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My physics teacher claimed in class today that for a 3-dimensional cube with discrete point charges, it matters not where any of the charges are distributed when finding the electric flux through any one of the six faces of the cube. This seemed... off to me, but I couldn't come up with any counterexamples on the fly, so I went with it.

I understand Gauss' Law pretty well (at least I think I do), and while it makes total sense to me that the sum of the flux through the the six faces is constant wherever the charges are located, it doesn't make sense to me to say that the flux through each face is equal to every other face regardless of how the charges are distributed.

Mathematically, what she was saying was that for a 3-dimensional cube with arbitrary charge distribution, it must be true that the flux through any particular face is $$ \Phi_{face}=\frac{Q_{enclosed}}{\epsilon_0*6}$$

and that $\Phi_{face}$ is constant regardless of which face is selected.

Is this true and how would one show it?

Edit: one fairly easy counterexample is putting a point charge in the corner of a cube, and then the flux is $\frac{Q_{enclosed}}{\epsilon_0*24}$ for some sides and 0 for others. However, this isn't a true counterexample since the charge isn't inside the cube, just on it.

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My physics teacher claimed in class today that for a 3-dimensional cube with discrete point charges, it matters not where any of the charges are distributed when finding the electric flux through any one of the six faces of the cube.

If you are accurately stating your teacher's statement, I don't believe she is correct. The total net outward electric flux for all the surfaces cube will be the same, regardless of where the charge is located within the cube. But the flux associated with a specific face of the cube will depend on the distribution of the charge within the cube.

See the diagrams below. Note that with the positive enclosed charge located at the middle of the cube in the diagram to the right, the number of flux lines through each surface is the same. But with the charge located towards one side of the cube in the diagram to the left, the number of flux lines are greater through that side of the cube than the others. Nevertheless, the net flux out of the entire cube is the same for both diagrams.

Hope this helps.

enter image description here

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It can't be correct. Imagine one charge approaching the center of a face. In the limit that it reaches the face, half the flux is going out that near face. Using symmetry, the original statement means half the flux is going out of the opposite face. That leaves zero flux escaping from the four other faces, but that is manifestly wrong.

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