E&M, Fall '10

Homework 12

1. Write down Maxwell's equations in differential form. Now take the case of "statics" -- i.e., set all the time-derivatives equal to zero. One of the equations now reads curl(E) = 0. Argue that, by working in terms of the electrostatic potential (E = - grad V) instead of the electric field, this one of Maxwell's equations is satisfied *automatically*. What does the *other* Maxwell equation involving E look like when re-written in terms of V? Now, go through that same progression of thoughts with the *magnetic* field. Namely: one of Maxwell's equations says that div(B) = 0. Is there a way to introduce a "magnetic potential" so that this equation will be satisfied *automatically*? What kind of object (scalar field, vector field, what?) will this "magnetic potential" be? What will the other Maxwell equation involving B look like when re-written in terms of this "magnetic potential" -- i.e., what equation should this "magnetic potential" satisfy?

2. Consider waves on a string (with tension and mass/length such that the wave velocity is v). Suppose Alice on the left sends sinusoidal waves of amplitude A propagating down the string to the right, while at the same time Bob on the right is sending sinusoidal waves of amplitude A propagating down the string to the left. Write an expression for the time-dependent shape of the string, i.e., y(x,t). Hint: the real point here is to mathematically *simplify* the expression you'd write down based on the idea of superposition. Use, for example, the fact that sin(A+B) = cos(A)sin(B)+sin(A)cos(B).

3. A string of mass m = 8.0 grams and length L = 2.0 meters is strung up horizontally. Its tension, measured by a force probe attached to one end of the string, is T = 40 Newtons. If you pluck it such that it moves up and down in the middle with no nodes (i.e., a standing wave mode with a wavelength twice the length of the string), what will be its frequency of oscillation?

4. In class we showed that electric fields (in empty space) obey the wave equation. Show that magnetic fields (in empty space) also obey the wave equation, and argue that one can't have electric field waves without magnetic field waves, nor vice versa -- i.e., argue that there is really just one kind of wave: electro-magnetic waves.