Are magnetic fields 'conservative'

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Discussion Overview

The discussion revolves around the nature of magnetic fields and whether they can be classified as conservative fields. Participants explore the implications of magnetic fields on work done on charges, the definitions of conservative fields, and the role of scalar and vector potentials in this context.

Discussion Character

  • Debate/contested
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • Some participants assert that the work done by magnetic fields on charges is zero, suggesting a conservative nature, despite the curl of the magnetic field being non-zero in the presence of currents.
  • Others caution against labeling magnetic fields as conservative, noting that the integral of the magnetic field around a closed path is not zero if the path encircles a current.
  • A participant questions the concept of non-conservative forces as described in a Wikipedia article, seeking clarification on the degrees of freedom that might be neglected in the context of magnetic forces.
  • One participant mentions that a static magnetic field can have a scalar potential, which is multivalued and undefined at points with current, leading to a nuanced view of conservativeness under specific conditions.
  • Another participant argues that the scalar potential can simplify algebraic manipulations and is useful, particularly for permanent magnets, despite its multivalued nature.
  • There is a discussion about the comparison between scalar and vector potentials, with some participants suggesting that the multivalued nature of scalar potentials does not necessarily complicate their use compared to vector potentials.

Areas of Agreement / Disagreement

Participants express differing views on whether magnetic fields can be considered conservative. While some argue for a conditional conservativeness under specific circumstances, others maintain that the presence of currents and the resulting non-zero curl complicate this classification. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Limitations include the dependence on the definitions of conservative fields and the specific conditions under which magnetic fields are analyzed. The discussion also highlights the complexity introduced by multivalued functions in the context of scalar potentials.

pardesi
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surely \vec{\nabla} \times \vec{B} \neq 0 in general
but the work done by magnetic field on any charge is 0 hence is independent of the path taken
So can we call such a field conservative
 
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Be careful in using "surely" and "general". Curl B is not zero in the presence of current.
The usual textbook definition of a "conservative field" is that
\oint{\bf B}\cdot{\bf dr}=0, which is not true if the path circles any current.
 
want to follow up a question. i read in wiki and it says that non-conservative force is due to negligence of certain degrees of freedom. Is this a well accepted concept? if yes, then what sort of degrees are neglected in case of magnetic force so that it becomes non-conserving? thankyou :smile:

http://en.wikipedia.org/wiki/Conservative_force
 
luben said:
want to follow up a question. i read in wiki and it says that non-conservative force is due to negligence of certain degrees of freedom. Is this a well accepted concept? if yes, then what sort of degrees are neglected in case of magnetic force so that it becomes non-conserving? thankyou :smile:

http://en.wikipedia.org/wiki/Conservative_force

I'm thinking wiki is just wrong.
 
pardesi said:
surely \vec{\nabla} \times \vec{B} \neq 0 in general
but the work done by magnetic field on any charge is 0 hence is independent of the path taken
So can we call such a field conservative

Indeed, if you have a static magnetic field, you can have a scalar potential (search for magnetic scalar potential on Google). This potential is multivalued, and undefined at points with current. Thus actually a magnetic field is kinda conservative --- if the field is static and the region you're interested in has no currents. However, it's not a very useful point of view, and doesn't actually simplify the algebra --- multivalued functions aren't very friendly.
 
genneth said:
However, it's not a very useful point of view, and doesn't actually simplify the algebra --- multivalued functions aren't very friendly.
That is Griffith's, somewhat naive, point of view, but the scalar potential does actually simplify the algebra, and can be quite useful, especially for permanent magnets.
The multi-valued part is no real problem. Are you also going to exclude logarithms and roots?
 
pam said:
That is Griffith's, somewhat naive, point of view, but the scalar potential does actually simplify the algebra, and can be quite useful, especially for permanent magnets.
The multi-valued part is no real problem. Are you also going to exclude logarithms and roots?

I believe the idea isn't that the scalar potential isn't simpler vs direct manipulation of fields, but rather that multivalued scalar potential isn't simpler than vector potential, plus the latter is generally true with no restrictions.
 

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