:−
10019:−
9996:−
9975:β
9972:α
9878:γ
9803:α
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9750:α
9733:τ
9721:α
9626:×
9620:˙
9569:˙
9552:−
9540:˙
9482:ϕ
9424:−
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9394:∂
9389:−
9336:ϕ
9278:−
9252:∂
9248:∂
9173:˙
9156:−
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9117:−
9085:∂
9080:ϕ
9077:∂
9068:−
9062:˙
9051:⋅
9040:∂
9030:∂
9010:∂
9002:∂
8937:spacetime
8933:arclength
8906:ϕ
8900:−
8894:˙
8883:⋅
8839:˙
8822:−
8801:−
8761:×
8755:˙
8676:×
8670:˙
8611:×
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8602:×
8596:˙
8537:×
8534:∇
8525:˙
8516:−
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8495:∇
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8429:∂
8414:∂
8408:−
8399:∂
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8332:∂
8326:−
8317:∂
8302:∂
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8242:∂
8227:∂
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8219:∂
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8142:∂
8127:∂
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8085:∂
8073:˙
8058:∂
8043:∂
8020:∂
8015:ϕ
8012:∂
8003:−
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7974:∂
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7947:˙
7932:∂
7917:∂
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7890:∂
7875:∂
7860:∂
7845:∂
7825:¨
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7773:∂
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7746:˙
7731:∂
7716:∂
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7674:∂
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7634:−
7622:∂
7614:∂
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7565:∂
7550:∂
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7523:∂
7508:∂
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7436:∂
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6975:ϕ
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6923:˙
6898:˙
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6820:˙
6773:ϕ
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6761:˙
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6735:˙
6726:⋅
6720:˙
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6663:˙
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6590:⋅
6579:−
6576:ϕ
6559:is then:
6523:ϕ
6493:˙
6484:⋅
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6352:⋅
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6297:ϕ
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6146:∂
6137:^
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6096:^
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6064:∂
6055:^
6039:˙
6028:∇
6000:∂
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5987:^
5968:∂
5964:∂
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5932:∂
5923:^
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5866:⋅
5860:˙
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5836:˙
5825:∇
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5785:˙
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5668:−
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5329:−
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5271:×
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5220:−
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5112:−
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5103:−
4859:×
4791:ℓ
4756:×
4733:Σ
4730:∂
4726:∮
4694:⋅
4690:ℓ
4670:Σ
4667:∂
4663:∮
4623:⋅
4619:ℓ
4599:Σ
4596:∂
4592:∮
4570:ℓ
4555:×
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4525:∮
4490:∂
4486:∂
4481:⋅
4457:Σ
4453:∫
4449:−
4413:⋅
4409:ℓ
4389:Σ
4386:∂
4382:∮
4365:and that
4319:⋅
4295:Σ
4291:∫
4268:−
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4225:ℓ
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4198:∮
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4098:−
4067:⋅
4063:ℓ
4043:Σ
4040:∂
4036:∮
3992:∂
3982:∂
3976:−
3965:×
3962:∇
3923:NB: Both
3838:⋅
3834:ℓ
3812:Σ
3809:∂
3805:∮
3628:⋅
3604:Σ
3600:∫
3587:Φ
3547:Φ
3535:−
3371:×
3367:ℓ
3358:∫
3305:ℓ
3251:×
3247:ℓ
3022:×
2922:B (III),
2844:μ
2834:ε
2786:μ
2777:π
2722:ε
2718:π
2680:ε
2676:π
2590:×
2493:⋅
2479:∂
2469:∂
2455:×
2452:∇
2325:ρ
2294:×
2280:∂
2270:∂
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2253:∇
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2209:∇
2206:−
2197:ρ
2153:⋅
2129:(flow of
2074:∂
2064:∂
2038:−
2034:σ
2030:⋅
2027:∇
1976:σ
1915:σ
1861:ρ
1818:×
1802:ρ
1794:∫
1754:×
1738:ρ
1702:ρ
1666:ρ
1610:×
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1474:×
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1324:⋅
1258:The term
1198:in which
1159:×
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1068:˙
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886:−
793:−
692:×
529:force law
436:| ≈
420:magnitude
395:×
158:×
36:particles
15124:Archived
14944:12949728
14842:(2003).
14702:(1910).
13982:oersteds
13771:Wiechert
13726:Poynting
13616:Einstein
13463:DC motor
13458:AC motor
13293:Lenz law
13078:Electret
12984:See also
12947:Railguns
11918:bivector
10678:tensor.
10585:′
6614:and the
6508:. For a
6190:Momentum
4928:-field,
3420:motional
3099:electron
2910:for the
1960:used in
1895:, using
1227:and the
562:Equation
550:and the
119:SI units
14656:, 1895.
13978:amperes
13756:Thomson
13731:Ritchie
13721:Poisson
13706:Neumann
13701:Maxwell
13696:Lorentz
13691:Liénard
13621:Faraday
13606:Coulomb
13433:Voltage
13408:Ohm law
13030:History
12892:is the
12789:is the
12160:In the
11057:yields
10888:yields
10674:is the
10565:is the
9786:is the
6390:For an
5465:not on
5189:is the
3662:is the
3445:induced
3171:or the
3167:or the
2914:of the
2898:History
2877:is the
2628:is the
2420:is the
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2316:where:
2113:is the
1678:is the
1654:is the
595:). The
212:), the
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48:physics
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14022:
13930:teslas
13741:Singer
13736:Savart
13716:Ørsted
13681:Larmor
13671:Kelvin
13626:Fizeau
13596:Ampère
13518:Stator
13025:Optics
12872:where
12756:where
12398:where
11994:where
11379:= 2, 3
10665:where
9780:where
6309:where
6212:energy
5894:where
5177:where
5068:scalar
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3908:is an
3863:where
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3732:moving
3578:where
3431:motion
3330:torque
3271:where
3107:plasma
3093:of an
2868:where
2624:where
2131:energy
2093:where
1632:where
1496:where
717:where
430:|
108:and a
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14438:78–79
14128:–73.
13997:9 May
13944:gauss
13766:Weber
13761:Volta
13751:Tesla
13666:Joule
13651:Hertz
13646:Henry
13631:Gauss
13513:Rotor
11760:, as
8945:phase
5716:With
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3737:From
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3119:drift
3105:in a
1294:total
611:field
602:field
581:on a
322:on a
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62:on a
60:force
38:in a
15095:ISBN
15074:ISBN
15055:ISBN
15036:ISBN
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14940:OCLC
14930:ISBN
14902:ISBN
14877:ISBN
14850:ISBN
14820:ISBN
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14710:ISBN
14680:ISBN
14611:ISBN
14580:ISSN
14539:ISBN
14506:ISBN
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14177:ISBN
14155:and
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14050:ISBN
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13999:2012
13970:The
13957:2013
13686:Lenz
13611:Davy
13601:Biot
12558:and
11825:and
11387:and
10686:The
10242:and
9946:the
9674:and
8713:and
7083:and
6618:is:
6406:has
6315:and
6194:The
6019:and
5520:is:
5191:curl
5051:and
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4957:and
4367:div
3932:and
3770:and
3756:Let
3720:sign
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3425:(or
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3147:and
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2978:and
2890:the
2881:and
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1989:and
1873:and
1213:same
604:and
489:and
478:and
466:and
450:and
350:and
303:and
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10294:1
10290:U
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8455:=
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8000:=
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7404:=
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6701:=
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6403:ṙ
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6332:=
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6087:+
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5946:+
5939:x
5920:x
5914:=
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5832:x
5818:t
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5742:=
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5665:)
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5626:=
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5554:=
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5325:[
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5318:=
5314:F
5287:.
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5279:)
5275:A
5265:(
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5254:+
5248:t
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5216:[
5212:q
5209:=
5205:F
5179:∇
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5150:=
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5131:t
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5100:=
5092:E
5076:ϕ
5063:A
5054:B
5048:E
5008:B
5002:E
4992:B
4989:Φ
4985:)
4983:t
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4959:d
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4939:B
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4930:Φ
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4882:)
4879:t
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4871:r
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4850:q
4847:+
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4841:t
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4833:r
4829:(
4825:E
4820:q
4817:=
4813:F
4786:d
4781:)
4778:t
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4768:r
4764:(
4760:B
4752:v
4742:)
4739:t
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4722:+
4719:)
4716:t
4710:,
4706:r
4702:(
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4685:d
4679:)
4676:t
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4659:=
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4653:t
4647:,
4643:r
4639:(
4636:q
4632:/
4627:F
4614:d
4608:)
4605:t
4602:(
4565:d
4559:B
4551:v
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4538:t
4535:(
4521:+
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4515:t
4512:,
4508:r
4504:(
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4463:t
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4446:=
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4440:t
4437:,
4433:r
4429:(
4426:q
4422:/
4417:F
4404:d
4398:)
4395:t
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4347:.
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4327:(
4323:B
4315:A
4310:d
4304:)
4301:t
4298:(
4284:t
4280:d
4274:d
4265:=
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4259:t
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4249:r
4245:(
4242:q
4238:/
4233:F
4220:d
4214:)
4211:t
4208:(
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4161:t
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4153:r
4149:(
4145:B
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4124:d
4118:)
4115:t
4112:(
4095:=
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4089:t
4083:,
4079:r
4075:(
4071:E
4058:d
4052:)
4049:t
4046:(
4002:.
3995:t
3986:B
3973:=
3969:E
3936:A
3934:d
3928:ℓ
3925:d
3918:)
3916:t
3905:ℓ
3903:d
3889:q
3885:/
3880:F
3876:=
3872:E
3851:q
3847:/
3842:F
3829:d
3821:)
3818:t
3815:(
3801:=
3796:E
3784:)
3782:t
3776:)
3774:t
3767:v
3762:)
3760:t
3709:)
3707:t
3696:A
3694:d
3690:t
3686:)
3684:t
3678:)
3676:t
3669:B
3650:)
3647:t
3644:,
3640:r
3636:(
3632:B
3624:A
3619:d
3613:)
3610:t
3607:(
3596:=
3591:B
3563:t
3559:d
3551:B
3542:d
3532:=
3527:E
3481:B
3477:E
3441:E
3438:q
3415:B
3411:v
3408:q
3379:.
3375:B
3362:d
3355:I
3352:=
3348:F
3337:I
3325:ℓ
3300:d
3282:I
3274:ℓ
3259:,
3255:B
3243:I
3240:=
3236:F
3217:B
3161:B
3157:E
3149:B
3145:E
3140:B
3136:E
3132:q
3030:.
3026:B
3018:v
3012:2
3009:q
3004:=
3000:F
2928:V
2887:0
2884:μ
2874:0
2871:ε
2856:.
2848:0
2838:0
2829:1
2824:=
2821:c
2817:,
2810:I
2807:S
2801:B
2790:0
2781:/
2774:4
2769:=
2763:G
2757:B
2751:,
2745:I
2742:S
2736:E
2726:0
2715:4
2710:=
2704:G
2698:E
2692:,
2684:0
2673:4
2666:I
2663:S
2658:q
2652:=
2646:G
2641:q
2626:c
2612:,
2608:)
2601:G
2595:B
2585:c
2581:v
2575:+
2569:G
2563:E
2557:(
2550:G
2545:q
2541:=
2537:F
2501:.
2497:E
2489:)
2482:t
2473:P
2463:+
2459:M
2449:+
2444:f
2439:J
2433:(
2407:M
2384:f
2379:J
2352:P
2329:f
2302:.
2298:B
2290:)
2283:t
2274:P
2264:+
2260:M
2250:+
2245:f
2240:J
2234:(
2230:+
2226:E
2221:)
2216:P
2201:f
2192:(
2188:=
2184:f
2161:.
2157:E
2149:J
2119:∇
2101:c
2077:t
2068:S
2052:2
2048:c
2044:1
2024:=
2020:f
1998:S
1958:T
1940:S
1882:J
1839:.
1836:V
1832:d
1827:)
1822:B
1814:J
1810:+
1806:E
1798:(
1791:=
1787:F
1758:B
1750:J
1746:+
1742:E
1735:=
1731:f
1706:v
1699:=
1695:J
1641:f
1619:)
1614:B
1606:v
1602:+
1598:E
1593:(
1586:=
1582:f
1561:V
1557:d
1536:q
1532:d
1510:F
1505:d
1483:)
1478:B
1470:v
1466:+
1462:E
1457:(
1453:q
1449:d
1445:=
1441:F
1436:d
1401:J
1392:ρ
1386:(
1379:f
1352:.
1348:E
1340:v
1335:q
1332:=
1328:F
1320:v
1286:)
1284:B
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1276:(
1274:q
1264:E
1261:q
1252:F
1246:B
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1224:v
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1206:t
1201:r
1185:]
1181:)
1178:t
1175:,
1171:r
1167:(
1163:B
1156:)
1153:t
1150:(
1140:r
1133:+
1130:)
1127:t
1124:,
1120:r
1116:(
1112:E
1107:[
1103:q
1100:=
1096:)
1092:q
1089:,
1086:t
1083:,
1080:)
1077:t
1074:(
1064:r
1057:,
1054:)
1051:t
1048:(
1044:r
1039:(
1034:F
1007:.
1003:)
997:x
993:B
987:y
983:v
974:y
970:B
964:x
960:v
956:+
951:z
947:E
942:(
938:q
935:=
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922:F
914:,
910:)
904:z
900:B
894:x
890:v
881:x
877:B
871:z
867:v
863:+
858:y
854:E
849:(
845:q
842:=
833:y
829:F
821:,
817:)
811:y
807:B
801:z
797:v
788:z
784:B
778:y
774:v
770:+
765:x
761:E
756:(
752:q
749:=
740:x
736:F
720:×
701:)
696:B
688:v
684:+
680:E
675:(
671:q
668:=
664:F
645:B
639:E
633:v
628:q
620:F
608:B
599:E
592:v
587:q
578:F
481:B
475:E
469:B
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453:B
447:E
438:c
433:v
425:v
403:)
399:B
391:v
387:+
383:E
379:(
376:q
373:=
369:F
354:B
346:E
338:v
330:q
318:F
306:B
300:E
260:B
256:q
202:v
196:B
190:E
185:q
171:.
167:)
162:B
154:v
150:+
146:E
141:(
137:q
134:=
130:F
114:B
105:E
96:v
91:q
20:)
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