Flux through a Box

Since the previous surface is not closed, we cannot use the divergence theorem to check our answer (although you may be able to guess the correct answer if you have a good understanding of the geometry of the electric field). In this problem you will compute the flux through a square box of edge length 2 that is centered about the charge at the origin.

The key to this problem is to recognize that the total flux my be broken into six pieces: the flux through

the right side of the box, {x=1, -1 < y,z < 1};
the left side of the box, {x=-1, -1 < y,z < 1};
the top of the box, {z=1, -1 < x,y < 1};
the bottom of the box, {z=-1, -1 < x,y < 1};
the front of the box, {y=-1, -1 < x,z < 1};
the back of the box, {y=1, -1 < x,z < 1}.

Question #3

Compute the flux across the right side of our box. Hint: this computation is identical to the computation involving flux across the infinite plane, except now we are only integrating over a piece of the plane. You should get a very simple expression for the flux across this region.

Question #4

Compute the flux across the left side of our box. Hint: your parametrization and unit normal will be different than for the previous problem. The flux through this surface is a number involving Pi, so you'll want to use evalf to get a numerical approximation to the flux.

Question #5

Compute the flux across the top of our box. Again, your parametrization and unit normal will be different than for the previous problem, but the basic idea remains the same. Again, you'll want to use evalf to get a numerical approximation to the flux.

Question #6

This problem has certain symmetries. Argue that these symmetries imply that we do not need to compute the flux across the three remaining sides, because the flux is identical to the flux across the top of the box. Hint: You must show three equalities. There are really two main points to argue, and each requires thought. The first argument is that the flux across the top and bottom is identical. For this, look closely at the dot product of the electric field with the normal vector. The second argument involves permuting the y and z variables.

Question #7

Use the previous problems to compute the total flux through our box. Compare this answer to the flux through the plane x=1. Can you explain the relationship between your two answers?
Use the Maple command diverge(EField,[x,y,z]); to compute the divergence of the electric field. (Yes, it's pretty messy.)
Use Maple to calculate the triple integral of the divergence over the rectangle x=-1..1, y=-1..1, z=-1..1, and compare your answer to Question #6. Use the divergence theorem to explain why your answers agree, or, if your answers do not agree, explain why the divergence theorem doesn't apply in this case.

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Previous: 3D Flux through a Plane

Robert E. Thurman<thurman@geom.umn.edu>

Document Created: Fri Mar 31 1995
Last modified: Mon Apr 14 11:14:23 1997