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can be neglected in a plasma as it is negligible compared to the current carried by the free charges. The only exception to this is for exceptionally high frequency phenomena: for example, for a plasma with a typical electrical conductivity of
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227:
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993:. For example, it can be of order 10 in a typical star. In this case, the fluid can be called a perfect or ideal fluid. So, the induction equation for an ideal conductive fluid such as most astrophysical plasmas is
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891:{\displaystyle \eta \nabla ^{2}\mathbf {B} \sim {\eta B \over L^{2}},\nabla \times (\mathbf {v} \times \mathbf {B} )\sim {VB \over L}.}
661:{\displaystyle {\partial \mathbf {B} \over \partial t}=\eta \nabla ^{2}\mathbf {B} +\nabla \times (\mathbf {v} \times \mathbf {B} ).}
1163:
cannot be applied. This means magnetic energy is dissipated to heat and other types of energy. The induction equation then reads
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More generally, the equation for the perfectly-conducting limit applies in regions of large spatial scale rather than infinite
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very large such that the diffusion term can be neglected. This limit is called "ideal-MHD" and its most important theorem is
907:
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1139:, the diffusive term overcomes the convective term. For example, in an electrically resistive fluid with large values of
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1056:{\displaystyle {\partial \mathbf {B} \over \partial t}=\nabla \times (\mathbf {v} \times \mathbf {B} ).}
397:/m, the displacement current is smaller than the free current by a factor of 10 for frequencies below 2
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1069:, used to explain the magnetic field evolution in the astrophysical environments such as
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989:, the first term in the induction equation vanishes. This is equivalent to a very large
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772:
752:
1219:{\displaystyle {\partial \mathbf {B} \over \partial t}=\eta \nabla ^{2}\mathbf {B} .}
1342:
222:{\displaystyle \nabla \times \mathbf {E} =-{\partial \mathbf {B} \over \partial t},}
1276:
which is the time scale for the dissipation of magnetic energy over a length scale
139:
900:
The ratio of these quantities, which is a dimensionless parameter, is called the
463:{\displaystyle \mathbf {E} +\mathbf {v} \times \mathbf {B} =\mathbf {J} /\sigma }
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Maxwell's equations describing the
Faraday's and Ampere's laws read:
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271:{\displaystyle \nabla \times \mathbf {B} =\mu _{0}\mathbf {J} ,}
573:, yields the induction equation for an electrically resistive
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1159:, the magnetic field is diffused away very fast, and the
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746:, is often identified with the magnetic diffusivity).
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is the magnetic diffusivity (in the literature, the
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1228:It is common to define a dissipation time scale
138:of an electrically conductive fluid such as a
1341:(2nd ed.). Cham: Springer. p. 468.
954:
529:Combining these three equations, eliminating
1127:(also called the frozen-in flux theorem).
1065:This is taken to be a good approximation in
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106:Learn how and when to remove this message
749:If the fluid moves with a typical speed
945:{\displaystyle R_{m}={LV \over \eta }.}
707:{\displaystyle \eta =1/\mu _{0}\sigma }
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1269:{\displaystyle \tau _{d}=L^{2}/\eta }
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44:adding citations to reliable sources
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128:partial differential equation
566:{\displaystyle \mathbf {J} }
544:{\displaystyle \mathbf {E} }
488:{\displaystyle \mathbf {v} }
377:{\displaystyle \mathbf {J} }
320:{\displaystyle \mathbf {B} }
296:{\displaystyle \mathbf {E} }
150:, and plays a major role in
7:
1338:High-Energy-Density Physics
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769:and a typical length scale
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1119:), as this also makes the
1112:{\displaystyle \eta \to 0}
982:{\displaystyle \eta \to 0}
959:For a fluid with infinite
955:Perfectly-conducting limit
1137:magnetic Reynolds numbers
739:{\displaystyle 1/\sigma }
142:. It can be derived from
1121:magnetic Reynolds number
991:magnetic Reynolds number
902:magnetic Reynolds number
349:{\displaystyle \mu _{0}}
1335:Drake, R. Paul (2019).
514:{\displaystyle \sigma }
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166:Mathematical statement
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1152:{\displaystyle \eta }
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1091:electric conductivity
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120:magnetohydrodynamics
55:"Induction equation"
40:improve this article
1316:Maxwell's equations
358:vacuum permeability
144:Maxwell's equations
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124:induction equation
1348:978-3-319-67711-8
1289:{\displaystyle L}
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782:{\displaystyle L}
762:{\displaystyle V}
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130:that relates the
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1306:Alfvén's Theorem
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1161:Alfvén's Theorem
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1125:Alfvén's theorem
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1085:Convective limit
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1135:For very small
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410:current density
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497:velocity field
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406:electric field
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152:plasma physics
132:magnetic field
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1067:dynamo theory
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729:
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718:, defined as
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525:of the fluid.
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160:dynamo theory
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57: –
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51:Find sources:
45:
41:
35:
34:
29:This article
27:
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18:
17:
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899:
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528:
403:
401:10 Hz.
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156:astrophysics
123:
117:
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96:October 2023
93:
83:
76:
69:
62:
50:
38:Please help
33:verification
30:
1322:References
412:using the
66:newspapers
1264:η
1237:τ
1200:∇
1196:η
1184:∂
1174:∂
1147:η
1104:→
1101:η
1093:, (i.e.,
1040:×
1029:×
1026:∇
1014:∂
1004:∂
974:→
971:η
935:η
868:∼
857:×
846:×
843:∇
821:η
815:∼
801:∇
797:η
734:σ
702:σ
693:μ
678:η
645:×
634:×
631:∇
614:∇
610:η
598:∂
588:∂
509:σ
458:σ
437:×
414:Ohm's law
338:μ
252:μ
240:×
237:∇
208:∂
198:∂
192:−
181:×
178:∇
148:Ohm's law
1363:Category
1300:See also
1075:galaxies
393:10
136:velocity
789:, then
521:is the
495:is the
470:where
356:is the
278:where:
80:scholar
1345:
140:plasma
122:, the
82:
75:
68:
61:
53:
1071:stars
670:Here
575:fluid
126:is a
87:JSTOR
73:books
1343:ISBN
1077:and
904::
551:and
404:The
388:The
229:and
154:and
146:and
134:and
59:news
395:mho
118:In
42:by
1365::
1296:.
1081:.
1073:,
963:,
577::
416::
162:.
1351:.
1284:L
1260:/
1254:2
1250:L
1246:=
1241:d
1214:.
1210:B
1204:2
1193:=
1187:t
1178:B
1107:0
1051:.
1048:)
1044:B
1036:v
1032:(
1023:=
1017:t
1008:B
977:0
940:.
931:V
928:L
922:=
917:m
913:R
886:.
881:L
877:B
874:V
865:)
861:B
853:v
849:(
840:,
833:2
829:L
824:B
811:B
805:2
777:L
757:V
730:/
726:1
697:0
688:/
684:1
681:=
656:.
653:)
649:B
641:v
637:(
628:+
624:B
618:2
607:=
601:t
592:B
560:J
538:E
499:.
482:v
454:/
449:J
445:=
441:B
433:v
429:+
425:E
399:×
371:J
360:.
342:0
314:B
290:E
266:,
262:J
256:0
248:=
244:B
217:,
211:t
202:B
189:=
185:E
109:)
103:(
98:)
94:(
84:·
77:·
70:·
63:·
36:.
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