§ {Activity: Flux through a Cube} asks you to compute the flux of the electric field through the faces of a cube. If you use technology to examine the integrand, you will discover that the integrand is largest near the center of the face. Why? Yes, the center of the face is closest to the charge; is that the only reason? Don't forget that the direction of the electric field also affects the integral, not merely its magnitude.

When using technology to evaluate these integrals, it is worth noticing that some of the integrals can be evaluated exactly, but others only numerically — and that this process requires choosing values for the various constants.

Moving the charge away from the center of the cube changes the flux through each face, but not the total flux — at least so long as you keep the charge inside the cube. If the charge is outside the cube, the flux through each face will be nonzero, but now the total flux will be zero.

So what happens when the charge is located at a corner of the cube? If you imagine the charge as a small but finite sphere, then it is easy to see that an eighth of the charge is “inside” the cube, so that the flux is $1/8$ of the flux due to a charge located at the center. Equivalently, how much of the electric field due to a charge at the corner of a cube points through the faces of the cube? Precisely $1/8$. Yet another way to describe this result is to consider electric field lines, not all of which pass through the cube.