Simple Nature - Light and Matter

Discussion Questions

g/Discussion question A-D.

A One question that might naturally occur to you about Gauss’s law

is what happens for charge that is exactly on the surface — should it be

counted toward the enclosed charge, or not? If charges can be perfect,

infinitesimal points, then this could be a physically meaningful question.

Suppose we approach this question by way of a limit: start with chargeq

spread out over a sphere of finite size, and then make the size of the

sphere approach zero. The figure shows a uniformly charged sphere

that’s exactly half-way in and half-way out of the cubical Gaussian sur-

face. What is the flux through the cube, compared to what it would be if

the charge was entirely enclosed? (There are at least three ways to find

this flux: by direct integration, by Gauss’s law, or by the additivity of flux

by region.)

B The dipole is completely enclosed in the cube. What does Gauss’s

law say about the flux through the cube? If you imagine the dipole’s field

pattern, can you verify that this makes sense?

C The wire passes in through one side of the cube and out through

the other. If the current through the wire is increasing, then the wire will

act like an inductor, and there will be a voltage difference between its

ends. (The inductance will be relatively small, since the wire isn’t coiled

up, and the∆Vwill therefore also be fairly small, but still not zero.) The

∆Vimplies the existence of electric fields, and yet Gauss’s law says the

flux must be zero, since there is no charge inside the cube. Why isn’t

Gauss’s law violated?

D The charge has been loitering near the edge of the cube, but is

then suddenly hit with a mallet, causing it to fly off toward the left side

of the cube. We haven’t yet discussed in detail how disturbances in the

electric and magnetic fields ripple outward through space, but it turns out

that they do so at the speed of light. (In fact, that’s what light is: ripples

in the electric and magnetic fields.) Because the charge is closer to the

left side of the cube, the change in the electric field occurs there before

646 Chapter 10 Fields