Random HW3 hints

Massive hints for your imminent HW set:

HW2 number 8

For question 8, you will need to sum an infinite series. It is in fact a very famous and neat little result, which you can find here.

HW2 / today

This afternoon, from 2pm onward, I'll be in or around my Bevill office (room 2050) if you have homework questions. You can also email/text me if you like.

Some of the problems on HW2 are well known (i.e., you could potentially google solutions or hints), some I have asked before in either PH106 or PH126. Just throwing that out there.

Lastly, in general if a HW problem suggests a particular method of solution, but you think you have a better way to get the same result, you can use your own method if you like. For instance, on #2 of HW2, you might decide it is just easier to superimpose the fields of two lines an a semicircle, since the field from those objects are well-known results, rather than setting up the problem as suggested. In general, any consistent method is fine, solutions by any means necessary. This means writing code or numerical solutions are fair game in general [but you should turn in the code with the rest of your HW].

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.

Friday's exam

You're ready. Get some sleep.

If you don't believe me, and want to cram anyway, I'd spend some time on Ch. 7, sections 2&3 in Griffiths, and then probably review the sections on Maxwell's equations (sans vector potential).

You will be rewarded if you can quickly recognize what to do with Maxwell's equations when (for instance) given an E field. You will also be rewarded if you have subjugated div, grad, and curl in spherical coordinates (formulas given).

Finally, you will be rewarded with bonus points if you remember what I said about tensors on Wednesday. Specifically, conductivity tensors.

PS - If you are unsure what a question means, or how to go about it tomorrow, don't hesitate to ask. More than likely, I will be willing to clarify the problem a bit or give you a hint to get you started. Also, show and turn in all your work, even if you think it illegible or unimportant. Partial credit is key.

Magnetic Dipoles

This will come in handy next week when we get to the vector potential. For now, it contains the solution to one of your homework problems, albeit using a method we have not discussed yet ... still, it may help you set up the problem.

Coefficients of capacitance

Hard stuff to find online.

HW2 #9

If you do #9 by building a plate out of thin rods, you should get this:

E_z=4k\sigma \tan^{-1}\left[\frac{a^2}{2z\sqrt{2a^2+4z^2}}\right]

This can be shown to be equivalent to Griffiths' result, along with the scary arctan identity I posted earlier

\tan^{-1}{\left(\frac{2u}{u^2-1}\right)}=2\tan^{-1}{\left(\frac{1}{u}\right)} \pm n\pi

For instance, try

u^2 = 1 + \frac{a^2}{2z^2}

to make the identity more 'obvious'. Keep in mind the freedom to add or subtract pi from arctan, the boundary conditions (e.g., field is zero at r=infinity) will tell you whether to add or subtract or not.

Griffiths result is

E_z = 8k\sigma\left[\tan^{-1}\left(\sqrt{1+\frac{a^2}{2z^2}}\right)-\frac{\pi}{4}\right]

Problem 9 / HW 2

Problem 9 on homework 2 is the same as Griffiths problem 2.41, by the way. However, I think it is conceptually easier to tackle the problem by first finding the field from a short line charge, and then building a plate out of line charges. If you do this, you will need an obscure identity to recover the same form as Griffiths.

\tan^{-1}{\left(\frac{2z}{z^2-1}\right)}=2\tan^{-1}{\left(\frac{1}{z}\right)} \pm n\pi

Here n is an integer. Just saying ... if you solve the problem the way I demonstrate in class (which is, I think, conceptually easier and leads to the appropriate limits more easily), there is some work involved to check that is the same as Griffiths' result.

I'm sure you realized that you can't use Gauss' law by this point. The fields of a finite square plate have an icky symmetry to them, as does anything square-ish when you're dealing with radial fields.

Also, problem 10 is the nearly same as a PH106 problem I assigned last year. Excepting that the integrations involved are more painful.