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.

What we covered today / what comes next

Today, what I basically derived was the Drude model of conductivity (or from an alternate viewpoint, the complex dielectric function). Griffiths covers essentially the same material in Ch. 9, treating the problem from a dielectric-centric viewpoint. The Feynman Lectures (vol II) also does a great job of covering the same material, in much the same way that we started today (i.e., from harmonically-oscillating charges). I would suggest reading Feynman, then Griffiths if you are curious. Follow that with Jackson Ch. 7 (link below) if you're really excited.

Here are a couple links you might find useful, if you are interested in going through what we did today a less brutal pace:

• Summary of the Drude model
• Ch. 1 of Solymar & Walsh does an excellent intro job (awesome book,I have a copy)
• Jackson Ch. 7 is fairly readable (when compared to the bulk of the book ...)
• Feynman Lectures II.32 (must read; dielectric-centric)
• Brief Derivation (fast-paced, but good)

Anyway: now we know why metals are shiny, and insulators are mostly transparent.

This brings up a question: we have one more recitation (Fri) which will be devoted to the RFID project, and one more lecture (Mon). What do you want to hear for your last lecture in PH126?

Anything you want, within reason and physics-related (if tangentially), I'll do my best. Leave your suggestions in the comments.

Next homework

Here's the exam you just took. It is also your next homework, due 20 Nov 2009.

Recall that you have an outstanding homework at the moment. It is OK if you turn that in on Monday.

Graduate research competition!

Upcoming event: see what our grad students are up to, and what you could get involved in.

Exam formula sheet

Here it is. Subject to some proofreading ...

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.

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. ;-)

Exam III

As you are probably aware, exam III is this Friday. There will be five problems, you must solve any two. The topics are

• ac circuits
• relativity
• induction
• Maxwell's equations / EM waves

Obviously, one of these sections will have two problems, the others a single problem. Here's what you might want to study/reread before the exam:

• ac circuits - my PH102 notes
• relativity - PH102 notes, Griffiths 12.1 (all) and 12.3.1-2
• induction - PH102 notes, Griffiths 7.2
• Maxwell - Griffiths 7.3, 8.1, 9.2.1

The corresponding example problems in Griffiths and end-of-chapter problems in my PH102 notes are particularly worth reading through.

This is assuming that you have already read the relevant homework solutions on these topics, which are also helpful. The exam problems will not be as difficult as the usual homework problems, however.

HW8

Finally, it is out. Relatively short, due at the end of the week.

Answers to the last quiz

Here you go.

HW 6 & 7 solutions

Homework 6 and 7 solutions are now complete.

Lab Today

Today, we'll do a lab on resonant LCR circuits, which should help out with our ongoing project.

Slides from today's lecture

You can find the slides Prof. Harrell used today (and many more he did not) here if you are interested.

Monday, we'll finish off magnetic induction ... so finish reading the appropriate chapter in Griffiths over the weekend if you would.

Tomorrow's recitation

Tomorrow, owing to an unavoidable funding-related meeting, I will not be able to make the recitation. Instead, Prof. Harrell will give a lecture about magnetism in real materials - where does permanent magnetism originate from, and why do magnets stick to your fridge? It will be mostly qualitative, but highly practical and interesting stuff.

And, yes, it might show up on the next exam or final, if you need another reason to show up ;-)

HW 7.1 hint

Check this. The first problem is an awkward one, but valuable for precisely that reason. The first time you see it, you scratch your head wondering why the ring would rotate, but once you understand why, you've learned something important.

When it rains it pours: HW6 solutions

Problem 6 does not have a full solution, but based on the general velocity transformation we went over in class you probably got it ...

Anyway: HW6 solutions are out.

Homework 5 solutions

Very late ... but here they are. You'll get your graded HW5 back tomorrow.

Tomorrow, we'll continue with magnetic induction, with a heavy emphasis on the homework problems. Friday, Prof. Harrell will give a lecture on magnetic materials. Monday, we'll finish up induction and Maxwell's equations so we can start ac circuits on Wednesday.

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.

HW 7 is out

Here it is. Eight problems in total, with a couple that should be pretty quick.

Relativistic time dilation

... how the last minute of a football quarter seems to take half an hour watching it on TV, in spite of what the on-field clock says. (The second quarter will not seem to end ...)

Homework

Back to serious stuff. On number 9, remembering your calculus is a big help. For instance, note

\frac{dv_x^{\prime}}{dt^{\prime}} = \frac{dv_x^{\prime}/dt}{dt^{\prime}/dt}

and use the Lorentz transformations along with velocity addition. I'll spell this out more tomorrow in recitation.

For number 6, the x component of the velocity is just what you think it is. However: the y component of the velocity in the second reference frame would be

u_y^{\prime}=\frac{dy^{\prime}}{dt^{\prime}}

Since

y=y^{\prime}

the numerator is trivial. However, there is still time dilation, so you'll need to use the Lorentz transformation to relate dt and dt'. Again, remember the calculus trick above, and the main point is this: the velocity along the direction of relative motion follows the addition formula we derived, but along the orthogonal direction, there is still a transformation because while distance is uncontracted, time is still dilated.

We'll go over the rest tomorrow, but you will find many of the other questions in my PH102 notes or previous PH102 homework sets. I'll give some hints on where to look ... but start with the problems at the end of Ch. 1.

Randomness

Since we're about to finish up relativity, I've been thinking a lot about what I think is the 'correct' way to cover mechanics, E&M, and relativity. In doing some reading, I ran across this quote, which made me think even more that our usual approach is severely lacking:

"The influence of the crucial Michelson-Morley experiment on my own efforts has been rather indirect. I learned of it through H.A. Lorentz's decisive investigations of the electrodynamics of moving bodies (1895) with which I was acquainted before developing the special theory of relativity . . . What led me more or less directly to the special theory of relativity was the conviction that the electromotive force acting on a body moving in a magnetic field was nothing else than an electric field. - Albert Einstein

Just something I find interesting: despite what you might hear in intro courses or popular accounts, E&M played a big role in inspiring relativity, it was not merely the speed of light and the aether.

Problem is, the history is in a way conceptually out of order, and harder to teach (IMHO). My evidence being how relativity is often presented in intro-physics sequences: as a total non sequitur, just sort of shoved in there. Moreover, in spite of the preserved historical ordering of topics, the motivation given usually starts out with the Michelson-Morley experiment and the aether, and quietly ignores EM forces in different reference frames. Thus, we keep the historical ordering, but throw out crucial parts of the original (and exceedingly insightful) motivation!

I prefer to follow the Mechanics -> Relativity -> Electromagnetism ordering, which I guess I've made obvious now. Relativity is hard conceptually, but I find students have a harder time with electromagnetism at first, particularly magnetism. Having relativity under your belt at least makes the magnetic field seem less arbitrary, which is reassuring I think. Introducing relativity after mechanics and E&M, while historically accurate, sometimes makes it seem like an ugly hack, which it wasn't at all. If you are going to do relativity after E&M anyway, why not cover the E&M aspect too? Purcell's book does a wonderful job.

On the other hand, doing relativity after mechanics is harder to motivate sometimes, and one has to resort to strange little thought experiments to find anything wrong with Newtonian physics. Thinking about relativity right after a mechanics course, though, has your brain in the right mode and the kinematics fresh in your mind.

Of course, I suppose it is just difficult either way, reality is a harsh mistress. And Poincare probably deserves more credit than he gets.

A note from the Russian club

The Russian Club will be hosting Dr. Anthony Vanchu, Director of the Johnson Space Center Language Education Center. He will be giving a lecture entitled "Teaching NASA Astronauts to Speak Russian, or How I Came to Love the FGB" on Thursday, October 22, at 6:30 pm in ten Hoor room 125. The Russian Club would like to invite you and your students to attend this lecture and encourages you to spread the word of this upcoming opportunity.

-Jonathan Williams
Vice President
UA Russian Club

Friday

Friday, we'll worry about energy & momentum in relativity, and if there is time, work out some reasonably general transformations of the E&B fields between reference frames. Basically, finish reading the first chapter of the ph102 notes, and you'll be ready for that. See here for an overview of much of what I did today with E&B, along with some additional notes on radiation & accelerated charges.

You might also find Lecture 12 here useful. For that matter, the rest of those notes are great too. If there is a model for how I've been trying to plan the course thusfar (if it is not apparent, I am planning things to a degree!), it would be 8.022 at MIT.

We'll make sure to spend some time on the homework problems that are still nagging you by then too ...

Also, Drew inadvertently reminded me about something: every time I make a crack about engineers, you should remind me that my undergrad degree is in Materials Science & Engineering ;-)

Tomorrow

Our Arduino boards for the project have arrived. We can start playing with them tomorrow. If you want to bring your laptop to make things easier, that might be good, otherwise, we'll use the PCs in the modern physics laboratory, which are not locked down by security software ...

Next week, we'll cover magnetic induction, which will explain the basics of how RFID works and allow you to get started more seriously on the project.

Tomorrow, we'll work on deriving magnetic fields from electrostatics + special relativity, and review the special relativity we covered on Monday. Friday, we'll finish of what we need of relativity by talking about relativistic energy, momentum, and force.

This week

Owing to popular demand, this week we'll cover relativity. This brings up two key points:

(1) Your next homework is out, and covers relativity. All problems are due this coming Friday.
(2) You'll want to read Ch. 1 of these notes, which cover relativity at the level we require. I'll try to have printed copies of that chapter for you tomorrow.

Monday's lecture should bring us all the way through time dilation and length contraction, while Wednesday's lecture will cover Lorentz transformations, spacetime diagrams, and energy & momentum.

Friday, we'll derive E from B and apply what we've learned to tie E&M together. For that, you should read section 6.1.2 in the notes linked above, which derives E from B for the special case of a point charge moving parallel to a current-carrying wire at constant velocity.

Misc

First, I ordered two of these for us to use in the RFID project. We'll probably have to bring in our own laptops to play with them, though, since the lab PCs are heavily locked down.

Second, if you have a preference for covering relativity, or just moving forward, please express it before Monday. If a quorum decides for relativity, we'll start that on Monday.

Third, there is really a new homework set coming out, due a week from today, but its content depends a little bit on what you choose to do next ...

Midterm Grades

Reminder ... your midterm grade is just the average of your two exam scores. I am not including homework, quizzes, or labs.

Quick Poll

Looking at the rest of the semester's schedule, I'd like to give you a (small) choice in how to proceed.

We could probably cover special relativity, quick and dirty, in two lectures. This would tie together E & B very nicely, and fill in a nagging gap in our curriculum thus far. Plus, it is just cool.

In order to squeeze this in, some things would have to be left for you to mostly read on your own, likely geometric optics. The geometric optics we cover is pretty simple, I have notes on the subject, and we would still spend a recitation period for Q&A to make sure you 'got it.'

So here's the question: would you be interested in taking a detour to cover relativity, at the expense of a bit more reading, or shall we stay the course? One argument in favor is that we can derive B from E and make things less mysterious, something you probably will not see again as an undergraduate. One argument against is that there would be (relatively easy) final exam material that we would not cover extensively in class.

Leave a comment with your preference, if you have one ... you have until the weekend to decide collectively. If you choose not to cover relativity, you will see it in PH253, though possibly without the connection to electricity and magnetism.

Warm-up project ...

If you read the last links, you'll realize that RFID seems really complicated. It is, and we probably shouldn't be messing with it in a 100-level class ;-) However, given that you are all exceedingly clever, I think we'll be OK.

As a sort of warm-up project, while we're figuring out how RFID workst, we'll first consider how shoplifting tags work. Not, and I cannot stress this enough, how to defeat them, but how they are implemented.

Two common types of anti-shoplifting measures, implemented in DVDs and books for example, essentially use magnetic induction and resonant circuits, which we'll get to next week. If you can figure out how these work, the basic idea behind RFID is not so far off. Another topic we'll touch on along the way to RFID is wireless power transmission, or how to charge your phone without cables. Ostensibly less potential for misuse, similar physics.

The main point here is that I think these things are worth understanding, highly relevant, and involve some nice applied physics. We will not delve into the nefarious uses of what we learn, just as we did not really discuss practical weaponry in mechanics. Learning enough to figure out what's going on around us is a Good Thing, learning how to use that knowledge is another.

Vonnegut had an interesting take on this in Timequake (see the top of the page), which I thought made a good point if a bit too extreme. We'll focus on physics, and leave the philosophizing to more qualified departments.

Anyway: just fair warning, I'm not going to help you figure out how to read other people's IDs, but I will help you figure out that the relevant technology is not in fact magic, but merely clever, Clarke's 3rd law notwithstanding.

Wednesday's class

Wednesday, we'll cover magnetism in a bit more depth. Specifically, we'll look at some tricks for how to solve for the field due to an arbitrarily-shaped wire with the Biot-Savart law. A good reference is this paper (you can only download the paper from on campus), which shows a cute trick for calculating the field from any wire whose geometry you can express in polar coordinates.

We'll also delve deeper into Ampere's law, and figure out how to find the field from solenoids and current sheets, which will let us derive some boundary conditions on the magnetic field at current sheets, similar to what we did for the electric field near sheets of charge.

Finally, we'll look at the most general equations we have for magnetostatics (i.e., only steady currents) and contrast the current situation to electrostatics.

Once we're done with the lecture part of class, we'll start in on the project for the rest of the semester: figuring out how to read RFID. There will be many sub-projects: coding, building circuitry, antenna design, and more. It will not be easy, but I think it will be a lot of fun, illustrate how one must be careful with this technology (morally speaking), and require all of you to pool your diverse expertise to work on a large project. Tomorrow, your task will be to figure out (1) what is RFID anyway, (2) what basic physics is involved in it, and (3) come up with more specific information-gathering missions and delegate them.

If all goes well, at the end of the semester you will know how to read my campus ID while it is still in my wallet. Of course, we'll have to be very careful about what we do with the knowledge we gain: knowing is one thing, but doing is another. There are rules.

Online Matrix Calculator

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

Exam II

Here's the exam you took today. Note that two of the magnetism problems are example problems from Griffiths (Ex. 5.9), the third is from Purcell. Two of the circuit problems are from Purcell, the third is based on a Serway problem (ph106 textbook).

I hope to have solutions out on Wednesday, and you will get your graded exams back during Wednesday's class if all goes well. So far, it seems the results will be very good. The exam was much easier than the homework, and your performance seems to be commensurate ...

Exam II

Seriously:

• field due to a bunch of straight or circular wires (e.g., #8 here)
• dc circuits with batteries and resistors (e.g., #9 here)

Partial HW4 solution

Here you are.

I will try to finish up a bit more of this tonight yet ...

UPDATE: only number 3 is missing a solution now.

Monday's exam

The exam will be low-key, and you should have plenty of time. Here's the basic format:

• six questions total
• three questions on circuits (dc; resistors and batteries only)
• three questions on the magnetic field from wires (straight or circular; superposition)
• answer any three of the six

I'll provide a full formula sheet, you're free to bring in one sheet of your own paper just as last time.

I think you can probably solve three problems and be done in less than 90 minutes, some of you certainly in about 60 minutes. They are not hard problems compared to the homework, and a few of them are drawn from PH102 homework in fact.

If you're cramming now, I would suggest reviewing the solution to the field from a finite segment of current-carrying wire, and the battery charging problem from Hw4. HW4 partial solutions will be out in a couple of minutes.

HW5

Quick google search turned this up.

Better: see homework 12 solutions here.

I don't think my method of solving the problem in class was that great, I'll try to post some quick notes on the dipole problem this afternoon.

Homework

Maybe I'm just loopy from preparing my tenure dossier, but here are some thoughts on homework.

1) By way of Drew: "Problems worthy / of attack / prove their worth / by fighting back."
2) I showed some grad students and faculty your homework, and they now fear you. You will be rocking PH331 and ECE340 when the time comes, because you're doing their homework already ;-)

Problem 1 HW 5

Take the spacing of the coils d to be the same as their radius R to make things easier. Strictly speaking, this is the Helmholtz arrangement.

Also, see here for clarification.

Latest Homework

Some of your problems are here.

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.

Power

Ahem.

HW5 is out

Here you go. First problems due this Wednesday.

Some notes on electrical measurements

This is an unfinished document that is part of another project - the start of some notes on how to perform electrical measurements in general, and specifically on samples of real, live materials. After our next meeting, it might be of interest.

Basically, the more interesting part at the end shows you how to calculate the resistivity (or conductivity) of a conducting material from experimental data, the so-called 'four point probe' technique. Moreover, you can figure it out for conductors of various interesting shapes, like thin films, using very general symmetry-related arguments. We'll cover the necessary background in Friday's recitation.

If you're studying or planning to study anything materials- or device-related, you will see the four-point probe technique again. It is not hugely difficult, but not commonly covered in any depth, and usually just taken on faith. So, when you do see the four-point probe expressions again, you can smile and know that they are not, in fact, magic.

And one more ...

Another is here. Be sure you know how to do the problems if you use these hints ... such problems will likely reappear at inconvenient times.

Friday's homework

The answers to one of your problems is here.

Friday's recitation

Tomorrow, we'll consider some of the microscopic physics behind conduction in metals, which will lead us to the general version Ohm's law among other things. I will have a handout in some form for this, since I'd like to do a bit more than is in the textbook.

In the remaining time (I hope our discussion will take about 30-35min ...) we'll go over the homework problems due on Friday.

Also: next week we'll start magnetism, so please read sections 5.1 and 5.2 in Griffiths before Monday. I'll also explain how the op-amps we used work during one of the lab sessions.

dc circuits slides / lab circuits

You can find all that stuff here:

http://faculty.mint.ua.edu/~pleclair/ph126/

Go to the "Media" directory for the circuits slides I've been using, and go into the "Labs" directory for images of the circuits we're building.

From time to time, other goodies will be deposited there as well. (I drew all the pictures and wrote all the text, so consider all this stuff to be freely distributable.)

Wednesday's circuit ...

We'll add a little amplifier to our photodiode to boost the signal. If everything works out, we should be able to pick out the blinking LED from a meter away or better without any additional cleverness. Increased cleverness can boost the range to several meters, which we'll worry about next week.

Here's the circuit. Basically, the voltage applied to the photodiode will lead to a current when light is incident. This current is converted into a voltage and amplified by the little triangle in the diagram :-) Explanations will be in order ... and it might take more than one class period to get it going (both on the board and in your head), but it will make sense shortly.


Other details: the voltage "Vcc" will need to be about 12V, so you will need an additional power supply.

(If you have taken a real circuits class, you'll see some problems with this thing ... but it does work, and we want to start reasonably simple!)

HW 4 ... more information

Problem 1: the power in a circuit element is in general current times voltage. Using Ohm's law, for a single resistor we can write this as

P=I^2R

The total power in the circuit is the sum of the power in the two resistors,

P=I_1^2R_1+I_2^2R_2

Using the conservation of charge equation, you can put power in terms of the current in a single resistor. Minimize.

Problem 2: now the resistors are in series, the internal resistance r and the external "load" resistance R. You can easily figure out the current through both resistors in this case, and again find the power in each resistor using the formula above. Maximize with respect to R.

Problem 3: see previous post; we'll do some more examples of Thevenin equivalents tomorrow.

Problem 4: break the cone up into tiny disks. If the current is running along the axis of the disk, the resistance of a tiny disk of radius r and thickness dx is

dR = \rho\,dx/\pi r^2

The radius is position-dependent for a cone, you might write it as

r=a+(b-a)x/l

in this case. Integrate over all such disks with resistance dR to find the total R. We'll set up a similar problem in class tomorrow.

Thevenin equivalents

The wikipedia has a nice page on this, including an example suspiciously like your homework. That is not by accident I guess.

More hints on Wednesday's problems will follow tonight or tomorrow afternoon.

Today's circuit slides / oral exam

Here's what I presented today, along with some stuff we'll get to next time. Powerpoint format, I'll try and make a PDF later today just in case.

Also: good work on the circuit today. I was very happy that everyone got it to work within the allotted time! Wednesday, we'll try to add an amplifier to the output stage (i.e., the voltage on the resistor you probed with the scope) to make the thing more sensitive.

Lastly, if you want to schedule an oral exam to improve your Exam I score, here are some time blocks that are good for me. I can meet you in Gallalee or Bevill, whichever is easier, but I'll list my preference.

Tues: 12-5 (Bevill preferred)
Wed: 1-2:30 (Gallalee preferred)
Thurs: 12-5 (Bevill preferred)
Fri: 10-11, 12-1 (Gallalee preferred)

If possible, I would like to do the oral exams this week, or this coming Monday at the latest.

Monday's lab & lecture

Monday, we'll start with basic dc circuit analysis. The lecture will mainly cover current, voltage, and resistance, but we will discuss quite a few general rules for circuit analysis.

After the lecture portion of the class, we'll finally get to that lab I wanted to do Wednesday, constructing an opto-isolator. Here's the circuit diagram you'll need (click for a larger version).
Monday's lecture will be mostly practical knowledge, though we will cover some general things like Kirchhoff's rules and Thevenin equivalents briefly.

Wednesday, we'll discuss more general aspects of electrical conduction, dissipation, and circuit networks. We'll also attempt to build an amplifier. As mentioned in the previous post, the reading for this week is from my ph102 notes, or any decent book on dc circuits you have handy.

HW 4 is out

Here you go. More than you ever wanted to know about circuits.

The topics for Wednesday's problems will all be covered in tomorrow's lecture. For this week, we are more or less abandoning Griffiths, since he doesn't cover circuits very much.

You might find my ph102 notes useful in the mean time. Chapter 3, section 3.6 covers capacitors, chapter 4 covers current & resistance, and chapter 5 covers basic dc circuits.

Update to HW3 solutions

See here.

I have written a relatively long solution to number 7, which tries to explain the idea behind the coefficients of capacitance. Hopefully this, combined with the short handout from the Purcell book, will make things clearer. If I have some time tonight, I will try to produce a few more examples of calculating coefficients of capacitance and make my own little handout.

I also updated the solution to number 8 (dipole) to try to make the derivation of the potential a bit more clear.

Lastly, HW4 is coming out this evening, and will have problems due on Wednesday and Friday of this week.

Exam I full solutions available.

Here they are. Let me know if you find any typos, or anything which could use further clarification.

You'll get your graded exams back on Monday.

Using analogies on the internet is like ...

I thought this was a nice read. My own personal data suggests it to be true.

Exam I and its (partial) solution

Find them here. I have solutions for problems 1 and 2, and the first two parts of 4.

I should have full solutions for 3 & 4 this evening, and you will get your exams back on Monday.

Do not try this at home

Wow. Relax and get some sleep before the exam.

Feynman lectures & videos

In case you missed it, this comment is worth a read, from the editor of the Feynman Lectures on Physics. Yay internet!

Also, you should really check out the video lectures by Feynman himself, made available to the public by a very generous gift from Bill Gates. These are lectures from 1964 at Cornell, and were not really available to anyone until Mr. Gates took it upon himself to make it possible. Many of the lectures tie in to the Feynman Lectures textbook, and are well worth checking out.

No matter how you feel about his software, the man has done physics a solid ...

More last minute thoughts on the exam

UPDATE: I made corrections & additions to the HW3 solutions today, might be worth a last-minute check.

So I've actually made it now, and I think you will be fine. I am going to do the problems again myself just to be sure the timing is somewhat reasonable. A few of random thoughts:

• As promised, the exam has 4 problems, you can solve any 2.
  
• There are no terrible integrals involved. Just polynomials and so forth, no weird arctans or anything.
• For most of the problems, there are at least two straightforward methods of attack. This is on purpose, with the hope that you'll see one of them quickly.
• Two problems will favor those of you that remember the basics of mechanics, two problems will favor those of you that like the math.
• The binomial approximation is just about the coolest thing ever. Know when you can get away with it.
• There are no numbers on the exam (except things like pi and 2, possibly a 3). A calculator is not useful unless it happens to do symbolic math.
• It is fine with me if you bring an complex calculator that does symbolic math. The basic rules are no pc's, no cellphone calculators, no PDAs that have wireless communication capabilities.

Anyway: if you've been able to follow the homework so far, the exam problems will seem almost laughably easy. The only real issue is time pressure: you get about 20 minutes per problem, so

• Use your time wisely, and watch the clock.
• Don't get stuck on anything - if you find yourself stuck, see if you can make a simplifying assumption to move on (this might entail some lost points, but many less than not finishing the problem at all), or pick another problem
• In spite of the time pressure, spend a few minutes reading and thinking about all the problems before starting. Make sure you really know what is being asked, and have a physical picture in your mind before moving on. If you can't at least see the 'flavor' of the answer, math is probably not going to help.
• Please, please make sure you read the problems you choose to answer a second time, there are multiple parts to some of the problems.
  

I'll be up another couple of hours if you have questions.

Draft of formula sheet

Here is a first draft of what I'll give you tomorrow with the exam. I have a bunch of other formulas to add yet, and some useful integrals, but I think you can get the idea of what will be there and what you don't need to include on your sheet.

I'll post again when I've finished a more complete draft, probably in a couple of hours.

UPDATE: a few changes to the formula sheet, this is a nearly final draft.

Exam on Friday

A few stray hints for the exam on Friday.

• It is only an hour exam, so there will be four problems. You solve any two of them. Heavy partial credit is possible.
• There will be a formula sheet with all the basics. You are additionally allowed to bring in one sheet of standard 8.5x11 inch paper with whatever you like on it.
• Your formula sheet will contain fundamental constants and integrals you will need.
• Understand the derivation of the (approximate) dipole potential ...
• Reading through my old PH106 homework solutions might be helpful, just for some examples of worked problems. HW 1-4 are relevant, mostly.
• If need be, the exam will be scaled ... so relax :-)

I'll be around Bevill most of the day tomorrow if you want to drop by the office with questions. If you're busy in classes tomorrow, I'll try to respond to email questions rapidly.

Feynman lectures

Just to make you aware, I should not have posted that link to the Feynman lectures earlier - it was a lapse in judgment and the post has been removed. That work is still under copyright, and still begin actively edited and maintained. Further, some of the royalties on those and related volumes actually go toward maintaining undergraduate lab equipment, which is not an easy thing to get money for. I wasn't really thinking when I posted that, and I'm sorry to the people who work on maintaining the Feynman lectures.

So: as I mentioned in class a few times, go buy the lectures. They are well worth having, and probably among the most-opened books in my office at home. Failing that, you'll find copies in our undergrad library (SPS room) and at the Rogers library.

Keep in mind nothing published before 1923 is likely to be out of copyright. And, yes, you can actually find good physics books from before that date which are available online.

HW 4

Let's hold off on any new homework until Friday. I'd rather not have you worrying about that with an exam coming up on Friday.

I'll put out HW4 on Friday, after the exam.

HW4 / HW 3 solutions updated

There is a HW4, I just haven't finished it yet. Coming soon, only two problems, due by the end of Friday.

They are still here, but the last problem has been updated a bit. I hope to type up the solution to the coefficients of capacitance problem tomorrow ...

Reading for next week

Next week, we'll start to discuss circuits, having learned enough about electrostatics to do something practical. The Griffiths book has next to nothing on circuits, so I'm supplementing this material with my own notes, which you can find here. [23Mb PDF]

For Monday (21 Sept), please read sections 3.6 & 3.7 along with the whole of Ch. 4, covering capacitors and electric current. It should be light reading - almost no math.

For Wednesday (23 Sept), please read Ch. 5, covering basic dc circuits.

(Chapters 2 and 3 might be worth skimming to make sure you've got the qualitative aspects of electrostatics down, if you have time. It is all stuff we covered already, but sometimes the math hides the qualitative understanding ...)

Wednesday's class & lab

Once again, I'm adjusting the schedule. I think our brief discussion on dielectrics last time is enough to get us by for a while, so we will spend most of the first part of class tomorrow just going over problems - particularly, the last homework set - in preparation for Friday's exam.

What this basically means is that we'll skip most of Ch. 4 for now, and move on to circuits next week. The things we really need from Ch. 4 to move forward is what we covered on Monday - roughly how dielectrics work microscopically, and the effect they have on electrostatic energy. After we have finished circuits, we'll touch back on dielectrics a bit again before moving on to magnetic forces & fields.

For the lab tomorrow, we'll learn how to use LEDs and photodetectors to make an opto-isolator, or from another viewpoint, the basic guts of a remote control. This will also illustrate some neat aspects of signal modulation, and lead us into amplifiers, triggers, and comparators next week.

If you don't know what most of those things are, we'll make an LED flash and pick it up from across the room on the scope, without any wires ;-)

Partial HW3 solutions

Here you go. I haven't typed up number 7 yet, and number 8 is incomplete. The rest are there, probably in more detail than you would like ...

If you didn't get every part of every problem, don't panic. They were very hard problems, and for the most part chosen to illustrate a certain point. If you mostly understand how these problems work out, but didn't quite get every last detail, I'll be happy with that (and partial credit will be generous).

For example, some of the problems (e.g., 5, 6, 8) have a relatively straightforward part followed by a much harder part. If you got the first part reasonably well, you will be fine on the exam, and I'd say you know what you are doing. The harder parts were mainly there to see what you could do, and make you think about things a little more. Again, if you didn't quite get everything, but found yourself thinking carefully about what the problem means, then mission accomplished.

Exam-wise, you can expect stripped-down versions of the easier parts of these problems. I can't stress enough: if you feel like you know what you are doing on the HW problems, but don't quite get them all the way worked out, you have no reason to panic.

On the other hand, panic should set in, a little, if you are not doing the homework (or at least reading the solutions after the fact).

SPS study session

"Society of Physics Students is hosting a homework help session Wednesday, September 16 at 6:00pm in 109 Gallalee. Anyone needing help with physics is welcome to attend."

Quiz 2 and its solution

Find them here.

Office hours today

Unexpected free time ... I'm here in Bevill from now until 4:45 if you need help on the homework.

Coefficients of capacitance

Hard stuff to find online.

Stray HW3 hints

(1) We set this one up in class. Put one charge at the origin, and write the distance from the second charge to the field point in terms of the distance to the first charge and the separation of the charges. E.g., if b is a vector pointing from one charge to the other,

\vec{r}^{\prime}= \vec{r}-\vec{b}\\
\hat{r} \equiv \frac{\vec{r}}{|\vec{r}|}\\
\vec{E}_1 = \frac{kq_1q_2}{r^2}\,\hat{r} = \frac{kq_1q_2\vec{r}}{r^3}\\
\vec{E}_2 = \frac{kq_1q_2}{r^{\prime\,}^2}\,\hat{r}^{\prime} = \frac{kq_1q_2 \left(\vec{r}-\vec{b}\right)}{|\vec{r}-\vec{b}|^3}

Do not forget that your volume element in spherical coordinates has an r-squared, or you will get a nasty integral.

(2) The field is nonzero only between the two spheres ... the result is the energy of a spherical capacitor, if you want to check your answer.

(3) Set this up in class. Solve for the potential at an arbitrary point, and either (a) show that at least 4 points from the same ellipse give the same potential, or (b) plug in the equation for an ellipse and show that it results in V=constant.

(4) See previous hint ...

(5) You can use the boundary conditions we derived for this one. From one side of a given plate to the other, the difference in the normal components of E has to give you the surface charge density.

E_{\mathrm{above}} - E_{\mathrm{below}} = \sigma/\epsilon_0

Above the top plate, or below the bottom plate, the field is zero. Thus, the field in between any two plates can be related to the surface charge on the upper or lower plate. How does that relate to the charge on a given side of the middle plate? Two parallel plates will have the same charge on the sides that are facing. Just like we talked about on Wednesday, if one plate has charge Q, it will induce -Q on the adjacent plate.

The only thing left is to realize that since the electric field must be constant inside (infinite plate, no spatial dependence!), the electric potential must just be

V = Ed = \sigma/\epsilon_0

This gets you the surface charge. Once you have that, you know the field. Once you know the field, you can use the method of problem 1 to find the total energy. Which can then be optimized. I bet you can guess the optimum spacing already though.

(6) Capacitance is just total charge divided by potential for a given conductor. If you know the potential for a prolate spheroid with a charge Q (problem 3!), C=Q/V. You can fancy that up by noting that the eccentricity (epsilon) is just d/a. After that, note

\ln{\left(\frac{1+x}{1-x}\right) = \ln{\left(1+x)} - \ln{\left(1-x)} \approx 2x

7. Here's the basic scheme for finding the coefficients. The self capacitance terms C_ii is just the capacitance of the system when the two elements are joined together, so they have the same charge and the potential of both is zero. The C_ij off-diagonal terms are found by setting the potential of one of the conductors to zero, and having a charge Q on the other. You'll need the formula for the capacitance of a spherical capacitor, which you can find in your text or online.

We'll go over that in class on Friday, but here's what you should come up with:

C_{11}=\frac{ab}{k\left(b-a\right)} = -C_{21}=-C_{12} \\ 
C_{22}=\frac{b^2}{k\left(b-a\right)}

8. Textbook problem. Try checking in some textbooks :-) Feynman (Vol. II) does a great job.

Equipotentials of a charged rod

Here are three of them, just for fun.

Oblig

If this is relevant, let me know. Won't be a problem.

The Jewish high holiday season is upon us. Rosh Hashana, the celebration of the New Year, begins Friday evening, September 18, and continues through the end of the weekend, while Yom Kippur, the Day of Atonement, begins Sunday evening, September 27, and continues all day Monday the 28th. If at all possible, please give your Jewish students who are observing these holidays some consideration should the observance conflict with papers, exams, or other class assignments. Feel free to contact me if you have any questions. Much thanks in advance.

Cube of charge

Two things:

1) The MIT course analogous to ours is 8.022. Open courseware is awesome.
2) Problem 4 on your homework is the same as Purcell problem 2.30.

Actually, most of your problems so far have been from the Purcell book, as it turns out, which is also favored for 8.022 at MIT during certain semesters. Looking through the 8.022 content on the open courseware site is highly recommended. Typically very thorough and lucid solutions.

Not all their content is on the open courseware site yet, but it can be found. try googling "purcell 2.30 MIT 8.022" and look at the first couple of links.

Rescheduling

After looking at the semester's schedule a bit, I have decided to adjust the schedule. Mainly I would like to cover circuits and more 'practical' matters sooner rather than later. This is partly to give you a brief interlude from the hardcore vector calculus, and partly to facilitate labs getting more interesting sooner rather than later.

Next week, we'll discuss dipoles & dielectrics (Ch. 4 Griffiths). On Friday you'll have your first exam, which will primarily cover chapter 2 in Griffiths, with a bit on dielectrics from chapter 4. More on this as it gets closer; the exam will have 3-4 problems for you to solve, much easier than the homework.

After dielectrics, we'll take a week's interlude to cover ohm's law and (non-inductive) circuits. Mathematically it should be a welcome relief. Most of this will be taught from my own notes (which you will receive), and the focus will be on analyzing and designing basic circuits.

After that week-long 'break,' it is on to magnetism and induction for about two weeks. This will give us what we need to discuss general ac circuits (damped harmonic motion from PH125 will make a return).

Exam 2 will take place during the week leading up to mid-semester study break (the Monday before the break), and it will cover magnetism and circuits. Your midterm grades, due that Wednesday, will reflect exams 1 & 2.

This is just a re-shuffling of the previous schedule (delaying magnetism by a week and moving up circuits). If any more time is needed here or there, we will 'adjust' by shortening geometric optics. Given the significant time we will spend on EM waves, geometric optics will be all but a foregone conclusion by that point.

Reading this week

Wednesday, we'll mostly be talking about sections 2.5.1-2.5.3 in Griffiths (conductors), which is really just applying what we have learned already to a specific situation involving mobile charges. Hopefully this will not take a huge amount of time, and we can spend some time setting up the homework problems for this week.

Friday, we'll begin with capacitance and discuss the method of images briefly. Before then, have a look through sections 2.5.4 and 3.2.1-3.2.4.

Next week, it is on to chapter 4 for the most part - except for a bit on image charges (Friday) and dipole moments (this week's homework), we will skip the bulk of chapter 3. If you want to get a head-start, for Monday it would be good to read sections 4.1.1-4.1.4. These will make more sense after you finish the last of this week's homework problems on dipoles ...

HW2 solutions

They are done, though probably not without errors. (UPDATE: a few stray typos fixed.)

In particular, in problem 11 showing that Coulomb's law is recovered in the limit of small lengths of the rods is stupidly difficult. I think it can be done much more quickly (or at least in many fewer steps) than I took in the solutions, but I wanted to try and make it as clear as I could. Given that it is not trivial to take the small length limits, you will be given a wide berth on that part of the question ...

Now, for HW3: tomorrow we'll set up several of the problems in lecture, I will post some hints tonight. If you did HW2 in a reasonably general way (or at all, really), you will save yourself some work on HW3. In particular, you can re-use the solution for the electric potential due to a line charge and a number of geometric insights.

HW3 #3

Here's a journal article which covers question 3 on homework 3. It is a bit terse in the beginning, but it should be big help if you can get through it ... (link will only work on campus).

Updated HW2 solutions

Here you go. All but the last two, I'll try and post those solutions tomorrow.

HW 3 is out

UPDATE: some typos in the homework have been corrected. We'll go over quite a few of these on Wednesday. I'll be in Bevill most of tomorrow afternoon if you want some help, send me an email as a heads-up.

Here you go. The descriptions are much longer than usual, but at least a few of the problems are quite short once you see how to do them. The extra length is mainly additional hints on how to approach the problems.

The Wednesday problems are all things we covered already, if you remember that an equipotential surface is just a surface on which the electric potential is constant.

By the way, most of these are from the Purcell book, which you can find in the library. There are no solutions in the book, but it is a good read, and might give you some extra insight into these problems. Some hints to follow.

(Purcell, Edward M. "Electricity and Magnetism." Part of the Berkeley Physics Course. 2nd ed. Vol. 2. New York, NY: McGraw-Hill, 1984. ISBN: 9780070049086.)

Just one more I swear

I think you'll appreciate this more when we get to current.

Urinary Protocol Vulnerability / xkcd

Seriously. With maths.

Restoring old moon footage

Fascinating stuff. Apparently, incredibly high-resolution image data of the moon (taken by an orbiter in 1966 to survey possible Apollo landing sites) has been sitting in a barn in California for about three decades.

Related presentation by a project team member.

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]

Office hours today

If you're having trouble with homework, I'll be in Bevill from 3:30 onward today (until about 5:45).

If I'm not in my office, Room 228, try my lab, Room 180.

HW 2 partial solutions

Solutions to Wednesday's problems from HW2.

Nice article on general relativity

Link. Mostly accessible, I think, if you remember your mechanics.

UPDATE: Broken link fixed.

Grading / Quiz 1

I will have graded things back to you on Friday, sorry for the delay.

Also: quiz 1 and a solution.

Update: having graded the quiz, I'm just going to throw out the first question. Apparently I was a bastard for asking it ... we'll go over the quiz tomorrow in recitation.

Pig Flu / Aporkalypse Now

If, for some reason, you end up missing an exam due to the pig flu, I will weight the final proportionally more.

If you miss something else due to the apparent pig flue epidemic (e.g., homework, quiz, lab), you can either make it up (if done within a short time) or receive a 'bye' on that particular assignment.

I do not become ill while teaching, as it turns out. I did teach with nearly a dozen hour-old stitches in my right hand and a head full of painkillers, but illness is right out. So I'll be there.

Just sayin'. Apparently, this is a Big Deal. Wash your hands, go to the med center, procure notes, etc.

(I am not belittling the public health threat here. I am, in fact, mocking the general response to and hysteria surrounding the public health threat.)

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.

Question 5 / HW2

I misspoke in class today ...

We calculated the potential due to a semicircle of charge, that was OK. I forgot about the line segments ... they do not cancel each other out, since they are both positive, and potential is a scalar (so there is no 'opposing direction').

What you need to do is superimpose on the semicircle result the potential due to two line charges a distance r away (along the line axis). Since both line segments are the same, find the result for one line and double it ...

See, e.g., PH106, F08, HW3, Q5.

Since I misspoke in giving you a hint, I won't count off if you miss the line bits.

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.

Wednesday's lab

Wednesday, we'll do a lab designed to get you more familiarized with wiring and analyzing (qualitatively) simple circuits. We'll figure out how resistors work, and how to combine resistors and capacitors.

In a couple weeks time, after we've gone over some of the basic hardware and know-how you need to build circuits, we will begin a semester-long project on electronics. At the moment, my plan is to have you learn how to build simple amplifiers and oscillators, and finally make an FM radio transmitter/receiver by the end of the semester.

Must crawl before walking ... so the first labs will go a bit slowly until we've covered all the basics.

Impending quiz

Wednesday, there will be a short quiz. It will cover finding the electric field from continuous charge distributions. You will not have to do any nasty integrals (or even solve an integral), you will just have to 'set up' the expression for the electric field from a given charge distribution.

Reading 2.1-2.2 in Griffiths and glancing at the homework will be enough to get you by easily. It should take only 15min or so.

Wolfram Integrator

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

Reading for Wednesday

For Wednesday, make sure you have read Griffiths Chapter 2 through the end of section 2.4.

Today, we'll cover 2.1 and 2.2.

Lab today

Today, we'll start learning a bit about circuits. The way we'll start out is by trying to get a feeling for what current and voltage are, and how different components behave when you attempt to supply current or voltage to them.

The lab procedure will be mostly qualitative, and it is mainly designed to get you used to wiring things up and doing hands-on work with circuits.

You can find the files here. There are two parts: first, a brief introduction to the hardware and software you'll be using; second, a small procedure for the lab itself.

Homework 2 is out

Here you go. Probably best not to delay in getting started ... we'll go over a few of these tomorrow to get you started.

Additional reading that might help

You might also find Chapter 2 in my PH102 notes useful, for a more qualitative introduction to electric forces and fields.

HW1 solutions

Here you go. Sources for most of the problems are indicated as well. Let me know if you find any typos/errors.

Homework 1 problem 9

Reminder: you do not have to do problem #9 on HW1. Full solutions to HW1 will be out tomorrow.

Homework 2 will come out tonight or tomorrow, I'll post it here. It will be less tedious, and at least raise the possibility of cleverness saving you some work.

For Monday, please read sections 2.1-2.2 in Griffiths. Study the problems in those sections carefully (this should be easier now that you have your little present from this morning). Warning, the solutions guide can be quite terse.

Looking at my old PH106 content might also be clever. For now, you might want to look at the first HW set, it is very much like what we started on today. (My solutions are sometimes much more lengthy than they need to be, I was trying to be as explicit as possible.)

Friday recitation

I think we've had enough math for now. We'll cover line integrals once we actually need them in a serious way ... not for a little while at least. Instead, we'll start on electric fields and forces tomorrow, the beginning of Ch. 2 in Griffiths.

I will go over HW 1 problem 9, though, so you know how to go about it.

HW1 partial solutions

Here are the solutions to the first four problems. Let me know if you find any typos ...

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).

Reading for the next days

By Friday, it would be good if you had read the first chapter of Griffiths. We have basically covered sections 1.1, 1.2, 1.4, and 1.5 at some level, so you should be able to skim those. Friday we'll worry about section 1.3, which is the last thing we need to get started with proper E&M ... so read section 1.3 with a little more care. We will not cover section 1.5 just yet, but will introduce the Delta function once we need it ... but please at least skim that section in the mean time.

For Monday, we'll start on Chapter 2 of Griffiths. Please have a look through sections 2.1 and 2.2, we should get through that much on Monday.

We will also start doing labs on Monday, details to follow soon.

Some useful notes from PH125 for future reference

Look here. A bit on motion along curved paths, and another little bit on central forces.

The bit on curved paths I will assume you are familiar with at this point, or can pick it up quickly. We spent quite some time on it in PH125. If you were not in PH125, you will see it in Cal III very soon, or I'd be happy to go over it with you in office hours. It will come up gradually during the semester.

The bit on central forces is very incomplete, and follows the central force notes. Most of what you actually need to know about central forces this semester we will go over in class anyway, the skeleton notes I have up might be a useful reference down the road.

At some point, perhaps this semester, both will be merged into the 'math guide' anyway I suppose.

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.)

Lab safety

Remind me next time that you each need to print & sign one of these.

It carries no real legal obligation whatsoever, it is more of a 'gentleman's agreement' than anything.

Help desk schedule

Once again this semester, all 100-level physics teaching assistants are pooling their office hours to better assist you. Here is the current schedule. Office hours are in 203 Gallalee.

Wed class / Stray hints on HW1

Well ... since we elected to re-boot today instead of going over new material, we're not quite as far as I wanted to be. That is OK, though, since I explicitly budgeted extra days just in case we wanted to spend more time on math background.

What this means is that on Wednesday, we'll start out by covering vector derivatives (div, grad, curl) and partial derivatives, and probably get to line integrals and the remainder of the vector calculus we'll need on Friday. By the end of Wednesday's homework, you should know how to do all but the very last of Friday's homework problems.

Homework 1 has 9 problems: 1-4 are due Wednesday, and 5-9 are due Friday (in both cases, by the end of the day). Here are some stray hints on the first 4 problems:

(1) The 'separation vector' is just the result of subtracting the two vectors. A unit vector can be constructed by dividing any vector by its magnitude.

(2) Body diagonal = opposite corners of a cube; there are four possibilities. Pick any two, the angles are the same. You could define two of these as (1,1,1) and (1,-1,1) for instance. If it helps, draw two cubes stacked on top of one another ...

(3) Just grind through it ... easy but tedious. A good chance to practice the cross-product-as-determinant rule! You can double-check some of your answers by verifying that axb is perpendicular to both a and b (meaning a-dot-b is zero).

(4) The first and last are not so bad, just remember how to take the derivative of a vector function, and that a and b are constant vectors. For the second, keep in mind that a and b are not necessarily perpendicular, so your normal rule for taking cross products does not work. Either you need to write the a and b vectors in a cartesian basis (in terms x & y unit vectors, say a=a_x xhat + a_y yhat), or you need to define a unit vector perpendicular to both a and b as well as the angle between a and b.

Probably the former option, writing a and b in terms of cartesian components, is easiest. This is still general, since you have the freedom to pick the x-y plane as the same plane formed by a and b for any two vectors a and b. You should not need to do this for the first and last parts, however.

Wednesday, we'll go over problems 5-9 a bit so you know how to get started.

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.

First homework, syllabus

Syllabus ... it includes the all-important grading information.

Homework 1 ... mostly math. Your first problems are due Wed 26 Aug, the rest are due Fri 28 Aug.

First week of class

As you probably know by now, I will be out of the country for our first two scheduled meetings: Thursday, 20 Aug 2009 and Friday, 21 Aug 2009.

If you got this far, then Dr. Harrell has covered the course introduction for me, and pointed you here. Peruse the posts here for interesting information, including your first homework set ...

I will be out of the country from 19 Aug 2009 until sometime on 23 Aug 2009. I will have sporadic email access. Feel free to email me with questions during this time, but keep in mind I may not be very responsive until 23 Aug.

What does this really mean for you?

• There is no recitation on Friday, 21 Aug 2009.
• I will be back for the second regular class period on 25 Aug 2009
• You still have homework problems due on 25 Aug 2009 (see posts above)
  
• Please read the course syllabus carefully (see posts above)
  
• Please read Chapter 1 of the textbook before 25 Aug 2009