Zero Electric Flux Through a Closed Surface

[student14]

I'm relearning basic electricity concepts and I can't find an answer to a situation I've thought up.

Imagine a cube with no enclosed charge and an electric field through it parallel to two of its faces. Gauss's law says that the flux should be zero because there is no enclosed charge.

Every example of this that I have seen goes something like this: The four perpendicular faces have zero flux due to the sides being perpendicular to the field, and the flux from the other two sides cancel each other out. I.e if E is the strength of the electric field, A is the area of the sides of the cube, and \Phi is the total flux, then

\Phi =EA+(−EA)=0.

But what if the strength of the electric field is not the same at both sides? Then there won't be any cancelling out and there will be nonzero flux, which is contrary to Gauss's law?

 

[cwasdqwe]

I guess in that case you would get a non zero net electric flux, therefore the net electric charge enclosed in your closed surface wouldn't be zero. That is, if the strength of E is different in both sides, then maybe the presumption of no enclosed charge breaks down.

 

[jaydnul]

A field can only be terminated ON a charge, so ANY field line that enters the cube must exit the cube if there is no charge inside for the field line to terminate on.

 

[student14]

Hello and thank you for the responses.

A field can only be terminated ON a charge, so ANY field line that enters the cube must exit the cube if there is no charge inside for the field line to terminate on.

I've seen this as the standard explanation and I am not quite satisfied with it, due to the fact that it boils the problem down to single field lines. By definition of flux, the contribution of a single field line is zero because it enters the surface at a point, which has area zero.

I guess in that case you would get a non zero net electric flux, therefore the net electric charge enclosed in your closed surface wouldn't be zero. That is, if the strength of E is different in both sides, then maybe the presumption of no enclosed charge breaks down.

This is what Gauss's law tells us. I'm under the assumption that my field cannot be a real physical electric field, but don't really have a way to verify it at the moment. My main problem is that in every case of the cube example that I have seen, the instructor assumes that the field strength is equal at both sides, which is a very specific case. My "counterexample" is a very simple situation, and I'm wondering why exactly it is breaking down.

I wrote up a better explanation of the situation: We have a cube and an electric field that runs perpendicular to four of its faces and parallel to the other two faces, which I'll call face 1 and face 2. The electric field strength at face 1 is the constant value of E_1 and the strength at face 2 is the constant value of E_2. Let A denote the area of each face.

Thus, the flux through face one is \Phi_1 = -E_1 A (assuming outward facing orientation for the faces) and the flux through face two is \Phi_2 = E_2 A. The total electric flux is

\Phi = \Phi_1 + \Phi_2 = -E_1 A + E_2 A = ( E_2 - E_1 )A.

If E_1 \not= E_2, then we have nonzero flux.

 

[nasu]

If the field is not uniform then the field lines are not parallel and you will have nonzero flux through more than two faces.
Parallel field lines means uniform field.

 

[BruceW]

Imagine a cube with no enclosed charge and an electric field through it parallel to two of its faces. Gauss's law says that the flux should be zero because there is no enclosed charge.

Every example of this that I have seen goes something like this: The four perpendicular faces have zero flux due to the sides being perpendicular to the field, and the flux from the other two sides cancel each other out.

Gauss' law should not be an obvious thing. It is an experimentally observed law. So don't worry if an intuitive explanation is not fully convincing to you. I prefer to think of it like the charges are a source (or sink) for the electric field. So, if there are no enclosed charges, we get no net flux. I fully realize that this is not a convincing logical argument for why Gauss' law should be true. It is not meant to be. It is just meant to serve as a way to intuitively think of the problem.

 

[jaydnul]

If the field is not uniform then the field lines are not parallel and you will have nonzero flux through more than two faces.
Parallel field lines means uniform field.

This