Reference material / upcoming material

One quick thing, before you stop reading: we need to have another exam soon. We will postpone it until next week, even though I had nominally scheduled it for this coming Friday. Plenty of time, and I'd rather we get through a bit more material first.

Some time ago I wrote a short math guide for intro PH courses. You might find it useful, though it isn't as complete as I would like. Actually, you might find it more useful when you get to 300-level PH courses that use vector calculus fluently.

Also, I came across a great electronics text online from Texas Instruments geared specifically toward op-amps. We've covered enough material now that the 'Review' should make sense now, so if you want to do a little reading on what we're doing now, this is a good start. My own favorite reference is The Art of Electronics by Horowitz and Hill, which is in the Rogers library. Our coverage of transistors, signals, and op-amps has been based largely on their treatment. Great book if you can find a decently priced copy, highly recommended if you plan to continue fiddling with circuits.

For the next few classes, we'll continue our discussion of magnetism. Our focus is going to be calculating the magnetic field due to various distributions of currents and the effects of magnetic fields on moving charges and currents, without much regard to *why* moving charges create a magnetic field for the moment. We'll also cover permanent magnets a bit. Later, after we've gotten through induction and time-varying E and B fields, we'll return to the subject of why magnetic fields arise in the first place, which will require a crash-course in special relativity. For that, you might find my PH102 notes useful. Light on math by construction, but they cover special relativity, deriving B from E, and at least touch on most of the topics in PH126.

A quick survey

You all have a pretty diverse background, and I want to make sure that I'm not getting you in over your heads. It would be helpful for planning purposes if I knew a little about your math, programming, and electronics experience. Can you answer the following (very short!) survey?

The survey is totally anonymous, and the results will not be seen by anyone but me. It is really just to get a grasp of the baseline skills you're coming in with so I can tailor course topics (and especially the labs) to what most of you are comfortable with. Please answer as honestly as possible to make this as useful as possible to me; if you really don't feel like you understand one of the topics, don't select it (even if you think you're supposed to understand it).

The survey is here if you wouldn't mind ...

[If you're not in the class, and just saw this on Twitter/Facebook/etc, please answer "no" to the last question.]

Notes for today's lecture

It won't be on the exam, but if you are curious to go through today's material on fluid dynamics at a bit more leisurely pace, I've written up some notes.

UPDATE: I did some cleaning up of the notes. I added quite a bit on fluid rotation (like what curve describes the shape of water going through a drain), and also added a few examples. Specifically, I showed how the Hall effect (current flow in the presence of a magnetic field) requires a tensor conductivity, and treated the case of steady flow through a cylindrical pipe. The latter is a rare example of an analytical solution to the Navier-Stokes equations (given a good number of reasonable assumptions), and quite practical.

The updated notes are in the same location linked above.

No, it is still not going to be on the final, it is just cool. ;-)

Relativity

If you find relativity interesting, and you've had a bit of math, you'll probably find these lecture notes very nice. In fact, they've been turned into a book by Prof. Carroll, which has been well-received (it is what we use for our grad relativity course).

There is also a "non-nonsense" introduction to general relativity, the first bits of which should be familiar. Don't be scared by the tensors later on, the barrier is mostly the notation.

Anyway: good book, free preview online. Can't beat that.

Online Matrix Calculator

Neat stuff. Very handy for those circuit problems ...

Homework 2 #10

Just for fun, here's a picture of the electric field surrounding the two rods for HW2.10. The lines correspond to contours of constant electric field. I guess you can figure out where I placed the rods ...


Yes, this is what I do when I have some free time.

It could be worse.

A truly pathalogical function. Continuous everywhere and differentiable nowhere.

A nice quote:

While it's not very common that badly-behaved functions arise in physics, there are functions which at least don't always remember to say please and thank you. They have to be gently corrected, but they're good at heart. The mathematicians are the ones who have to deal with the truly shady functions, the ones who form prison gangs and don't play by the rules and obey the laws. Or theorems.

I have an image of rogue mathematical symbols ganging up on me now. Great.

He's no Wolfram, but then again, who is?

Look, a derivative calculator!

Also, Dr. Wolfram & Co. have more tricks up their collective sleeves. No identity too obscure, no function too pathalogical.

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.

Wolfram Integrator

For all your integration needs. If this thing can't figure it out, you are probably doing something wrong ;-)

HW1 #4c

You need the magnitude of r and dr/dt squared. For this, you can use the dot product - the magnitude of a vector squared is the vector dotted into itself.

|\vec{r}|^2 = \vec{r}\cdot\vec{r}=\left(\vec{a}\,\cos{\omega t} + \vec{b}\,\sin{\omega t}\right)\cdot \left(\vec{a}\,\cos{\omega t} + \vec{b}\,\sin{\omega t}\right) \\
|\vec{r}|^2 = |\vec{a}|^2\,\cos^2{\omega t} + 2\vec{a}\cdot\vec{b} \,\sin{\omega t}\cos{\omega t} + |\vec{b}|^2\,\sin^2{\omega t}

You have terms with "a dot b" in them. Just leave them be. Find dr/dt, repeat ... and those "cross terms' will drop out anyway. The rest should be easy.

MathWorld

It is the awesome. If you can't remember some bit of math, or need to pick up something new quickly, it should arguably be your first stop (the other argument being in favor of wikipedia, which in my opinion can be too terse).

Math stuff so far

So ... after 'rebooting' a bit on the math background today, how are things looking? Of the things we covered today, are there topics that still seem very mysterious?

A good gauge is probably to look at the first four homework problems. If you basically know how to do them, but they seem tedious, things are OK. If you aren't sure how to even start one or more of them, we might need to review a bit more. I'll work out some of the first homework problems in detail on Wednesday's class (but not quite all the way, since they are due at the end of the day).

Unless you have specific requests/thoughts, I planned to move on to 'vector derivatives' on Wednesday, but not yet get in to line integrals and so on. Div, Grad, Curl and so forth, continuing a bit more carefully until we have that down.

Drop a comment to let me know what you think.

(Also: please stop me in the lectures if things start to seem mysterious to you. Often, you are not alone if you feel this way, and often, it is because I skipped something I shouldn't have. You're doing everyone a favor if you ask me to clarify something, rather than letting it go.)

Slides from the first lecture / schedule

I've uploaded the slides [~6Mb PDF] that Prof. Harrell will be using for the first lecture, which is mainly a course overview and a math review.

Prof. Harrell will (probably) go through all of this material, I'll pick up where he left off on Monday 24 Aug and we'll go further into derivatives of vector fields. Wednesday 26 Aug, we'll go through integration over vector fields (line integrals) and start doing some actual E&M.

Don't worry too much if the math seems frightening during the first few lectures. After our brief tour of vector calculus, the math will get less scary again. The quick overview in the beginning is meant to give you an idea of what we'll need during the semester, and as these topics come up in real situations, we will review them in more detail at a slower pace. Any math that is not part of a prerequisite course I will cover in class, and I will try hard to bridge any gaps between our textbook and what is covered in, e.g., Cal II.

A short math guide

Clearly, a work in progress, but meant to be a quick reference for things we'll need this semester.