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.