Does Gauss' Law Hold for Infinite Gaussian Surfaces?

[Ahmes]

Hi,
From what I understand the proof of Gauss' law applies only to finite surfaces.
Can anyone give an example of a charge distribution and an infinite Gaussian surface, where the total flux on it is not proportional to the enclosed charge?

Thanks!

 

[JustinLevy]

This is similar to what you asked for.
An example I heard was this:
Imagine the universe filled with a constant non-zero charge distribution.

By symmetry, we know the electric field is zero everywhere.

But any closed surface we draw, the enclosed charge is non-zero, but the surface integral is zero.

[my memory is rusty on the conclusion, someone please correct this if I am wrong]
Therefore yes, Gauss' law does depend on an assumption at infinity... it only applies to vector fields that vanish at infinity.


I believe what you asked for though "a charge distribution and an infinite Gaussian surface", will always work if the charge distribution is finite in extent so that the vector field vanishes at infinity. So with that one caveat, I believe infinite Gaussian surfaces are no problem.

 

[charng]

I can confirm that Gauss' law does indeed apply to finite surfaces. However, the concept of an infinite Gaussian surface can still be useful in certain situations. For example, in the case of a uniformly charged infinite plane, the electric field is constant and perpendicular to the surface, making it a useful application of Gauss' law. In this case, the total flux through the infinite Gaussian surface would be proportional to the enclosed charge.

However, as you have rightly pointed out, there are situations where the total flux on an infinite Gaussian surface may not be proportional to the enclosed charge. This can occur when the charge distribution is not symmetric or when the electric field is not constant. An example of this could be a point charge located at the center of an infinite conducting sphere, where the electric field is not constant and the total flux through an infinite Gaussian surface would not be proportional to the enclosed charge.

In conclusion, while Gauss' law is most commonly applied to finite surfaces, the concept of an infinite Gaussian surface can still be useful in certain cases. It allows us to simplify calculations and understand the behavior of electric fields in certain situations. However, it is important to keep in mind that it may not always hold true and other factors such as symmetry and the nature of the charge distribution must be taken into account.