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This collection is from problems that explicitly deal with time dependence. Other that, they range widely, from LRC-type circuits to wave propagation, radiation and antenna. So it's a bit of an eclectic collection.

Problem 1: Implicit in this is that a defind voltage source is connected between "a" and the ground and that a volt meter (alone) is connected between "b" and the ground.

Problem 2: First show the sign convention explicitly in a diagram. The given equations make sense only with a particular sign convention (as to the positive directions of the currents and those of the induced potentials).

Problem 6: If you use a complex representation for E and B, i.e.,

E=E₀e^i(k·r-ωt), B=B₀e^i(k·r-ωt), then E₀ and B₀ are generally complex, mutually perpendicular vectors. For dissipating medium like a conductor, we must allow k to have an imaginary part (say, k=k₀+iκ). Then use Maxwell's equations to relate the phase between B and E to that of k, and solve for that.

Problem 10: The power flux per unit solid angle from a dipole antenna in the far-field region is given by dP/dΩ ~ (proportional to) k⁴ |p|² sin²θ where p is the dipole moment and θ is the angle the line of sight from the antenna to the observation point makes with the vertical.

Since I said I would provide the answers to all problems but ran out of time in class today, I give the answers here for the remaining problems.

#7: (a) (use Faraday's law), (b) (use (d/dt)Φ_B=–∫(del)xE da) (surf integral) = –∫E dl (line int) where the last line integral is through the perfectly conducting loop, (c) (use Ampere's law with the displacement current included)

#10:  T=1/(∂S/∂E)=(E₁/k_B)[1/(ln(NE₁/E – 1)]. Since E can be measured accurately, this equation then provides T accurately.

First, Happy Thanksgiving to you all! Since you have plenty of time to work this set (till 12/4), You may wish to hold off on reading the following comments/hints and try them by yourself first. Then come back here for some hints.

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This set is mostly a collection of problems that involve a current loop (or a magnetic dipole) with a couple of problems on superconductors and Stat Mech thrown in. 

Problem 1: While you could work this problem with brute force as an application of the Biot-Savart's law, you can do it a bit, or a lot more easily using some, or much, symmetry and knowledge from related geometry. In the latter case, you either don't need the second integral, or neither of the integrals provided. If you are interested in saving efforts, think about what simple device this problem reminds you of.

Problem 4: I had corrected the formula given in this problem, whereas Griffiths has a general dipole field equation (5.87) which would reduce to the originally given equation in the problem when r lies in the xy-plane. The reason why this correction was made is as follows:

The original formula is from the multi-pole expansion, which applies far from the dipole, or equivalently, for a "point dipole". In contrast, this current problem is looing "inside" a dipole (analogous to looking for fields between the two charges that make up a macroscopic electric dipole). The distance scales are, then, such that the multi-pole expansion is meaningless. You can see the true solution of this problem in Jackson, 3rd Ed., p.183, which reduces to the corrected formula in the lowest non-vanishing order. 

To see how poor the original formula is, just consider this: if  "r" is the distance from the origin to the point where B is measured, then the original equation blows up at the origin where r=0 (both the one in the problem and (5.87) of Griffiths). However, we know from integrating Biot-Savarrt's law for a current loop, that B on the axis is B_z = µ₀/2 I R² / (z²+R²)^3/2. This gives a finite, well-defined B_z at the center of the loop where z=0.

The cross-sectional area of the wire is also needed part (b). So I defined it as A in my corrected text of the problem.

Please don't ask me how they handled this question at/after the realQualifying Exam - I just don't know. If you do encounter a problem that you think is providing incorrect formula, you owe it to yourself to bring this to the attention of the proctor. If you are still sure the problem is not correct or well-defined, you shuold write out why that is the case in your answer sheets.

Problem 6: Ordinarily, we consider B=0 inside a superconducting slab. However, in this problem, we are considering the evernescent penetration of B into the slab. Use known boundary conditions for B.

Problem 10: If E can be measured very accurately as they say, what would you need for the system to be usable as a thermometer? Yes, a theoretical relation between T and E which would allow the calculation of T from the measured E.

Our PHYS521 class for Tuesday, Nov. 25 will be an optional, review session. I intend this session to be mostly discussions of past homework problems (#7-#11) and related issues. I will stay as long as there are questions from you (up to 11:45 am). If no one shows up by 10:40 am, however, I will leave.

Most of the E&M problems in this set are related to Faraday's Law.

Problrm 5:  This problem is the magnetic version of the force between parallel plates of a capacitor while the potential difference is maintained. When the current is maintained, there will be a radially outward pressure on the solenoid. It is simplest to consider the energies (don't forget the external source) as a function of the radius and calculate the force from there.

Problem 8: (a) Estimate how many atoms are in the excited state with internal energy Δ vs 0 and use conservation of energy (including the translational kinetic energy too). (b) Total entropy is the sum of the internal entropy due to the two available states and the translational entropy.

Problem 9: Statistical temperature is given by dU = T dS.

Problem 10: Entropy of mixing. We will review the classical ideal gas in class, but in short, the partition function of an N-particle,monatomic, classical ideal gas is Z_N = z₁^N/N! where z₁ = n_Q V and n_Q = [(mk_BT)/(2π hbar²)]^3/2. F_N= –k_BT ln Z_N and S=−(∂F/∂T)|_T.

We spoke of the common errors involved in the energies when the capacitance of a capacitor (inductance of an inductor) is changed while maintaining the voltage (current) by an external source. When you neglect to take into account the work done by the external source, you make a sign error in the resulting forces because there is exactly double the amount of energy expended by the external source while the field energy in the capacitor (inductor) grows by a certain amount. This factor of 2 is exact and very simple to understand. I just posted an explanation on this site under "Syllabus and Reference Materials" for your convenience. Again, this is a very common mistake, committed even by very smart people (including faculty!).

Equation sheet that came with Fall 2006 Qualifier is posted here under Syllabus and Reference Materials. You may consult it for doing Problem 9.

OK, a bit late, but here goes the usual pointers for the problems of Homework 9. Most of the E&M questions in this set are related to the motion of charges in E and/or B fields.

By the way, only 3 people came to class today. What a shame! I will put some of the things we did in class here, but it is not possible to put it all down on "paper" like this. In addition to the current homework, we also discussed two nice problems of charges/currents in field, one of them from this Fall's Qualifier. You will be missing opportunity for live exchanges if you don't come. I am here to help you pass the Quals and only for that purpose. Don't forget that!

Problem 1:  This is, in my view, the hardest problem in this set. Though you could solve for the velocity as a function of time for both k>0 (with dissipation) and k=0 (no dissipation), the situation is rather different in the two cases. For part (a), while you can solve for v_max if k=0, you cannot solve for it in closed from if k>0. In that case, please replace the question as "obtain v_term" where v_term is the terminal velocity. You will see that the trajectory is a cycloid if k=0.

Problem 2: For consistency with SI unit system, please add "µ_0" to the right of the given equation: Div B = µ₀ ρ_m.

Problem 3: The physics here is that the momentum and angular momentum in the E and B fields initially are transferred to the mechanical momentum and angular momentum of the charges as time goes on. E and B are related by Faraday's law.

Problem 5: It's probably best to absorb c_0 and c_n in the Hints into F_0 and F_n. Then Laplace equation applied to the Fourier series allow you to solve for F_0 and F_n. Here, don't forget that Laplace equation is valid only in upper and lower half spaces and NOT on the xy-plance. So the solutions for F_0 and F_n in general involve different coefficients for the two half spaces. Then use symmetry to settle the differences.

Problem 9: In the actual Qualifier, the equation sheet contained the Lorentz transformation of the E and B fields. For part (b), you should note the relationship of your computed answers and your expectations based on what should be a Lorentz invariant.

Problems 1-6 are from magnetostatics. Problem 7 is one where we investigate a possible implication of magnetic monopoles.

Problem 1: The power transported by the cable is the integral of the Poynting vector over the cross sectional area.

Problem 2: It's a problem of Ampère's law, but in which form is it most useful here? Then, what is continuous across the "small gap"?

Problem 3: Principle of superposition. Then, use geometry to simplify the final B (both direction and magnitude).

Problems 4 and 5: Both of these problems are actually similar thoubh one is electrical and the other magnetic. In both, you have to include the work being done by the external power source just as we discussed in class for the parallel-plate capacitor problems (when the plate separation is changed and when dielectric media is made available to the space between the plates).

Problem 7:  It's simplest to first simplify the integrand for L, using both spherical and Cartesian coordinates as well as symmetry. A lot of terms drop out, and the integral should reduce to the one provided. The answer will be independent of d!

Problem 9: Don't worry about factors of 2's or 3's. This is an order of magnitude calculation.

Problem 10: It's for a classical particle.

Here are some pointers for Homework 7. All of the E&M problems here are related to Gauss's Law. The two stat mech problems are from classical statistical mechanics.

Problem 1: Gauss's Law in the two forms (one for E and the other for D) and the relationship between the two give you all you want to know.

Problem 2: A number of ways to do this. If you wish to do it in the order of the question parts (i.e., (a) first, then (b), etc.), then you can calculate the overall capacitance by modelling the capacitor as the series combination of thin slices of capacitors (with varying capacitance). Then use the two forms of Gauss's Law.You could also do (b) first, then (a), (c).

Problem 7: Don't forget that the pressure acting on a surface charge density σ is σ²/2ε₀ (factor of 2 there!). Do the integral carefully (taking into account the directions of the force).

Problem 9: Remeber what we discussed about the need to include the "battery" in order to analyze correctly when one remains connected to the capacitor while some changes are made (such as a dielectric inserted, or, in this case, the voltage is changed).

Problem 10: Straight application of classical stat mech to canonical ensemble.

Because there is no class on Tuesday, I am giving more detailed and numerous hints here than usual. Please only look at them as necessary. The more you can work out without first looking at them, the better for yourself when you face the real exams.

Also for basic relativistic formulae, please refer to a short excerpt of the notes for Special Relativity that I posted under Syllabus and Reference Materials section. The notes come from my class on mechanics at the junior/senior level and are based on the text by Thornton and Marion.

Problem 1: If you cut a spring of spring constant k and length L in half, what is the spring constant of each half?

Problem 2: Assume that the student loses contact with the floor when s/he is stretched to full height. Think about the energetics.

Problem 3: K (he kinetic energy) = E (total energy) -  U. Classically, a particle cannot be at a position where U>E since K>=0. A phase diagram (in x-p space) is a set of trajectories and as such you should not forget to draw arrows to indicate the directions of flow as well.

Problem 4: Likely the hardest (and an oddball) problem of the lot. It may simplify algebra to set up a coordinate system translated by R in both horizontal and vertical directions from the floor/wall. In principle, this is a simple problem of energetics (with a geometric constraint). A key is to express the velocity of the bottom cylinder solely in terms of one trigonometric function of an angle.

Problem 5: Think about time dilation and relativistic Doppler effect. Also, what is the relationship between the total number of pulses Ulysses emits and that of the pules Homer receives?

Problem 6: Understanding this problem leads to that of the basic problem of transverse relativistic Doppler effects. In particular, this problem addresses what's called the superluminal velocity - an apparent velocity that is greater than c.

This problem takes you from (a) to (b). Just follow its course and try not to jump ahead by using some relations that you pull out of the air.

(a) First note that we are talking about the light that was emitted exactly at the instant (say t=0) that the blob passed the location shown in the diagram (not before or after). Take a small time Δt (in the lab=observer frame) and calculate the position of the blob at t=Δt and from it the time it takes for the light emitted at Δt to reach the observer. Then you know the time duration between the time when the observer saw the light emitted at t=0 to the time when s/he sees the light emitted at t=Δt. Then the observer will use the transverse distance the blob travelled in Δt and the observed time duration (as seen by him/her) to calculate the apparent transverse velocity of the blob. This velocity can be larger than c! Think about when that could happen and why.

(b) Think about the number of pulses emitted by the blob during time  Δt in the observer frame (taking into account the time dilation). Since you would have calculated the time the observer actually measures between the start and end of this time period, now you have the frequency of pulses the observer measures. Is this red-shifted or blue-shifted?

Problem 7: (a) Apply the Lorentz transformation to (E/c, p), and obtain p'_x in the CM frame as a function of the angle the π's make with the x direction (in CM frame).  (b) Use momentum conservation. (c) Another Lorentz transformation between the lab frame and the observer frame will be useful. Also, first relating the lifetimes to the "proper" lifefime (i.e., in its own rest frame) will be useful.

Problem 8: Use equipartition theorem. For (b), use an argument (without any calculation). Think about how quatum and classical calculations of things like heat capacity are related.

Problem 9: You can assume that the entropy S_s in the superconducting phase goes to zero as T→0. Also, since the entropy S_n in the normal phase would asymptotically be linear in T as T→0 IF there were no intervening superconducting phase transition, you can also assume that S_n goes to zero as T→0.

Problem 10: The formula given in the problem for the chemical potential μ_s of the solvent molecule in a solution comes from the (extensive) Gibbs free energy of a dilute solution (cf. Landau and Lifshitz in the chapter for solutions, e.g.):

G(T,P,N_s,N_b) = N_s μ_0 + N_b k_B T ln[(N_b/e) (f(T,P)/N_s)]

where μ_0 is the solvent chemical potential in a pure solvent (not a solution).

Thus, μ_s = (∂G/∂N_s)|_(constant N_b,T,P) = μ_0 - (N_b/N_s) k_B T as given in the problem. Anyway, you can just use this relation in this problem, not derive it. 

In this problem, use the fact that, in equilibrium, the solvent chemical potential in the pure solution (on the right) and that in the solution with solute molecules also present (on the left) must be equal. Since the solute cannot pass through the membrane, the solute chemical potential doesn't have to match; it's the solvent chemical potential that must match between the right and left halves.

Then, interpret that relation in terms of a Taylor series expansion of the solvent chemical potential in pressure and use the second relation that is also given in the problem.

Mathematically, this (and similar) problem is very simple, but if you are not used to this type of a problem, it may bewilder you as to the significance and origin of terms, relations, and results.

Midterm Exam will be given in Rm.298 (a conference room).

Friday, 7 - 9 pm, 10/17/2014

H. Nakanishi

Problem 3: Consider the lengths in this problem to be in units of AU (astronomical unit) and express velocities in units of AU/yr, accelerations in units of AU/yr^2, for similicity. The answer to part (c) should be M = (a number) x M_s.

Problem 4: In part (d), since the angle made by OP (where O and P are on the circle of radius R) and PA (where OA is a diameter) is always 90 degrees, r=OP=2R cos(theta).

Problem 5: Two bodies M1 and M2 orbit around their center of mass O with the same angular velocity omega (for central forces).  Then, a small object m at a Lagrange point must experience a force toward O and orbit with the same angular frequency omega in order to keep the same relative position with M1 and M2.

Problem 9: In part (b), assume that the 100W bulb left on inside the refrigerator is the dominant part of heat that needs to be removed to keep the inside of the refrigerator at constant temperature.

Problem 10: The result of (c) is identical to what is known as Clausius-Clapeyron equation. Usually, this relation is derived by equating the change in the Gibbs free energy per particle along the phase coexistence curve for just inside one phase (say A) and just inside the other phase (B), i.e., d mu_A = d mu_B. The current problem is cute in that it gives you the same result, but do note that it makes some bold assumptions (not justified) whereas the usual derivation is exact.

By the way, today, discussing the problem PI-8 of Fall 2014, I said you need to assume C_P and C_V remain constant, but actually that's clearly true since the problem is for the monatomic ideal gas. This assumption did not have to be stated and the problem is well posed.

Unfortunately, when discussing the moment of inertia of the two body system in response to Greg's question, I gave a wrong answer. The correct answer is yes, the moment of inertia about the center of mass is equal to mu r^2. This was in my face as I had done essentially the same calculation for the total kinetic energy of the same two body system.

My incorrect answer was (falsely) associated with another situation where a massless rod, with two point masses at two different points along it, is pivoted at one end, say, O. For this situation, the moment of inertia about O is NOT equal to the total mass times the distance from O to the center of mass. The correct answer for this case is I_tot = I_CM + mu r^2, where I_CM = (m1+m2) R^2 where R is the distance from O to the CM.

I apologize for the confusion.

We are tentatively scheduling our Exam I for 7 - 9 pm on Friday, Oct. 17 (location to be announced). This will cover all of mechanics (including special relativity, a topic covered in Homework #6), thermodynamics such as those covered in HW #1 through #6, and combinations of them. It will be closed book, no crib sheet, just like the real Qualifier, except that there will be 5 problems to be worked on in 2 hours - roughly the same amuont of time per problem. All problems will be from real, past Quals problems. So you would expect to be roughly in passing zone if you can get 50% or better overall. (In other words, they wll be just as tough as those problems you have been working on in homework and also on the real thing!) Please let me know ASAP if this time/day absolutely does not work. Thanks.

H. Nakanishi

As we discussed already, our class for this Tuesday, Sept. 30 has been cancelled. Also, the current homework (#5) is due on Thursday next week. This gives us some time to discuss heat cycles (engines and refrigerators), central force motion, and special relativity in class. Thanks.

Problem 1: Though no word "friction" appears in the problem, it does say "rough horizontal table". This implies that you should use (standard) kinetic friction that is proportional to the normal force. As I said, some force must cancel the tension in the string, and it would have to be the friction.

Problem 2: Use various arguments/equations to rule out unphysical assumptions. As an example, if you suppose all 3 blocks move together (the two upper blocks do not slip on the bottom one), then you can show that the whole piece (3 blocks) must accelerate opposite to the pulling force, which is a contradiction.

Problem 3: Assume that the static friction coefficient is greater than the kinetic one, which is a reasonable assumption.

Problem 6: As I said in class, let the angular momentum vector to be the sum of the (unperturbed) omega_3 in z-direction and small perturbations each in x- and y-directions. Note all these "directions" are relative to the body coordinates (fixed to the body). Then apply the Euler equations.

Problem 8: The particle flux onto the hole can be related to the pressure and momentum change of a particle if it hits the wall and were to be reflected back.

Problem 9: Just draw an isotherm, starting from where all of the material is a gas at a low density/pressure, and decrease volume (increase pressure) at constant temperature. At some point, you enter a two-phase region where part of it starts to become liquid. Eventually, the density (volume) is high (low) enough that all material becomes liquid. While in the two-phase region, the pressure remains constant; otherwise, pressure increases when volume is decreased. This is the analog of the well known situation when you cool liquid at constant pressure and it starts to freeze. While the freezing is going on, the temperature remains constant, but it starts to decrease again after all material has become solid.

Problem 10: The relationship between the ambient pressure, the pressure inside the bubble, and the interface tension is called the Young-Laplace equation. The interface tension is the excess surface free energy per unit area when the interface is present.

Good luck!

Problems 1, 3, and 6: Don't forget that the net force is equal to the rate of change of mementum (F=dp/dt).

Problem 4: As in more complex collision problems, think about the process in multiple stages, i.e., the initial state, the duration of the "collision" where A is sliding over B and C, and the final state where A is exactly on top of C.

Problem 7: You need to define what your "system" is first. It might be simplest if you treat the wheel without the added bit of paint as your "system" and the paint as external to it, though it's up to you how you solve the problem.

Problem 8: Deceptively hard! You may want to consider the system to be made up of 3 components (glass, the ground, and "outer space"), each of which remains in equilibrium.

Problem 10: I did this problem in class but assuming temperature for both halves to be the same initially. This would be the case if A and B are the same type (e.g., monatomic) of ideal gas, but not if they aren't. So you need to come up with some thermodynamic argument for the case where T_A and T_B are different initially.

Good luck!

Hisao Nakanishi

Hisao