Clarifications

There were a couple of typos on number four: the argument of the sin functions should have a + sign, not a - sign, and the B_z component should be divided by c. The posted version of the exam has been corrected.

Here are some clarifications I sent one of you by email, for all to see:

In the formulas for problem 1, what does the i term indicate? Also, I
think the little e is Euler's number, and just wanted to check and
make sure.

It is good to be certain before you start. The "e" is indeed Euler's number, the base of the natural logarithms. The "i" is the imaginary unit, so the formula for E is just writing a sine wave in complex exponential notation. Check the chapter in Griffiths on EM waves for similar notation & problems.

On problem 4, I'm not sure what the k and x stand for in the wave
equations. Also, I think you left out the units you wanted the energy
density in.

Here k is the wavevector, which relates to the spatial periodicity of the wave (k = 2pi/lambda). Omega is the angular frequency. You can basically treat them as constants. Since these are wave equations, you can guess that omega/k should give the velocity of the wave ... The energy density should be in joules per cubic meter, or energy per unit volume.

For problem 5, are we supposed to assume the dipole is in a specific
orientation for the first two parts? It seems as though you meant for
it to be perpendicular to the z axis. I'm assuming total force for
part (a) is not the same as net force, because that would depend on
the direction of the field. Also, I'm not sure to what distance the
distance d in part (b) refers.

First, the distance d is the separation of the two charges, I should have noted that.

You can assume that the dipole is along the z axis, so the two charges are sitting on the z axis with the origin at the center of the two charges. You can also let the E field be along the z axis if you like. It isn't perfectly general then, but close enough; just work the first parts of the problem as if it is one dimensional, and generalizing for the last part is not hard.

There need not necessarily be a net force for the first part ... For the second part, you are to pretend that the E field is slightly different at one charge than the next. For instance, say E is a little bit bigger at the positive charge sitting at z=(d/2) than at the negative charge at z=(-d/2). You could say then that the field at the negative charge is just Eo, and the field at the positive charge is roughly that plus the gradient of the field times the separation distance: E = Eo + (dE/dz)*(d). If the separation distance d is small enough, it is pretty good to approximate the variation in the field as a constant plus distance times a gradient.

For problem 8, I have a certain vision for how this date would ideally
play out, and want to check that it is appropriate. I imagine that
the couple starts on the level of the earth, then, at the moment the
top of the sun disappears over the horizon, the cherry picker lifts
them until the bottom of the sun is level with the horizon. Also,
should we take into account diffraction due to the atmosphere?

You are correct, they wait until the sun is level with the horizon and then rise up until they are again level with the sun & the horizon. You can ignore the atmospheric refraction, since we already did a problem on that. You can also assume that the cherry picker is sitting in a little hole, so they start out exactly at the ground level.

Ok, go ahead and look here if you like, but your solution should be a bit more thorough and elegant.

I didn't recognize this until I got into the problem. On problem 3,
to which nodes exactly does "successive nodes" refer? Is it nodes
across the top, from top to bottom, across the bottom, or something a
bit more specific?

If you move from left to right, it is probably easiest to say that a node starts at the left of an R1, and the next node starts after the R2 to its right. It doesn't matter too much, so long as you are consistent and move from left to right (along the top) as the figure is drawn. More hints on this one later. As an aside, this sort of circuit is useful for quick & dirty digital-analog conversion. Think about that: power of 2 ratio of voltages at every node ... just meant for binary.