Gauss's Law and conservative fields

[abhi3142]

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.

 

[ZealScience]

(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.

 

[abhi3142]

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?

 

[ZealScience]

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.

 

[pmacias]

I can confirm that not all conservative fields follow Gauss's Law and not all fields that follow Gauss's Law are necessarily conservative in nature. This is because the conditions for a field to be conservative and to follow Gauss's Law are different.

A conservative field is one in which the work done by the field on a particle depends only on the initial and final positions of the particle, and not on the path taken between them. This is equivalent to saying that the curl of the field is zero. On the other hand, Gauss's Law states that the flux of the electric field through a closed surface is equal to the enclosed charge divided by the permittivity of free space.

While it is true that for all laws of nature with a 1/R^n dependence, the field would be conservative, it is only when n=2 that it would also follow Gauss's Law. This is because for n=2, the field is proportional to the inverse square of the distance, which is the form required for Gauss's Law to hold.

Therefore, there can be fields that are conservative but do not follow Gauss's Law, and there can also be fields that follow Gauss's Law but are not conservative. An example of the latter is a non-conservative field created by a changing magnetic field, which follows Gauss's Law but does not satisfy the condition of being conservative.

As for rigorously proving these statements, they can be mathematically derived and verified through experiments and observations. However, it is important to note that in science, theories and laws are constantly being revised and refined as new evidence and understanding emerge. So while these statements hold true based on our current understanding, they may be subject to change in the future.