Christmas comes early

You can skip one problem on the final exam, since it seems to be plenty time-consuming at 7 problems. Which one you skip is up to you, they all carry equal weight.

Homework (partial) Amnesty Day

Any late homework you get to me by 5pm this Wednesday will be graded at 75% of full points. Last chance.

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.

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.

Final Exam

Ok, a couple of days later than planned, but here it is. You probably will not like it. Do not delay in starting, it is not something you can bang out in a couple of hours over coffee.

I will post hints over the next days on various problems, with increasing helpfulness as the deadline gets closer. Feel free to ask for clarification if you aren't sure how to get started. They are not easy problems, but I think you can handle them. The final exam is due back to me by 5pm next Thursday, 10 Dec 2009.

You're allowed to use your textbook and notes (which includes posted solutions/notes from this page), and I would consider Wikipedia fair game, but random googling for answers is not. You will need to sign your exam, stating that you've played by the rules. A bit much, I know, but those are the breaks when you get a take-home exam ... anyway, with most of these problems you would not have much luck googling anyway.

Also, HW8 solutions are out. Exam III solutions should follow this weekend, since they might be of some utility for the final.

Also.