Sufficient Condition for Existence of Vector Potential

Click For Summary

Discussion Overview

The discussion centers around the sufficient condition for the existence of a vector potential in the context of vector fields, particularly in relation to magnetostatics. Participants explore the necessary and sufficient conditions for such existence, touching on mathematical concepts and implications.

Discussion Character

  • Technical explanation, Debate/contested

Main Points Raised

  • Some participants propose that a divergence-free field is a sufficient condition for the existence of a vector potential.
  • Others argue that while being divergence-free is a necessary condition, it may not be sufficient in all contexts.
  • A later reply suggests that in Euclidean space, the condition may be both necessary and sufficient, particularly in the context of magnetostatics.
  • One participant provides a mathematical example involving a magnetic field and the application of Poincare's Lemma to illustrate the relationship between closed forms and vector potentials.

Areas of Agreement / Disagreement

Participants express disagreement regarding whether being divergence-free is sufficient for the existence of a vector potential, with some asserting it is necessary but not sufficient, while others claim it is sufficient in specific contexts.

Contextual Notes

The discussion includes references to mathematical concepts such as closed forms and Poincare's Lemma, which may introduce limitations based on the assumptions of the space being considered.

ManuelF
Messages
7
Reaction score
0
Hello!
Can you tell me the sufficient condition for the existence of the vector potential?
Thank you very much!
 
Physics news on Phys.org
ManuelF said:
Hello!
Can you tell me the sufficient condition for the existence of the vector potential?
Thank you very much!

when a field is divergence free it has a vector potential.
 
That is a necessary condition, but I do not think is enough!
Are you sure?
Thank you.
 
ManuelF said:
That is a necessary condition, but I do not think is enough!
Are you sure?
Thank you.

It is necessary and sufficient in Euclidean space.

For instance, in magnetostatics a magnetic field always has a vector potential.

If the field is B = (u,v,w) consider the 2 form udy^dz -vdx^dz + wdx^dy

Since B has zero divergence, the 2 form is closed. By Poincare's Lemma it is therefore the exterior derivative of a 1 form adx + bdy + cdx. The curl of (a,b,c) equals B.
 
Thank you very much!
 

Similar threads

Undergrad How to prove that a scalar potential exists if the curl of the vector point function is zero?
  • · Replies 2 ·
Replies
2
Views
2K
Undergrad Vector field and Helmholtz Theorem
  • · Replies 20 ·
Replies
20
Views
4K
Undergrad Curl operator for time-varying vector
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad How to picture the vector potential?
  • · Replies 3 ·
Replies
3
Views
795
Undergrad Find potential integrating on segments parallel to axes
  • · Replies 2 ·
Replies
2
Views
1K
Undergrad Is Cauchy Reimann condition sufficient for complex differentiability?
  • · Replies 4 ·
Replies
4
Views
2K
Undergrad Magnetic vector potential of infinite straight wire
  • · Replies 2 ·
Replies
2
Views
3K
Graduate Vector Potential and Zero Divergence
  • · Replies 3 ·
Replies
3
Views
7K
Undergrad How to use geometrical symmetries -- general advice? (Vector potential)
  • · Replies 2 ·
Replies
2
Views
2K
Graduate Introduction to Electric Vector Potential and Its Applications
  • · Replies 17 ·
Replies
17
Views
4K