31:
180:
640:
This statement does not mean that the field lines of a solenoidal field must be closed, neither that they cannot begin or end. For a detailed discussion of the subject, see J. Slepian: "Lines of Force in
Electric and Magnetic Fields", American Journal of Physics, vol. 19, pp. 87-90, 1951, and L.
376:
427:
225:
590:
312:
160:
90:
321:
253:
530:
174:
gives an equivalent integral definition of a solenoidal field; namely that for any closed surface, the net total flux through the surface must be zero:
612:
281:
668:
131:
393:
185:
545:
469:
658:
315:
622:
596:
480:
641:
Zilberti: "The
Misconception of Closed Magnetic Flux Lines", IEEE Magnetics Letters, vol. 8, art. 1306005, 2017.
686:
37:
691:
434:
259:
112:
476:
233:
502:
171:
664:
266:
and a solenoidal field. The condition of zero divergence is satisfied whenever a vector field
379:
271:
162:
A common way of expressing this property is to say that the field has no sources or sinks.
30:
654:
617:
536:
96:
493:
462:
452:, which is σωληνοειδές (sōlēnoeidēs) meaning pipe-shaped, from σωλην (sōlēn) or pipe.
371:{\displaystyle \nabla \cdot \mathbf {v} =\nabla \cdot (\nabla \times \mathbf {A} )=0.}
680:
263:
118:
125:
486:
429:(Strictly speaking, this holds subject to certain technical conditions on
17:
449:
262:
states that any vector field can be expressed as the sum of an
318:(as can be shown, for example, using Cartesian coordinates):
660:
Vectors, tensors, and the basic equations of fluid mechanics
274:
component, because the definition of the vector potential
422:{\displaystyle \mathbf {v} =\nabla \times \mathbf {A} .}
220:{\displaystyle \;\;\mathbf {v} \cdot \,d\mathbf {S} =0,}
585:{\textstyle {\frac {\partial \rho _{e}}{\partial t}}=0}
307:{\displaystyle \mathbf {v} =\nabla \times \mathbf {A} }
548:
505:
396:
324:
284:
236:
188:
134:
40:
584:
524:
421:
370:
306:
247:
219:
154:
84:
255:is the outward normal to each surface element.
8:
155:{\displaystyle \nabla \cdot \mathbf {v} =0.}
190:
189:
613:Longitudinal and transverse vector fields
559:
549:
547:
510:
504:
411:
397:
395:
354:
331:
323:
299:
285:
283:
240:
235:
203:
199:
191:
187:
141:
133:
41:
39:
34:An example of a solenoidal vector field,
85:{\displaystyle \mathbf {v} (x,y)=(y,-x)}
29:
633:
542:where the charge density is unvarying,
260:fundamental theorem of vector calculus
448:has its origin in the Greek word for
7:
567:
552:
405:
348:
339:
325:
293:
135:
25:
128:zero at all points in the field:
27:Vector field with zero divergence
412:
398:
386:there exists a vector potential
355:
332:
300:
286:
241:
204:
192:
178:
142:
42:
382:also holds: for any solenoidal
359:
345:
79:
64:
58:
46:
1:
314:automatically results in the
248:{\displaystyle d\mathbf {S} }
109:divergence-free vector field
525:{\displaystyle \rho _{e}=0}
105:incompressible vector field
708:
623:Conservative vector field
597:magnetic vector potential
481:incompressible fluid flow
470:Gauss's law for magnetism
435:Helmholtz decomposition
114:transverse vector field
101:solenoidal vector field
586:
526:
423:
372:
308:
249:
221:
156:
92:
86:
587:
527:
424:
373:
309:
250:
222:
157:
87:
33:
546:
503:
499:in neutral regions (
394:
322:
282:
234:
186:
132:
38:
582:
522:
419:
368:
304:
245:
217:
172:divergence theorem
152:
103:(also known as an
93:
82:
574:
16:(Redirected from
699:
673:
655:Aris, Rutherford
642:
638:
602:in Coulomb gauge
591:
589:
588:
583:
575:
573:
565:
564:
563:
550:
531:
529:
528:
523:
515:
514:
428:
426:
425:
420:
415:
401:
377:
375:
374:
369:
358:
335:
313:
311:
310:
305:
303:
289:
272:vector potential
254:
252:
251:
246:
244:
227:
226:
224:
223:
218:
207:
195:
182:
181:
161:
159:
158:
153:
145:
91:
89:
88:
83:
45:
21:
707:
706:
702:
701:
700:
698:
697:
696:
687:Vector calculus
677:
676:
671:
653:
650:
645:
639:
635:
631:
618:Stream function
609:
566:
555:
551:
544:
543:
537:current density
506:
501:
500:
458:
443:
392:
391:
320:
319:
280:
279:
232:
231:
228:
184:
183:
179:
177:
168:
130:
129:
97:vector calculus
36:
35:
28:
23:
22:
15:
12:
11:
5:
705:
703:
695:
694:
692:Fluid dynamics
689:
679:
678:
675:
674:
669:
649:
646:
644:
643:
632:
630:
627:
626:
625:
620:
615:
608:
605:
604:
603:
593:
581:
578:
572:
569:
562:
558:
554:
533:
521:
518:
513:
509:
494:electric field
490:
483:
473:
463:magnetic field
457:
454:
442:
439:
418:
414:
410:
407:
404:
400:
367:
364:
361:
357:
353:
350:
347:
344:
341:
338:
334:
330:
327:
302:
298:
295:
292:
288:
243:
239:
216:
213:
210:
206:
202:
198:
194:
176:
167:
164:
151:
148:
144:
140:
137:
81:
78:
75:
72:
69:
66:
63:
60:
57:
54:
51:
48:
44:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
704:
693:
690:
688:
685:
684:
682:
672:
670:0-486-66110-5
666:
662:
661:
656:
652:
651:
647:
637:
634:
628:
624:
621:
619:
616:
614:
611:
610:
606:
601:
598:
594:
579:
576:
570:
560:
556:
541:
538:
534:
519:
516:
511:
507:
498:
495:
491:
488:
484:
482:
478:
474:
471:
467:
464:
460:
459:
455:
453:
451:
447:
440:
438:
436:
432:
416:
408:
402:
389:
385:
381:
365:
362:
351:
342:
336:
328:
317:
296:
290:
277:
273:
269:
265:
261:
256:
237:
214:
211:
208:
200:
196:
175:
173:
165:
163:
149:
146:
138:
127:
123:
120:
116:
115:
110:
106:
102:
98:
76:
73:
70:
67:
61:
55:
52:
49:
32:
19:
659:
636:
599:
539:
496:
479:field of an
465:
445:
444:
430:
387:
383:
275:
267:
264:irrotational
257:
229:
169:
121:
119:vector field
113:
108:
104:
100:
94:
270:has only a
681:Categories
648:References
446:Solenoidal
390:such that
166:Properties
126:divergence
18:Solenoidal
663:, Dover,
568:∂
557:ρ
553:∂
508:ρ
487:vorticity
441:Etymology
409:×
406:∇
352:×
349:∇
343:⋅
340:∇
329:⋅
326:∇
297:×
294:∇
197:⋅
139:⋅
136:∇
74:−
657:(1989),
607:See also
477:velocity
456:Examples
450:solenoid
380:converse
316:identity
117:) is a
111:, or a
667:
433:, see
230:where
629:Notes
489:field
468:(see
124:with
665:ISBN
595:The
535:The
492:The
485:The
475:The
461:The
378:The
278:as:
258:The
170:The
107:, a
437:.)
95:In
683::
532:);
366:0.
150:0.
99:a
600:A
592:.
580:0
577:=
571:t
561:e
540:J
520:0
517:=
512:e
497:E
472:)
466:B
431:v
417:.
413:A
403:=
399:v
388:A
384:v
363:=
360:)
356:A
346:(
337:=
333:v
301:A
291:=
287:v
276:A
268:v
242:S
238:d
215:,
212:0
209:=
205:S
201:d
193:v
147:=
143:v
122:v
80:)
77:x
71:,
68:y
65:(
62:=
59:)
56:y
53:,
50:x
47:(
43:v
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.
