Actual hints (I)

I really think exams should be a learning experience. With normal exams, this should be so: you get to see the solutions and discuss. With finals, not so much. This is one more reason I like the take-home final: I can try to teach you a few last things, and I don't feel bad coaching you a little along the way, since most of the problems are brand-new for you. I'll be around campus until Thursday afternoon if you want to drop by. Anyway, some hints:

#1. Use the integral form of Faraday's law to get the first correction to the E field. Take a square contour which (looking from the sides) runs down the center, parallel to the plates, up the right side, and back to the center.

The original field E will have no contribution to the integral of E.dl around the line contour. The new contribution will. If the new contribution is due to time variation in B, you know its symmetry ... so all but one side of the square will give zero to the integral. Put another way, the flux of B only contributes to the new correction to the E field, so you can find the correction directly. After the exam, I'll tell you where I found this; brilliant discussion.

More massive hints to follow on this one later in the week; it is subtle.

#2. Build it out of rings. You know the field from a ring.

#3. If the network is infinite, one more element makes no difference at all. Terminate it at some arbitrary place, and the rest of the network continuing on can be represented by some Req. That Req has to be the same wherever you terminate, so pick some easy places: after just one instance of R1 and R2, and after none. The two have to give the same Req.

Next, imagine you're in the middle of the network somewhere. Now you can have a single R1 and R2 terminated on *both* sides by Req if it is an infinite network. Now you have a simple 4 resistor circuit; find the voltages. If the ratio holds for two arbitrary nodes like this, it holds for all.

#4. Download the final again so you get the correct equations without typos. Apply the curl equations for E & B in free space ... that's about it. Apply the divergence equations as a trivial sanity check. w/k should be the velocity of propagation, right? Energy density can be had from the field amplitudes.

#5. Just work it in one dimension until part c, it makes no difference really. Two dimensions if you like, one component of E is important, the other just gives a torque. For the last part, generalizing to three dimensions should not be too hard if you're careful.

#6. The chain rule thing is key: d(fg) = f*dg + df*g. Also note that at certain points you'll want to write v in terms of gamma (to simplify the final result) and gamma in terms of v (to do an integral).

#7. Force is the gradient of the potential energy (with a minus sign). Write the energy of the capacitor for fixed charge ...

#8. See previous post. You may neglect atmospheric refraction, as it is essentially the same at the top and bottom of the cherry picker.