Gauss's Law and conservative fields

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Discussion Overview

The discussion revolves around the relationship between conservative fields and Gauss's Law, exploring whether all conservative fields adhere to Gauss's Law and whether all fields that follow Gauss's Law are conservative. The scope includes theoretical considerations and mathematical proofs related to electric, magnetic, and gravitational fields.

Discussion Character

  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • Some participants propose that while all fields of the form 1/R^n are conservative, they only follow Gauss's Law when n=2, suggesting that a conservative field may not necessarily adhere to Gauss's Law.
  • Others question the definition of Gauss's Law, indicating that it may refer to different contexts, such as the potential following the Poisson equation, and that not all scalar potentials conform to Gauss's Law.
  • A participant mentions that magnetic fields, as described by Ampere's Law, do not exhibit conservative behavior, even though they follow a form of Gauss's Law.
  • There is a suggestion that a mathematical proof could be developed to explore the conditions under which a field that follows Gauss's Law might be conservative, with the idea that time dependence could be a necessary restriction.
  • Another participant notes that for a field to be conservative and satisfy Gauss's Law, it should be expressible as the gradient of a scalar potential, although they have not yet explored this idea in detail.

Areas of Agreement / Disagreement

Participants express differing views on the relationship between conservative fields and Gauss's Law, with no consensus reached on whether all conservative fields follow Gauss's Law or vice versa. The discussion remains unresolved with multiple competing perspectives.

Contextual Notes

Participants highlight the complexity of definitions and contexts for Gauss's Law, including distinctions between electric and magnetic fields, and the implications of time dependence on conservativeness. There are unresolved mathematical steps regarding the proof of the proposed relationships.

abhi3142
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1) Do all conservative fields follow Gauss's Law? For all laws of the nature 1/R^n the field would be conservative but would follow Gauss's Law only when n=2. So a field may be conservative but would not follow Gauss's law?

2) Do all field that follow Gauss's Law conservative in nature? Are there field distributions in nature which would follow Gauss's Law and not be conservative?

Can the above statements be rigorously proved.
 
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(1)What do you mean by Gauss Law? Is it the potential that follows Poisson equation? If so, it is basically a Green function problem, and since 1/R potential is the fundamental solution to the Green function problem in 3-D, other values of n which are not the solution in 3-D would not form Poisson equation (for instance, for n = 1, the potential is apparently that of a 2-D problem).

Conversely, the conservative field is defined by a scalar potential; as aforementioned, not all scalar potentials follow Poisson equation, they are conservative but do not follow Gauss Law.

(2)For the second one, the 'Gauss Law' is once again confusing. We all know that there is a magnetic Gauss Law which is always valid since magnetic monopoles have not been proven to exit. However, magnetic field is apparently not conservative when there is current or changing electric field as given by Ampere's Law.
 
Sry about that, I was referring to gauss's law for electric/magnetic/ gravitational field-http://en.m.wikipedia.org/wiki/Gauss's_law. So there are laws that might be conservative that do not follow gauss's law and also the other way around. I could not think of examples of a field that might follow gauss's law and is not conservative. So was thinking about a way to mathematically prove this fact. That is given a model or imagined field, area integral of which is a scalar, it would always turn out to be conservative or are there more restrictions I would need to pose on this field to make it conservative. One restriction can be that It should not be time dependent. Would that be enough to go about a mathematical proof?
 
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Well, the article in your link says thatit is the Gauss Law for elctric field. If you take Gauss Law for magnetic field into consideration, apparently magnetic field is not conservative due to Ampere's Law.

For the conservative field that satisfy Gauss law, I have suggested that the field should be fully described in terms of the gradient of a scalar potential. I have not gotten time to try that yet; perhaps, you could consider that.
 

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