2384:
2134:
2536:
2379:{\displaystyle 0=\nabla \cdot (\nabla \times \mathbf {B} )=\nabla \cdot \left(\mu _{0}\left(\mathbf {J} +\varepsilon _{0}{\frac {\partial \mathbf {E} }{\partial t}}\right)\right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +\varepsilon _{0}{\frac {\partial }{\partial t}}\nabla \cdot \mathbf {E} \right)=\mu _{0}\left(\nabla \cdot \mathbf {J} +{\frac {\partial \rho }{\partial t}}\right)}
157:, charge conservation means that in reactions that create charged particles, equal numbers of positive and negative particles are always created, keeping the net amount of charge unchanged. Similarly, when particles are destroyed, equal numbers of positive and negative charges are destroyed. This property is supported without exception by all empirical observations so far.
1189:
60:, implies that the change in the amount of electric charge in any volume of space is exactly equal to the amount of charge flowing into the volume minus the amount of charge flowing out of the volume. In essence, charge conservation is an accounting relationship between the amount of charge in a region and the flow of charge into and out of that region, given by a
937:
160:
Although conservation of charge requires that the total quantity of charge in the universe is constant, it leaves open the question of what that quantity is. Most evidence indicates that the net charge in the universe is zero; that is, there are equal quantities of positive and negative charge.
3020:
but there are theoretical arguments that such single-photon decays will never occur even if charge is not conserved. Charge disappearance tests are sensitive to decays without energetic photons, other unusual charge violating processes such as an electron spontaneously changing into a
852:
649:
2815:
2922:
Gauge invariance is a very important, well established property of the electromagnetic field and has many testable consequences. The theoretical justification for charge conservation is greatly strengthened by being linked to this symmetry. For example,
2061:
335:
1517:
at the same point. The equation equates these two factors, which says that the only way for the charge density at a point to change is for a current of charge to flow into or out of the point. This statement is equivalent to a conservation of
2532:
1184:{\displaystyle \int _{t_{0}}^{t}\left({\dot {Q}}_{\rm {IN}}(\tau )-{\dot {Q}}_{\rm {OUT}}(\tau )\right)\,\mathrm {d} \tau =0\;\;\forall t>t_{0}\;\implies \;{\dot {Q}}_{\rm {IN}}(t)={\dot {Q}}_{\rm {OUT}}(t)\;\;\forall t>t_{0}}
1794:
2881:
so gauge invariance is equivalent to the well known fact that changes in the overall phase of a wavefunction are unobservable, and only changes in the magnitude of the wavefunction result in changes to the probability function
2954:
on positive and negative particles must be extremely close to equal, differing by no more than a factor of 10 for the case of protons and electrons. Ordinary matter contains equal numbers of positive and negative particles,
1947:
2441:
2118:
1496:
687:
2623:
470:
1707:
1404:
1276:
are equal (not necessarily constant) over time, then the overall charge inside the control volume does not change. This deduction could be derived directly from the continuity equation, since at steady state
1584:
1980:
928:
221:
2564:
2734:
2963:, in enormous quantities. If the elementary charge on the electron and proton were even slightly different, all matter would have a large electric charge and would be mutually repulsive.
2876:
1863:
2705:. The full statement of gauge invariance is that the physics of an electromagnetic field are unchanged when the scalar and vector potential are shifted by the gradient of an arbitrary
373:
132:
2970:
that would be allowed if electric charge is not always conserved. No such decays have ever been seen. The best experimental test comes from searches for the energetic photon from an
1315:
1274:
462:
1230:
418:
96:
2703:
1719:
2917:
2938:
Even if gauge symmetry is exact, however, there might be apparent electric charge non-conservation if charge could leak from our normal 3-dimensional space into hidden
1876:
2389:
2066:
1444:
2450:
2726:
2678:
678:
1649:
3895:; et al. (1999). "Charge non-conservation restrictions from the nuclear levels excitation of Xe induced by the electron's decay on the atomic shell".
3333:
2127:
The invariance of charge can be derived as a corollary of
Maxwell's equations. The left-hand side of the modified Ampere's law has zero divergence by the
2657:. This is related to the fact that the electric and magnetic fields are not changed by different choices of the value representing the zero point of
2569:
3763:
3704:
1537:
2443:
By the Gauss divergence theorem, this means the rate of change of charge in a fixed volume equals the net current flowing through the boundary:
1638:
is the current density (charge per unit area per unit time) at the surface of the volume. The vector points in the direction of the current.
847:{\displaystyle Q(t)=Q(t_{0})+\int _{t_{0}}^{t}\left({\dot {Q}}_{\rm {IN}}(\tau )-{\dot {Q}}_{\rm {OUT}}(\tau )\right)\,\mathrm {d} \tau .}
1320:
644:{\displaystyle Q(t_{2})=Q(t_{1})+\int _{t_{1}}^{t_{2}}\left({\dot {Q}}_{\rm {IN}}(t)-{\dot {Q}}_{\rm {OUT}}(t)\right)\,\mathrm {d} t.}
185:
it is now discovered and demonstrated, both here and in Europe, that the
Electrical Fire is a real Element, or Species of Matter, not
1711:
Charge conservation requires that the net current into a volume must necessarily equal the net change in charge within the volume.
2128:
1826:
137:
This does not mean that individual positive and negative charges cannot be created or destroyed. Electric charge is carried by
2931:
be massless, so the good experimental evidence that the photon has zero mass is also strong evidence that charge is conserved.
4017:
3802:
3409:
3371:
3281:
3124:
3097:
859:
3314:
2810:{\displaystyle \phi '=\phi -{\frac {\partial \chi }{\partial t}}\qquad \qquad \mathbf {A} '=\mathbf {A} +\nabla \chi .}
3155:
2542:
4009:
3025:, and to electric charge moving into other dimensions. The best experimental bounds on charge disappearance are:
340:
104:
3339:
2834:
2131:. Expanding the divergence of the right-hand side, interchanging derivatives, and applying Gauss's law gives:
1280:
1235:
464:
is the amount of charge flowing out of the volume; both amounts are regarded as generic functions of time.
423:
3102:
2056:{\displaystyle 0=\iiint _{V}\left({\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} \right)dV.}
1194:
382:
330:{\displaystyle {\frac {\mathrm {d} Q}{\mathrm {d} t}}={\dot {Q}}_{\rm {IN}}(t)-{\dot {Q}}_{\rm {OUT}}(t).}
3108:
70:
4041:
3517:
S.L. Dubovsky; V.A. Rubakov; P.G. Tinyakov (2000). "Is the electric charge conserved in brane world?".
3119:
1410:
170:
57:
4036:
930:
corresponds to the absence of charge quantity change in the control volume: the system has reached a
3832:
2950:
Simple arguments rule out some types of charge nonconservation. For example, the magnitude of the
2649:
of the underlying physics. The symmetry that is associated with charge conservation is the global
2686:
19:
This article is about the conservation of electric charge. For a general theoretical concept, see
3643:
3482:
S.Y. Chu (1996). "Gauge-Invariant Charge
Nonconserving Processes and the Solar Neutrino Puzzle".
2885:
2658:
3596:
3306:
3114:
2527:{\displaystyle {\frac {d}{dt}}Q_{\Omega }={\frac {d}{dt}}\iiint _{\Omega }\rho \mathrm {d} V=-}
3361:
3271:
4001:
3757:
3698:
2654:
1866:
3952:
3904:
3858:
3781:
3735:
3662:
3611:
3559:
3532:
3491:
3444:
3233:
3180:
2711:
2663:
2638:
656:
8:
3623:
3592:
2646:
215:
209:
61:
3956:
3908:
3862:
3739:
3666:
3615:
3536:
3495:
3448:
3237:
3184:
3976:
3846:
3820:
3808:
3686:
3652:
3522:
3460:
3434:
3299:
3249:
3223:
2939:
1643:
138:
52:
3916:
3748:
3723:
3245:
16:
Fundamental physical law – electric charge is continuously conserved in space and time
4013:
3968:
3936:
3874:
3812:
3798:
3780:(1989). "Comments on Testing Charge Conservation and the Pauli Exclusion Principle".
3690:
3678:
3544:
3405:
3367:
3310:
3277:
3253:
3196:
3151:
3092:
2951:
1789:{\displaystyle {\frac {dq}{dt}}=-\iiint _{V}\left(\nabla \cdot \mathbf {J} \right)dV}
681:
174:
3980:
3464:
3960:
3912:
3866:
3790:
3743:
3674:
3670:
3619:
3540:
3499:
3452:
3397:
3267:
3241:
3188:
3068:
3051:
3011:
2932:
2924:
2681:
2650:
2642:
154:
150:
20:
3941:"Improved limit on charge conservation derived from Ga solar neutrino experiments"
1942:{\displaystyle {\frac {dq}{dt}}=\iiint _{V}{\frac {\partial \rho }{\partial t}}dV}
3932:
2436:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.}
2113:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.}
1491:{\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot \mathbf {J} =0.}
1429:
1414:
178:
99:
47:
43:
39:
35:
3870:
3214:
E. Masso; F. Rota (2002). "Primordial helium production in a charged universe".
3192:
2637:
Charge conservation can also be understood as a consequence of symmetry through
3794:
2967:
1601:
1501:
1418:
65:
3724:"Search for electron decay mode e → γ + ν with prototype of Borexino detector"
3503:
3456:
2618:{\displaystyle \mathbf {J} \cdot {\rm {d}}\mathbf {S} =-I_{\partial \Omega }.}
4030:
3964:
2999:
2966:
The best experimental tests of electric charge conservation are searches for
3940:
3921:
This is the most stringent of several limits given in Table 1 of this paper.
3789:. World Scientific Lecture Notes in Physics. Vol. 19. pp. 99–116.
4006:
The Life of
Benjamin Franklin, Volume 3: Soldier, Scientist, and Politician
3972:
3878:
3682:
3391:
3273:
Electricity in the 17th and 18th centuries: a study of early Modern physics
3200:
2825:
2706:
1519:
931:
3401:
3228:
3105:– includes further discussion of gauge invariance and charge conservation
3087:
2821:
1441:
per square meter). This is called the charge density continuity equation
2680:. However the full symmetry is more complicated, and also involves the
3527:
3338:. Vol. 3. Yale University Press. 1961. p. 142. Archived from
1508:
3516:
3425:
A.S. Goldhaber; M.M. Nieto (2010). "Photon and
Graviton Mass Limits".
1702:{\displaystyle I=-\iiint _{V}\left(\nabla \cdot \mathbf {J} \right)dV}
653:
The general solution is obtained by fixing the initial condition time
375:
is the electric charge accumulation rate in a specific volume at time
3997:
3783:
Comments on
Testing Charge Conservation and Pauli Exclusion Principle
3641:
Coll.) (2015). "Test of
Electric Charge Conservation with Borexino".
3636:
2629:
In particular, in an isolated system the total charge is conserved.
3892:
3777:
3719:
3717:
3657:
3638:
3022:
2975:
2971:
2960:
153:
can be created and destroyed in elementary particle reactions. In
142:
42:
never changes. The net quantity of electric charge, the amount of
3439:
1399:{\displaystyle {\dot {Q}}_{\rm {IN}}(t)={\dot {Q}}_{\rm {OUT}}(t)}
467:
The integrated continuity equation between two time values reads:
3072:
2641:, a central result in theoretical physics that asserts that each
1438:
1425:
214:
Mathematically, we can state the law of charge conservation as a
27:
3171:
S. Orito; M. Yoshimura (1985). "Can the
Universe be Charged?".
2979:
2956:
2928:
146:
2935:
also implies quantization of hypothetical magnetic charges.
1579:{\displaystyle I=-\iint _{S}\mathbf {J} \cdot d\mathbf {S} }
198:
Benjamin
Franklin, Letter to Cadwallader Colden, 5 June 1747
169:
Charge conservation was first proposed by
British scientist
934:. From the above condition, the following must hold true:
177:
in 1747, although the first convincing proof was given by
3111:– application of charge conservation to electric circuits
2820:
In quantum mechanics the scalar field is equivalent to a
34:
is the principle, of experimental nature, that the total
3424:
2063:
Since this is true for every volume, we have in general
3849:(1987). "Possible Nonconservation of Electric Charge".
3150:(3rd ed.). Cambridge University Press. p. 4.
3067:
charge non-conserving decays are less than 8 × 10 (68%
3930:
3145:
2547:
3321:
benjamin franklin william watson charge conservation.
2888:
2837:
2737:
2714:
2689:
2666:
2572:
2545:
2453:
2392:
2137:
2069:
1983:
1879:
1829:
1722:
1652:
1540:
1447:
1323:
1283:
1238:
1197:
940:
862:
690:
659:
473:
426:
385:
343:
224:
107:
73:
3366:. UK: Cambridge University Press. pp. 164–165.
3170:
420:
is the amount of charge flowing into the volume and
3591:
3558:Patrignani, C. et al (Particle Data Group) (2016).
3298:
2911:
2870:
2809:
2720:
2697:
2672:
2617:
2558:
2526:
2435:
2378:
2112:
2055:
1941:
1857:
1788:
1701:
1578:
1500:The term on the left is the rate of change of the
1490:
1398:
1309:
1268:
1224:
1183:
923:{\displaystyle Q(t)=Q(t_{0})\;\forall t>t_{0},}
922:
846:
672:
643:
456:
412:
367:
329:
126:
90:
3389:
2632:
2122:
4028:
3276:. University of California Press. p. 330.
3213:
2559:{\displaystyle {\scriptstyle \partial \Omega }}
1820:is the integral (sum) of the charge density in
3845:
3390:Sakurai, J. J.; Napolitano, Jim (2017-09-21).
3141:
3139:
1622:, the outward pointing normal of the boundary
203:
173:in 1746 and American statesman and scientist
3762:: CS1 maint: multiple names: authors list (
3703:: CS1 maint: multiple names: authors list (
3146:Purcell, Edward M.; Morin, David J. (2013).
1858:{\displaystyle q=\iiint \limits _{V}\rho dV}
3363:Introduction to Elementary Particle Physics
3136:
3027:
2984:
1417:can be used to express the law in terms of
368:{\displaystyle \mathrm {d} Q/\mathrm {d} t}
3557:
3551:
3296:
1525:
1161:
1160:
1087:
1086:
1082:
1081:
1061:
1060:
897:
127:{\displaystyle \mathbf {J} (\mathbf {x} )}
3776:
3747:
3656:
3526:
3438:
3290:
3227:
1507:at a point. The term on the right is the
1045:
832:
629:
56:. Charge conservation, considered as a
3891:
3481:
3266:
2945:
3359:
2871:{\displaystyle \psi '=e^{iq\chi }\psi }
1310:{\displaystyle \partial Q/\partial t=0}
4029:
3630:
1269:{\displaystyle {\dot {Q}}_{\rm {OUT}}}
457:{\displaystyle {\dot {Q}}_{\rm {OUT}}}
3996:
3305:. Rutgers University Press. pp.
1225:{\displaystyle {\dot {Q}}_{\rm {IN}}}
413:{\displaystyle {\dot {Q}}_{\rm {IN}}}
3385:
3383:
1870:
1713:
91:{\displaystyle \rho (\mathbf {x} )}
13:
3990:
2798:
2766:
2758:
2607:
2604:
2583:
2551:
2548:
2511:
2502:
2474:
2416:
2404:
2396:
2362:
2354:
2337:
2303:
2294:
2290:
2264:
2227:
2217:
2170:
2153:
2144:
2093:
2081:
2073:
2028:
2016:
2008:
1924:
1916:
1764:
1677:
1471:
1459:
1451:
1381:
1378:
1375:
1342:
1339:
1295:
1284:
1260:
1257:
1254:
1216:
1213:
1162:
1145:
1142:
1139:
1106:
1103:
1062:
1047:
1025:
1022:
1019:
986:
983:
898:
834:
812:
809:
806:
773:
770:
631:
609:
606:
603:
570:
567:
448:
445:
442:
404:
401:
358:
345:
309:
306:
303:
270:
267:
239:
229:
14:
4053:
3380:
3301:Physics in the Nineteenth Century
4010:University of Pennsylvania Press
3560:"The Review of Particle Physics"
3125:Franklin's electrostatic machine
2791:
2779:
2691:
2589:
2574:
2534:
2423:
2344:
2310:
2271:
2221:
2197:
2160:
2100:
2035:
1771:
1684:
1572:
1561:
1478:
117:
109:
81:
3924:
3885:
3839:
3770:
3711:
3585:
3510:
3475:
3471:Conservation of Electric Charge
3335:The Papers of Benjamin Franklin
2776:
2775:
3675:10.1103/PhysRevLett.115.231802
3624:10.1088/0954-3899/37/7A/075021
3519:Journal of High Energy Physics
3418:
3396:. Cambridge University Press.
3353:
3326:
3297:Purrington, Robert D. (1997).
3260:
3207:
3164:
3098:Conservation Laws and Symmetry
3042:mean lifetime is greater than
2899:
2890:
2633:Connection to gauge invariance
2164:
2150:
2123:Derivation from Maxwell's Laws
1428:per cubic meter) and electric
1393:
1387:
1354:
1348:
1157:
1151:
1118:
1112:
1083:
1037:
1031:
998:
992:
894:
881:
872:
866:
824:
818:
785:
779:
722:
709:
700:
694:
621:
615:
582:
576:
512:
499:
490:
477:
321:
315:
282:
276:
121:
113:
85:
77:
1:
3917:10.1016/S0370-2693(99)01091-6
3749:10.1016/S0370-2693(01)01440-X
3246:10.1016/S0370-2693(02)02636-9
1600:oriented by outward-pointing
3597:"Tests of Conservation Laws"
3545:10.1016/0370-2693(79)90048-0
3360:Bettini, Alessandro (2008).
3103:Introduction to gauge theory
2698:{\displaystyle \mathbf {A} }
1411:electromagnetic field theory
7:
3983:. Link is to preprint copy.
3871:10.1103/PhysRevLett.59.1510
3193:10.1103/PhysRevLett.54.2457
3081:
2912:{\displaystyle |\psi |^{2}}
1973:
1967:
1955:
1802:
204:Formal statement of the law
50:in the universe, is always
10:
4058:
3795:10.1142/9789812799104_0006
2993:e → ν + γ
207:
164:
18:
3504:10.1142/S0217732396002241
3457:10.1103/RevModPhys.82.939
3427:Reviews of Modern Physics
3148:Electricity and magnetism
2828:of the charged particle:
58:physical conservation law
4002:"Chapter 2: Electricity"
3965:10.1103/PhysRevD.53.4086
3484:Modern Physics Letters A
3393:Modern Quantum Mechanics
3130:
3109:Kirchhoff's circuit laws
3851:Physical Review Letters
3644:Physical Review Letters
3521:. August (8): 315–318.
3173:Physical Review Letters
3120:Relative charge density
3061:n → p + ν +
2927:also requires that the
2659:electrostatic potential
1526:Mathematical derivation
1511:of the current density
3637:Agostini, M.; et al. (
2913:
2872:
2811:
2722:
2699:
2674:
2619:
2560:
2528:
2437:
2380:
2114:
2057:
1943:
1859:
1790:
1703:
1580:
1492:
1400:
1311:
1270:
1226:
1185:
924:
848:
674:
645:
458:
414:
369:
331:
201:
128:
92:
3402:10.1017/9781108499996
2946:Experimental evidence
2914:
2873:
2812:
2723:
2721:{\displaystyle \chi }
2700:
2675:
2673:{\displaystyle \phi }
2655:electromagnetic field
2645:is associated with a
2620:
2561:
2529:
2438:
2381:
2115:
2058:
1944:
1867:Leibniz integral rule
1860:
1791:
1704:
1581:
1493:
1401:
1312:
1271:
1227:
1186:
925:
849:
675:
673:{\displaystyle t_{0}}
646:
459:
415:
370:
332:
189:by the Friction, but
183:
129:
93:
3718:Back, H.O.; et al. (
3604:Journal of Physics G
2886:
2835:
2735:
2712:
2687:
2664:
2570:
2543:
2451:
2390:
2135:
2067:
1981:
1877:
1827:
1720:
1650:
1646:this can be written
1538:
1445:
1321:
1281:
1236:
1195:
938:
860:
688:
657:
471:
424:
383:
341:
222:
105:
71:
46:minus the amount of
3957:1996PhRvD..53.4086N
3909:1999PhLB..465..315B
3863:1987PhRvL..59.1510M
3740:2002PhLB..525...29B
3667:2015PhRvL.115w1802A
3616:2010JPhG...37g5021N
3593:Particle Data Group
3537:1979PhLB...84..315I
3496:1996MPLA...11.2251C
3469:; see Section II.C
3449:2010RvMP...82..939G
3238:2002PhLB..545..221M
3185:1985PhRvL..54.2457O
3115:Maxwell's equations
1596:is the boundary of
1317:holds, and implies
962:
749:
546:
216:continuity equation
210:Continuity equation
139:subatomic particles
62:continuity equation
32:charge conservation
2909:
2868:
2807:
2718:
2695:
2670:
2615:
2556:
2554:
2524:
2433:
2376:
2110:
2053:
1939:
1855:
1845:
1786:
1699:
1644:Divergence theorem
1576:
1488:
1396:
1307:
1266:
1222:
1181:
941:
920:
844:
728:
670:
641:
518:
454:
410:
365:
327:
124:
88:
4042:Conservation laws
4019:978-0-8122-4121-1
3897:Physics Letters B
3857:(14): 1510–1512.
3804:978-981-02-0453-2
3728:Physics Letters B
3567:Chinese Physics C
3490:(28): 2251–2257.
3411:978-1-108-49999-6
3373:978-0-521-88021-3
3283:978-0-520-03478-5
3216:Physics Letters B
3179:(22): 2457–2460.
3093:Charge invariance
3079:
3078:
3018:
3017:
2952:elementary charge
2773:
2639:Noether's theorem
2495:
2467:
2411:
2369:
2301:
2234:
2129:div–curl identity
2088:
2023:
1963:
1962:
1931:
1898:
1836:
1812:The total charge
1810:
1809:
1741:
1613:is shorthand for
1466:
1370:
1334:
1249:
1208:
1134:
1098:
1014:
978:
801:
765:
682:integral equation
680:, leading to the
598:
562:
437:
396:
298:
262:
247:
175:Benjamin Franklin
151:Charged particles
4049:
4037:Electromagnetism
4023:
3985:
3984:
3951:(7): 4086–4088.
3928:
3922:
3920:
3903:(1–4): 315–322.
3889:
3883:
3882:
3843:
3837:
3836:
3830:
3826:
3824:
3816:
3788:
3774:
3768:
3767:
3761:
3753:
3751:
3715:
3709:
3708:
3702:
3694:
3660:
3634:
3628:
3627:
3601:
3589:
3583:
3582:
3580:
3578:
3564:
3555:
3549:
3548:
3530:
3514:
3508:
3507:
3479:
3473:
3468:
3442:
3422:
3416:
3415:
3387:
3378:
3377:
3357:
3351:
3350:
3348:
3347:
3330:
3324:
3323:
3304:
3294:
3288:
3287:
3264:
3258:
3257:
3231:
3229:astro-ph/0201248
3222:(3–4): 221–225.
3211:
3205:
3204:
3168:
3162:
3161:
3143:
3064:
3049:
3047:
3033:
3028:
3012:Confidence Level
3009:
3007:
3002:is greater than
2996:
2990:
2985:
2974:decaying into a
2940:extra dimensions
2933:Gauge invariance
2925:gauge invariance
2918:
2916:
2915:
2910:
2908:
2907:
2902:
2893:
2877:
2875:
2874:
2869:
2864:
2863:
2845:
2816:
2814:
2813:
2808:
2794:
2786:
2782:
2774:
2772:
2764:
2756:
2745:
2727:
2725:
2724:
2719:
2704:
2702:
2701:
2696:
2694:
2682:vector potential
2679:
2677:
2676:
2671:
2651:gauge invariance
2643:conservation law
2625:
2624:
2622:
2621:
2616:
2611:
2610:
2592:
2587:
2586:
2577:
2566:
2565:
2563:
2562:
2557:
2555:
2538:
2537:
2533:
2531:
2530:
2525:
2514:
2506:
2505:
2496:
2494:
2483:
2478:
2477:
2468:
2466:
2455:
2442:
2440:
2439:
2434:
2426:
2412:
2410:
2402:
2394:
2385:
2383:
2382:
2377:
2375:
2371:
2370:
2368:
2360:
2352:
2347:
2331:
2330:
2318:
2314:
2313:
2302:
2300:
2289:
2287:
2286:
2274:
2258:
2257:
2245:
2241:
2240:
2236:
2235:
2233:
2225:
2224:
2215:
2213:
2212:
2200:
2190:
2189:
2163:
2119:
2117:
2116:
2111:
2103:
2089:
2087:
2079:
2071:
2062:
2060:
2059:
2054:
2043:
2039:
2038:
2024:
2022:
2014:
2006:
1999:
1998:
1957:
1948:
1946:
1945:
1940:
1932:
1930:
1922:
1914:
1912:
1911:
1899:
1897:
1889:
1881:
1871:
1864:
1862:
1861:
1856:
1844:
1804:
1795:
1793:
1792:
1787:
1779:
1775:
1774:
1758:
1757:
1742:
1740:
1732:
1724:
1714:
1708:
1706:
1705:
1700:
1692:
1688:
1687:
1671:
1670:
1636:
1628:
1621:
1612:
1599:
1595:
1585:
1583:
1582:
1577:
1575:
1564:
1559:
1558:
1530:The net current
1516:
1506:
1497:
1495:
1494:
1489:
1481:
1467:
1465:
1457:
1449:
1436:
1423:
1405:
1403:
1402:
1397:
1386:
1385:
1384:
1372:
1371:
1363:
1347:
1346:
1345:
1336:
1335:
1327:
1316:
1314:
1313:
1308:
1294:
1275:
1273:
1272:
1267:
1265:
1264:
1263:
1251:
1250:
1242:
1231:
1229:
1228:
1223:
1221:
1220:
1219:
1210:
1209:
1201:
1190:
1188:
1187:
1182:
1180:
1179:
1150:
1149:
1148:
1136:
1135:
1127:
1111:
1110:
1109:
1100:
1099:
1091:
1080:
1079:
1050:
1044:
1040:
1030:
1029:
1028:
1016:
1015:
1007:
991:
990:
989:
980:
979:
971:
961:
956:
955:
954:
929:
927:
926:
921:
916:
915:
893:
892:
853:
851:
850:
845:
837:
831:
827:
817:
816:
815:
803:
802:
794:
778:
777:
776:
767:
766:
758:
748:
743:
742:
741:
721:
720:
679:
677:
676:
671:
669:
668:
650:
648:
647:
642:
634:
628:
624:
614:
613:
612:
600:
599:
591:
575:
574:
573:
564:
563:
555:
545:
544:
543:
533:
532:
531:
511:
510:
489:
488:
463:
461:
460:
455:
453:
452:
451:
439:
438:
430:
419:
417:
416:
411:
409:
408:
407:
398:
397:
389:
378:
374:
372:
371:
366:
361:
356:
348:
336:
334:
333:
328:
314:
313:
312:
300:
299:
291:
275:
274:
273:
264:
263:
255:
248:
246:
242:
236:
232:
226:
199:
155:particle physics
133:
131:
130:
125:
120:
112:
97:
95:
94:
89:
84:
21:charge (physics)
4057:
4056:
4052:
4051:
4050:
4048:
4047:
4046:
4027:
4026:
4020:
3998:Lemay, J.A. Leo
3993:
3991:Further reading
3988:
3945:Physical Review
3929:
3925:
3890:
3886:
3844:
3840:
3828:
3827:
3818:
3817:
3805:
3786:
3775:
3771:
3755:
3754:
3722:Coll.) (2002).
3716:
3712:
3696:
3695:
3635:
3631:
3599:
3590:
3586:
3576:
3574:
3562:
3556:
3552:
3515:
3511:
3480:
3476:
3423:
3419:
3412:
3388:
3381:
3374:
3358:
3354:
3345:
3343:
3332:
3331:
3327:
3317:
3295:
3291:
3284:
3265:
3261:
3212:
3208:
3169:
3165:
3158:
3144:
3137:
3133:
3084:
3062:
3045:
3043:
3031:
3005:
3003:
2994:
2988:
2968:particle decays
2948:
2903:
2898:
2897:
2889:
2887:
2884:
2883:
2853:
2849:
2838:
2836:
2833:
2832:
2790:
2778:
2777:
2765:
2757:
2755:
2738:
2736:
2733:
2732:
2713:
2710:
2709:
2690:
2688:
2685:
2684:
2665:
2662:
2661:
2635:
2603:
2599:
2588:
2582:
2581:
2573:
2571:
2568:
2567:
2546:
2544:
2541:
2540:
2539:
2535:
2510:
2501:
2497:
2487:
2482:
2473:
2469:
2459:
2454:
2452:
2449:
2448:
2447:
2422:
2403:
2395:
2393:
2391:
2388:
2387:
2361:
2353:
2351:
2343:
2336:
2332:
2326:
2322:
2309:
2293:
2288:
2282:
2278:
2270:
2263:
2259:
2253:
2249:
2226:
2220:
2216:
2214:
2208:
2204:
2196:
2195:
2191:
2185:
2181:
2180:
2176:
2159:
2136:
2133:
2132:
2125:
2099:
2080:
2072:
2070:
2068:
2065:
2064:
2034:
2015:
2007:
2005:
2004:
2000:
1994:
1990:
1982:
1979:
1978:
1923:
1915:
1913:
1907:
1903:
1890:
1882:
1880:
1878:
1875:
1874:
1840:
1828:
1825:
1824:
1770:
1763:
1759:
1753:
1749:
1733:
1725:
1723:
1721:
1718:
1717:
1683:
1676:
1672:
1666:
1662:
1651:
1648:
1647:
1630:
1623:
1614:
1605:
1597:
1587:
1571:
1560:
1554:
1550:
1539:
1536:
1535:
1528:
1512:
1504:
1477:
1458:
1450:
1448:
1446:
1443:
1442:
1432:
1430:current density
1421:
1415:vector calculus
1374:
1373:
1362:
1361:
1360:
1338:
1337:
1326:
1325:
1324:
1322:
1319:
1318:
1290:
1282:
1279:
1278:
1253:
1252:
1241:
1240:
1239:
1237:
1234:
1233:
1212:
1211:
1200:
1199:
1198:
1196:
1193:
1192:
1175:
1171:
1138:
1137:
1126:
1125:
1124:
1102:
1101:
1090:
1089:
1088:
1075:
1071:
1046:
1018:
1017:
1006:
1005:
1004:
982:
981:
970:
969:
968:
967:
963:
957:
950:
946:
945:
939:
936:
935:
911:
907:
888:
884:
861:
858:
857:
833:
805:
804:
793:
792:
791:
769:
768:
757:
756:
755:
754:
750:
744:
737:
733:
732:
716:
712:
689:
686:
685:
664:
660:
658:
655:
654:
630:
602:
601:
590:
589:
588:
566:
565:
554:
553:
552:
551:
547:
539:
535:
534:
527:
523:
522:
506:
502:
484:
480:
472:
469:
468:
441:
440:
429:
428:
427:
425:
422:
421:
400:
399:
388:
387:
386:
384:
381:
380:
376:
357:
352:
344:
342:
339:
338:
302:
301:
290:
289:
288:
266:
265:
254:
253:
252:
238:
237:
228:
227:
225:
223:
220:
219:
212:
206:
200:
197:
179:Michael Faraday
167:
116:
108:
106:
103:
102:
100:current density
80:
72:
69:
68:
48:negative charge
44:positive charge
40:isolated system
36:electric charge
24:
17:
12:
11:
5:
4055:
4045:
4044:
4039:
4025:
4024:
4018:
3992:
3989:
3987:
3986:
3931:Norman, E.B.;
3923:
3884:
3847:R.N. Mohapatra
3838:
3829:|journal=
3803:
3769:
3734:(1–2): 29–40.
3710:
3651:(23): 231802.
3629:
3584:
3550:
3528:hep-ph/0007179
3509:
3474:
3433:(1): 939–979.
3417:
3410:
3379:
3372:
3352:
3325:
3316:978-0813524429
3315:
3289:
3282:
3268:Heilbron, J.L.
3259:
3206:
3163:
3156:
3134:
3132:
3129:
3128:
3127:
3122:
3117:
3112:
3106:
3100:
3095:
3090:
3083:
3080:
3077:
3076:
3065:
3059:
3056:
3055:
3040:
3034:
3016:
3015:
2997:
2991:
2947:
2944:
2906:
2901:
2896:
2892:
2879:
2878:
2867:
2862:
2859:
2856:
2852:
2848:
2844:
2841:
2818:
2817:
2806:
2803:
2800:
2797:
2793:
2789:
2785:
2781:
2771:
2768:
2763:
2760:
2754:
2751:
2748:
2744:
2741:
2717:
2693:
2669:
2634:
2631:
2627:
2626:
2614:
2609:
2606:
2602:
2598:
2595:
2591:
2585:
2580:
2576:
2553:
2550:
2523:
2520:
2517:
2513:
2509:
2504:
2500:
2493:
2490:
2486:
2481:
2476:
2472:
2465:
2462:
2458:
2432:
2429:
2425:
2421:
2418:
2415:
2409:
2406:
2401:
2398:
2374:
2367:
2364:
2359:
2356:
2350:
2346:
2342:
2339:
2335:
2329:
2325:
2321:
2317:
2312:
2308:
2305:
2299:
2296:
2292:
2285:
2281:
2277:
2273:
2269:
2266:
2262:
2256:
2252:
2248:
2244:
2239:
2232:
2229:
2223:
2219:
2211:
2207:
2203:
2199:
2194:
2188:
2184:
2179:
2175:
2172:
2169:
2166:
2162:
2158:
2155:
2152:
2149:
2146:
2143:
2140:
2124:
2121:
2109:
2106:
2102:
2098:
2095:
2092:
2086:
2083:
2078:
2075:
2052:
2049:
2046:
2042:
2037:
2033:
2030:
2027:
2021:
2018:
2013:
2010:
2003:
1997:
1993:
1989:
1986:
1961:
1960:
1951:
1949:
1938:
1935:
1929:
1926:
1921:
1918:
1910:
1906:
1902:
1896:
1893:
1888:
1885:
1854:
1851:
1848:
1843:
1839:
1835:
1832:
1808:
1807:
1798:
1796:
1785:
1782:
1778:
1773:
1769:
1766:
1762:
1756:
1752:
1748:
1745:
1739:
1736:
1731:
1728:
1698:
1695:
1691:
1686:
1682:
1679:
1675:
1669:
1665:
1661:
1658:
1655:
1574:
1570:
1567:
1563:
1557:
1553:
1549:
1546:
1543:
1527:
1524:
1502:charge density
1487:
1484:
1480:
1476:
1473:
1470:
1464:
1461:
1456:
1453:
1419:charge density
1395:
1392:
1389:
1383:
1380:
1377:
1369:
1366:
1359:
1356:
1353:
1350:
1344:
1341:
1333:
1330:
1306:
1303:
1300:
1297:
1293:
1289:
1286:
1262:
1259:
1256:
1248:
1245:
1218:
1215:
1207:
1204:
1178:
1174:
1170:
1167:
1164:
1159:
1156:
1153:
1147:
1144:
1141:
1133:
1130:
1123:
1120:
1117:
1114:
1108:
1105:
1097:
1094:
1085:
1078:
1074:
1070:
1067:
1064:
1059:
1056:
1053:
1049:
1043:
1039:
1036:
1033:
1027:
1024:
1021:
1013:
1010:
1003:
1000:
997:
994:
988:
985:
977:
974:
966:
960:
953:
949:
944:
919:
914:
910:
906:
903:
900:
896:
891:
887:
883:
880:
877:
874:
871:
868:
865:
856:The condition
843:
840:
836:
830:
826:
823:
820:
814:
811:
808:
800:
797:
790:
787:
784:
781:
775:
772:
764:
761:
753:
747:
740:
736:
731:
727:
724:
719:
715:
711:
708:
705:
702:
699:
696:
693:
667:
663:
640:
637:
633:
627:
623:
620:
617:
611:
608:
605:
597:
594:
587:
584:
581:
578:
572:
569:
561:
558:
550:
542:
538:
530:
526:
521:
517:
514:
509:
505:
501:
498:
495:
492:
487:
483:
479:
476:
450:
447:
444:
436:
433:
406:
403:
395:
392:
364:
360:
355:
351:
347:
326:
323:
320:
317:
311:
308:
305:
297:
294:
287:
284:
281:
278:
272:
269:
261:
258:
251:
245:
241:
235:
231:
205:
202:
195:
171:William Watson
166:
163:
123:
119:
115:
111:
87:
83:
79:
76:
66:charge density
15:
9:
6:
4:
3:
2:
4054:
4043:
4040:
4038:
4035:
4034:
4032:
4021:
4015:
4011:
4007:
4003:
3999:
3995:
3994:
3982:
3978:
3974:
3970:
3966:
3962:
3958:
3954:
3950:
3946:
3942:
3938:
3937:Goldhaber, M.
3934:
3933:Bahcall, J.N.
3927:
3918:
3914:
3910:
3906:
3902:
3898:
3894:
3888:
3880:
3876:
3872:
3868:
3864:
3860:
3856:
3852:
3848:
3842:
3834:
3822:
3814:
3810:
3806:
3800:
3796:
3792:
3785:
3784:
3779:
3773:
3765:
3759:
3750:
3745:
3741:
3737:
3733:
3729:
3725:
3721:
3714:
3706:
3700:
3692:
3688:
3684:
3680:
3676:
3672:
3668:
3664:
3659:
3654:
3650:
3646:
3645:
3640:
3633:
3625:
3621:
3617:
3613:
3610:(7A): 89–98.
3609:
3605:
3598:
3594:
3588:
3572:
3568:
3561:
3554:
3546:
3542:
3538:
3534:
3529:
3524:
3520:
3513:
3505:
3501:
3497:
3493:
3489:
3485:
3478:
3472:
3466:
3462:
3458:
3454:
3450:
3446:
3441:
3436:
3432:
3428:
3421:
3413:
3407:
3403:
3399:
3395:
3394:
3386:
3384:
3375:
3369:
3365:
3364:
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3342:on 2011-09-29
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3048:10 years
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3008:10 years
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3772:
3758:cite journal
3731:
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3713:
3699:cite journal
3648:
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3632:
3607:
3603:
3595:(May 2010).
3587:
3575:. Retrieved
3570:
3566:
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3355:
3344:. Retrieved
3340:the original
3334:
3328:
3320:
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3272:
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3176:
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2965:
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2826:wavefunction
2819:
2707:scalar field
2636:
2628:
2126:
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1966:
1964:
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1811:
1800:
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1534:a volume is
1531:
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932:steady state
855:
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190:
186:
184:
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3088:Capacitance
2822:phase shift
1865:So, by the
1191:therefore,
4031:Categories
3658:1509.01223
3346:2010-11-25
3036:e →
1965:Equating (
1816:in volume
1509:divergence
208:See also:
3831:ignored (
3821:cite book
3813:124865855
3778:L.B. Okun
3691:206265225
3577:March 26,
3440:0809.1003
3254:119062159
3071:) of all
2961:electrons
2895:ψ
2866:ψ
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2802:χ
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2759:∂
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143:electrons
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4000:(2008).
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3893:P. Belli
3879:10035254
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3465:14395472
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3201:10031347
3082:See also
3038:anything
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2976:neutrino
2972:electron
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2647:symmetry
1977:) gives
1426:coulombs
196:—
141:such as
64:between
3953:Bibcode
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3859:Bibcode
3736:Bibcode
3663:Bibcode
3612:Bibcode
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3445:Bibcode
3234:Bibcode
3181:Bibcode
3075:decays
3073:neutron
2957:protons
2824:in the
2653:of the
1971:) and (
1629:. Here
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1439:amperes
187:created
165:History
147:protons
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337:where
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3809:S2CID
3787:(PDF)
3687:S2CID
3653:arXiv
3600:(PDF)
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3523:arXiv
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3250:S2CID
3224:arXiv
3131:Notes
3050:(68%
3010:(90%
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193:only.
4014:ISBN
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