Physics 460 Questions and Answers
 Date Question Answer 12/17 In part c of question 1 you ask for a sketch of time averaged power as a function of the angle theta.  Should it read "sketch time averaged power per unit solid angle as a function of the angle theta"? Yes of course.  I'm sorry about that omission. 12/13 12.45 ...am I using the right v's and gamma's? In the transformation equations, it's the velocity of the new frame with respect to the old one, right? And the velocity of the charges never enters in until the force calculation? You have to remember that when computing the velocities of the particles in each frame, you must use the relativistic transformation law, not the Galilean transformation law. This will be important in computing the speed of frame B with respect to C. 12/13 In 11.11c, are we meant to find

or P(t)?

12/6 I am working on 10.25 and I have a question regarding the surface of integration for the power. The total power is the energy stored in the fields integrated over a surface (integral of S.da). If we consider a spherical shell surrounding the fields as our surface of integration for the TOTAL power, then the plane x=a will intesect that shell as a circle. Is this circle our da for the total power passing through the plane? First, you must carefully distinguish between energy stored in the fields and power transmitted by the fields.  Notice that they are not dimensionally the same!  The product S dot dA gives the power flowing across the area element dA, so to get the total power flowing through any surface of inite size you need the integral of S dot dA over that finite surface.  A circle is a line and the area of it is therefore zero! 12/2 In 10.25, are we meant to assume the non-relativistic limit? No. 11/29 In your online notes, page 6,  why can't you allow your lower limit to be zero and multiply by 2, rather than using the full range of integration? I can see the answers wouldn't be consistent, why  not? I just added a diagram to these notes for someone else.  Maybe that will help you too.  The question is, why do you think you should be able to do that?  The fundamental rule is: if you can't show it's right, it probably isn't!  Now of course if we took z=0, then we would have an integral that is symmetric about the origin, and it would be OK. Can we take z=0?  Well, yes, if we are able to carefully argue that the potential should be independent of z (due to translational symmetry along the z-axis), then we can pick any convenient z, say z=0, to do the calculation.  This is basicaly what Griffiths does, without justifying it.  I wanted to show you that the result is independent of z, and the calculation is not really any harder. Does this help? 11/26 10.9  I am still confused! In this problem, I = k*t = k(tret + R/c). I can go ahead and integrate over the primed coordinates, but if I want to get A(t) and not A(tret), at some point I'll have to plug back in for tret = t - R/c, which puts the primed coordinate back in! (This is what I meant by your example not having a functional dependence on t: I = I0, not I(t) for t > 0) Do I just leave I=k*t? Or is there some other trick I'm missing? You are not reading what I am saying.  The expression for A at coordinates r, t is given as an integral over the source at r', tret.  Not t!!!  Think about how we got this expression and what it means. We have to evaluate the source term in the past! What you have done here is to simply write t as a function of  tret. But that's not right  You must replace t with tret.  After you integrate over the primed coordinates, what you have left is a function of t and the unprimed coordinates. 11/25 In 10.18, are we meant to assume that the particle is moving in the +x direction at a constant velocity speed v so that v-vector = v * x-hat? Yes. 11/25 Is 10.13 as simple as it seems? Is it basically a write-down aside from finding v-vector? It is pretty simple.  A takes a little more effort than V- I wouldn't call it a write-down, exactly. 11/25 On problem 10.9a, I'm confused about how to find the vector potential A. I know from equation 10.19 that I should be integrating I(s, t_r)dz, but t_r is a function of z? Do I ignore that during the integration? If so, do I rewrite it in terms of z before I plug in the limits? I think this isn't clicking for me for two reasons: 1) In your example and in Griffiths', there's no functional dependence on time, so I've just not seen how this should work and it's not intuitive for me. 2) In this problem, I've set up my axes so that there is no primed variable -- I just have the wire along the z-axis. It seems like this should make things easier, but maybe by doing that I'm confusing myself about what I'm integrating over... The expression for A at coordinates r, t is given as an integral over the source at r', tret.  The retarded time tret is a function of both the primed and unprimed coordinates. You integrate over the primed coordinates. Yes there is!  As I recall, the source was zero prior to t=0.  Check to see how that shows up in the solution.  A similar thing happens here. Yes there is!  But your source is zero except at x' = y' = 0. This choice of coordinates should indeed make things simpler. 11/15 10.2 When calculating the poynting vector in part b, should we consider both the positive and negative y hat directions for B? Or should we just consider the positive direction since at our point of interest (the box) the B field is traveling in that direction? First you need to review Example 10.1 to be sure you understand why the +/- symbol is there.  That should answer your question for you. This is yet another example of why it  is important to include the sentences that explain the "how" and "why", "when" and 'where". 11/14 For 10.2, am I suppose to use the equations for E & B that are given in ex. 10.1? Also do I need to take into account the potentials that are given in 10.1 as well? It is a little confusing. If so, I am plugging in my values for t1 and t2 in equations E & B to get the result of my fields. For E, I get ------> E= constant(d-x)2. When d = x, my equation goes to zero, but I don't know why? Am I assuming that there is a surface current in the yz plane? If so, what the heck is going on??? I am really confused if you can't tell. I think this set up of a box over a surface is confusing me. Help!! You only need the fields. Where did the "squared" come from? Notice that the surface current turns on at t=0.  All the fields are zero everywhere before t=0, and so the universe beyond x=ct does not yet know that the cuurent exists. Read Example 10.1 carefully to see how we know the surface current is there. 11/12 When Griffiths uses rho-dot in the third equation, does he mean the FULL derivative of rho with respect to time? I want to use equation 5.29 (page 214) but that involves the partial derivative of rho with respect to time. No, he means partial (in this case at least).  And you are correct to use charge conservation.  (Note this relation was also on your exam formula sheet.  It is pretty fundamental-- not just some random numbered relation!) 11/8 Is there some significance to the fact that (assuming my calculations are right), the maximum wavelength to excite one TE mode in the waveguide is exactly the long dimension of the waveguide's cross-section? I'm not sure how to interpret this. Absolutely! Think about the meaning of the normal modes in this system.  Again, comparison with a vibrating string may be helpful. 11/3 For problem 9.28, the second question, does Griffiths refer to one of the TE modes we found for the first question, T_30 for example, or to TE_mn modes in general. How many modes did you find at 1.7x1010 Hz?  Were there more than one? Why or why not?  Supose you want only one? Which one would it be, and why? 11/1 Part c) of 9.37 refers to the results of problem 9.16.  Does this mean we need to do 9.16? If you checked your lecture notes you wouldn't need to ask this! 11/1 Could you please clarify what prob. 37 wants? What does he mean by "do the same"? Repeat part a and b? Do you need to repeat a?  Do any of these results depend on the polarization? 10/28 A question about prob. 9.25. So, we're asked to find the group velocity of the waves described by eqn. 9.166 and 9.169. To find v_g, we need to differentiate an expression for w in terms of  k. Would this be eqn 9.169, rewritten in terms of k? If it is, this is confusing because k is a complex number, with a very complex complex part (no pun intended ;o)) I think the confusion will go away if you read the question more carefully.  What are you to assume?   What effect does that assumption have on eqn 9.169? 10/25 I'm wondering about the portion of the wave that is reflected off the boundary between medium 2 and medium 3.  I see that this reflected wave would eventually get back to the boundary between medium 1 and 2 (traveling to the left), and some of this wave would be transmitted though boundary 1 and some reflected and so on an so forth.  Is it safe to side step this issue by considering that the total reflected portion of the original incident wave (medium 1) is also composed of a portion of the second reflected wave? Yes. 10/25 I know you mentioned in office hours that you could "say interesting things" about the expression for T in problem 9.34, but I've done a number of checks and I'm not sure I'm thinking in the right direction. Could you give me a hint about what kind of analysis would be especially useful here? Well,  are there any conditions under which T=0?  T=1? When is T max?  When min? What happens if  n1 = n3? How would this change (or not) any of the answers to the above? 10/25 In problem 9.34, we are asked to find the transmission coefficient for the monochromatic plane wave that passes through a slab of material. I used the relation mag(E) = 1/c * mag(B),  and then applied the boundary conditions to find the mag(E) of the final transmitted wave as a function of the original incident wave mag(E). To get the intensity, we can use the poynting vector, but instead I used  I = eps * mu * mag(E)^2. (G 9.73). I took the ratio of the incident and the transmitted intensity, and got an answer for T as a function of the various indices of refraction. This does not match the answer provided in the problem. Where did I go wrong? This is true in vaccuum, not in a material where n>1. Should be OK. This is not 9.73!  Read it again, more carefully. 10/25 Am I right in assuming that in problem 9.17 we have a rotationally symmetric system, so that I can always arrange my point of view so that polarization is in the plane of incidence and therefore equation 9.109 is valid? No.  I'm not sure what you mean by " rotationally symmetric", but except, at normal incidence, the two polarizations are quite distinct.  You should be able to see this from a clear drawing.  That said however,  G's discussion, and hence Fig 9.16, refers to only one case: polarization in the plane of incidence.  Since your diagram should be "analagous to Fig 9.16", it should presumably refer to the same polarization. It would be instructive to also draw an equivalent diagram for the other polarization, using results from the class notes. 10/25 In office hours you pointed out an alternative way to compute (problem 9.11, pg. 382).  The last sentence in the paragraph above that equation says that this (trick) is only valid when the waves have the same "k" vector and omega. How can they have the same "k" vector if medium 1 and 3 are different? the "k" vector is defined as (omega/speed) or (omega*index of refraction) / (speed of light). The index of refraction is different in mediums 1 and 3. Please clarify this for me? The two waves G refers to are his f and g. In our case that would be E and B.   But in an EM wave the E and B waves always have the same k and omega.  It wouldn't make any sense to compute a Poynting vector using E from medium 1 and B from medium 3! 10/18 Ok, in 3.33 I need to find the B field from the E field.  I found it using the 3rd maxwell equation (E X B= -dB/dt) but I'm not sure if this is a good idea because I'm using the equation which I am trying to prove.  Should I use a different relation to get B? And what "different" relation would that be?   You still have three other Maxwell equations to verify. 10/18 Well, the boundary conditions we are using split E into the parallel and perpendicular components.  So can I say that the component of the E field perpendicular to the boundary is zero? That would depend on what you mean by "the" E field.  The incident field? The total field? The transmitted E field? 10/18 Can I assume that the incident wave in this problem (9.14)  is propagating in the z direction,  and thus the perpendicular component of the e field is zero? Yes Perpendicular to what? 10/18 9.33.  I'm not quite sure what he means by a spherical wave. I'm tempted to picture a circumferential E wave (because E is in the phi-hat direction) propagating outward in the +r-hat direction (for no reason other than that's how I picture a spherical wave). I think I'm basically thinking of it like the 3-D version of the waves created by a pebble dropped in a pond. Am I at all correct? Yes 10/18 In problem 9.33 part b, we're asked to find the intensity vector and state whether it points in the expected direction, but Griffiths never actually defines the intensity vector in the text. Am I correct in assuming that the intensity vector points from lower intensity to higher intensity? "Intensity" is defined in G page 381 eqn 9.63. and ny notes3 page 10.  It is easy to extrapolate from this to the intensity vector, since intensity is the magnitude of a vector. 10/16 Problem 9.33 asks us to "show that E obeys all four of Maxwell's equations, in vacuum, and find the associated magnetic field," but there's only one Maxwell equation that involves E only. Is Griffiths asking us to use Maxwell's equations to find the associated B field, or are we meant to assume equation 9.49 (which itself comes from Maxwell's) and then prove that it does satisfy Maxwell's, or is there some other, less circular, way of doing this? Equation 9.49 is in section 9.2.2, titled  "Monochromatic plane waves" .  Is the wave in problem 9.33 a plane wave?  If so, you can use eqn 9.49, but not otherwise. You should verify the one equation that involves E only, find the associated B, and then verify that the remaining two equations are also satisfied. 10/11 I thought that, because the wall is a fixed point that absorbs no energy, the total energy in equals the total energy out. Since the only difference then is the direction of the wave, and cos(wt)=cos(-wt), the equations cancel out entirely. I'm stuck on how else to interpret the boundary condition. Can you tell me where my logic is failing? Also, I was wondering if we can assume that the phi's are the same for our situation. I ask this by the same logic that the book uses on p510 which says that for harmonic waves at a free boundary the reflected wave has the same phase as the incident wave. What do you mean by "equations cancel out"?  This argument should allow you to find Ar. No.  You should use the boundary conditions to find  the phis. 10/11 In the LB problem the one with the waves on two different strings (#84). The part of the question that asks us to use conservation of energy. The equations we are given tell us about a point on the wave. I set my problem that at time t=0 the open end of the string at x= L2. The potential energy is easy to calculate. However which velocity would be used? The velocity of the point as it moves up and down or the wave phase velocity? The way I was going to use energy of the conservation the total energy on the string at any other time should be equal the to the energy that the point that was initally disturbed had at time t=0. I was thinking that at some later time t I wanted to look at the energy at the position x=0. But the energy of that point is not equal to the energy contained in the whole string. Does it? So, then I was thinking that to get the total energy of the system should I use the potential energy of the initially disturbed point and wave phase velocity of the incident wave. I don't know which velocity you are talking about. Look at the total energy flowing in toward the wall versus the total energy flowing out away from the wall. 10/11 The second part of 9.9 has an em wave moving from the origin to (1,1,1).  It is polarized parallel to the x-z plane.  I don't see how this is possible if the direction of polarization and propagation are orthogonal. It's possible. Use this condition to find the polarization vector. 10/11 In Griffith's page 380 in equations 9.51 and 9.52 what is the last term in the parentheses. Can these equations be used in part a of 9.9? Go back and review the derivation of these relations and it should be clear what that factor is.  Also review my notes3 section 2.1.  You may use these relations in 9.9, but only after you have come to understand where they come from. 10/11 Could you please clarify something for me on prob. 84? Where does the Hint end? I'm confused because I'm not sure if the basic question is the discussion part and the hint and the questions at the end of the problem are supposed to compliment this discussion, OR if the hint ends with the displacement equations and the rest of the problem are additional questions. The hint ends at "and B". The answers to the last two questions should form part of your discussion. 10/8 In 15.84, I'm confused as to why there isn't also a reflected wave in string 1. (And wouldn't that also transmit back to string 2?) Are we just neglecting that for this problem, or is there some reason that there is no reflected wave in string 1? By "string 1" I presume you mean the one on the left, that is attached to the wall.  Well, there is, of course.  But the combination of a leftward-going and a rightward-going  wave on a string of fixed length with zero displacement at the fixed end gives the standing wave described by yl.  And the sum of the two rightward going waves on string 2 is given by yr.   If you don't believe me, you can write out all 5 waves, put in the bc at the wall, do the algebra, and convince yourself it all collapses to the 3 displacements given. 10/8 I want to make sure I understand the question in 15.73: When you say "2 counter-moving wave pulses" do you mean one travelling around the ring in the +phi-hat direction and one going in the -phi-hat direction or do you mean 2 waves travelling in the +phi-hat direction, one polarized in the z direction and one polarized in the s direction? Yes These would not be "counter-moving", but "polarized perpendicularly". 10/4 8.8 I'm trying to calculate the magnetic field at r>R using the expression in example 5.11 page 236-237, is this way ok? Look at Example 6.1, pg  264/5 as well. 10/4 8.6.  In part c, I have found the impulse of my system per say. Though I have a Q in my expression. Usually we solve for Q. However, I am not sure whether I can use the relationship for the electric field between two capacitor plates. I have a feeling that my wire has its own magnetic field due to the current in the wire. Thus, there would be an induced electric field. Due to Lenz's law. However, I don't know if I have to account for this. ?? Yes. The wire is relevant only in part (b), not (c). In (b), you should consider only the intially given B, as we do not consider objects (here the wire)  to exert forces on themselves. 10/4 Question 1:  I am trying to compute the induced E field in part b) of 8.8.  I am using the integral form of faradays law.  In order to find the mag. flux I am using the integral of B dot dA.  First of all, how does the magnetic field depend on time?  I'm assuming I should make use of the statement that the field is uniformly decreased, but I'm not sure how to make my B field depend on time; let alone take the derivative in order to get the flux.  Any hints?  Question 2:  Same problem.  In computing the flux I have choosen my dA to be a spherical shell enclosing the sphere such that all mag. field lines pass through it.  Now when I plug this into faradays law I get the integral of E dot dL on the LHS.  Is the integral of dL a circle surrounding the sphere? You don't need to know how B depends on time., as long as ytou know the initial and final values. You don't need to take the derivative to get the flux! Review Ch 30 of LB for the method.  This surface is an open surface spanning your curve. A spherical shell would be a closed surface.  Start by choosing your curve; then find a suitable spanning surface. 10/4 Now I am stuck on part (c) of 8.8. I think I'm confused about which value for the current I need to plug into the Lorentz force equation. Whatever I do I seem to end up with the wrong set of sines and cosines when I go to find the torque on the whole sphere. If I find I over the entire sphere, it goes as cos(th) and I end up with sin(th)2*cos(th)2 in the z-hat direction. If I leave it as dI/d(th) that doesn't seem to make sense since the force integral is suppose to be I*dl x B. This also gets me a problem later on since the x-hat and y-hat terms will be the ones with the sin(th)2*cos(th)2 which won't integrate away. If I use dI, then it already has a d(th) term in it and I don't know what to do with dl, which I would otherwise think was R*sin(th)*d(th). Since the surface current depends on both theta and phi (Remember: that theta hat unit vector is a function of phi!) you will need to look at the surface current on a differential area dA of the sphere, compute the force and torque on that one little patch, and then integrate over the whole sphere.  It should be straightforward to get the surface current as a function of theta once you have I as a function of theta. The x and y components of the torque should have phi dependence. 10/4 8.8: After I obtain an expression for l_em in the Theta^ direction I then integrate to get the total angular momentum. You said in office hours that for theta^, only the Z^ component survives integration. I still am having trouble justifying this. Is there an important part of vector calculus I am forgetting? Any insight would make me eternally happy. You must express the (variable) unit vector theta hat in terms of the (constant) unit vectors x hat, y hat and z hat.  When you do this, you will find that there are cos and sin phi terms multiplying x hat and y hat, and these give zero when you integrate over phi. Look at a diagram to see what these vector sums are zero. 10/4 8.6:  I still don't totally get why E_induced points in the -phi-hat direction if B is pointing in the z-hat direction. I can see the negative sign in Faraday's law and all, but my (clearly wrong) instinct is that it should follow the right-hand rule.  Can you help me get at why a circulating E field creates B field in the "wrong" direction? This is Lenz's law (aka energy conservation) which causes the induced E-field to follow a left hand rule.  Think about what would happen with a right-hand rule. Let the curent flow through a resistor and use the power to do work.  You'll see that you could build a perpetual motion machine and make a fortune.  Alas, you can't! 10/2 in part a of 8.6 when dT is integrated we need to find the volume, may we assume the area of the plates is A and the volume of the space is d.A? Yes 9/27 I am working on 7.54 part a) and I am stuck. I have an equality between the two total fluxes of the coils. The equality contains, I1, I2, N1, N2, L1, L2 and M. I can't find a way to get rid of everything but L1,L2, and M. Am I missing a math trick somewhere? I think I understand the problem conceptually but I just don't know how to get things to cancel. Perhaps setting the two total fluxes to be equal is the wrong way to go...it said in the problem that the same flux passes through both the primary and secondary coil and I took that as my method for proving M2=L1L2. The expression you have is valid for all values of I1 and I2, including one or both of them being zero. 9/27 7.54b Should we start the problem by arbitrarily picking the direction of the currents in both coils, or should we set the the direction of the current in the primary coil and then determine which direction the current should be flowing in the secondary coil. As we discussed in class, you do not know or care which way the current are actually flowing ahead of time.  You must define your current variables,  specifying a direction that you choose to call positive. Then write your equations in a self consistent way. 9/27 In problem 7.44, I think the Dipole will orientate itself with one pole close to the conductor and the other away from it ( a straight line up) as the superconductor repels the field lines? Is this completly wrong? What is your reasoning?  You need to consider the torque on the dipole, and find the orientation for which the torque is zero. You will also need to consider stability.  Be sure you read through problems 7.42 and 7.43 so you get the necessary background. 9/27 I had been thinking that I needed to find an expression for B at every point so that I could find the total energy, but I think you said that I just need to know the B field at the dipole itself. That seems to imply that I can get away with just finding the energy density at the dipole. Is that true, or am I now making it too simple? What do you mean by "the total energy"?  Energy of what?  If the "total energy" includes all the energy stored in the fields, then I guess you need B everywhere.  But is that what you want? 9/27 Do we have to explain the reasoning why we're using an image dipole to model the superconductor in great detail? I mean, are a couple sentences enough to explain this part? You mentioned boundary conditions in office hours. Did that concept apply to this part? A couple of GOOD sentences are enough. And yes, boundary conditions are the whole issue here. (You will also need some math.) 9/27 The magnetic dipole wants to line up with the field, if it is displaced an angle theta from the z-axis, just like an electric dipole that spins wants to line up with the E field. This occurs when the torque equals zero. Is this analysis sufficient to prove that the final orientatin of the dipole a height z above the superconductor will be in the direction of the B field? We were also discussing potential energy --that it'd be zero when the system is stable (like in mechanics). Would that analysis be relevant here? Finally, are there several equilibria? My understanding is that there's only one point when the system is stable (like the ball at the bottom of a hill in mechanics). No. The issue here is, what do you mean by "the field"?  Your answer should give the orientation with respect to the superconductor surface. A minimum --- not necessarily zero.  Remember, energies may be positive, negative or zero. There can be local energy minima and global minima. The local minima represent stable states as long as the perturbation from the state is not too great.  Think of a roller coaster with several dips on the way to the bottom. 9/27 On problem 7.44 can/should I use the result for surface current in the previous problem.  Part c) of 7.33 asks to derive the surface current in the conductor. No.  The situation in 7.43 (not 7.33) is not identical to that in 7.44. 9/18 7.32-- I always get confused about surface current. In particular, I  think Griffiths' example has never totally made sense to me. My instinct tells me that the surface current due to I flowing onto the capacitor should be something like K(s) = 2*pi*s*sigma where s is the radial coordinate and sigma is the charge density which is constant in s, but I can't argue it. I am pretty sure that I want to look at dq flowing into a ring of area 2*pi*s*ds in time dt, but I think that dl_perpendicular = s*dphi and not ds, so I get stuck if I try to use K = dI/dl_perpendicular. It seems like either my instinct is wrong, or my assumption about dl_perpendicular is wrong or both. Is that expression even dimensionally correct?  Check it! Here you are getting formula-itis again!  Why are you trying to use that formula for K?  Think about what it is, in words! Make use of the symmetry!  Does anything depend on phi?  Use charge conservation to get a differential equation for K.  Solve, and put in the boundary conditions.  A good diagram will help. 9/18 7.34 -- I'm not sure how to read the expression for E: Is that theta(a function of vt-r) or theta(a function of x)*(vt-r). Also, since E = E(r-vector, t), are we taking v to be |d(r-vector)/dt| = constant? Well, I don't see any x!  You are told it is the theta FUNCTION, so it is a function of something.  That something is vt-r. I don't understand that "since".  v is a parameter in the given function, which you may assume to be constant.  E is a function of r (position vector) and t .  These are your variables. 9/18 7.30 -- For part (b) where we're comparing to equation 6.35, I'm not sure I understand what Griffiths means by the 'interaction energy' of  2 magnetic dipoles. Follow-up: Yep, I did! My printing has 'Intensity' and then 'Internal resistance' under I, and nothing that helped me under 'Energy' or 'Energy density'. But perhaps this is another thing that was fixed between printings? (There was a problem last semester that looked  different in my copy and yours, too, even though they're both 3rd edition.) OK, but that just gets me page 282 which doesn't actually define "interaction energy", it just gives formula 6.35 that we're supposed to comment on. (and page 165 which is the equivalent for electric dipoles). I still don't understand what he means by the term. Here's a short definition I found online: "The interaction energy is a potential energy of orientation associated with a magnetic torque." This makes me feel a little better, but it's still not something I feel totally comfortable with. I _think_ it's saying that the interaction energy is the energy that 2 dipoles have in each other's fields, which will depend on how they're oriented with respect to one another. Did you try using the book's index? Did you try "Dipole"? I found "dipole-energy of interaction of two, 165"  That entry is on page 564, top of second column. Page 165 gives you eqn 4.7 in problem 4.8.  As you can see, what we have is the energy of one dipole in the field of the other, just as the interaction energy of two point charges (discussed in 360, see 360notes3) is the energy of one point charge in the field of the other, ignoring self-energies. These references are to electric dipoles, but since the magnetic dipole field has the same form as the electric dipole field, and the energies both go as field squared, the results, and interpretation, are the same for magnetic dipoles. 9/18 31.79 -- 1) On Thursday, you reminded me that the charged loop also has an E field to worry about, but since we're looking at the center of the loop, wouldn't that be zero? Did you just want us to mention that, or should we be looking at points _just above_ the center of the loop so that z << r but z != 0? 2) How far back do we need to go in deriving the expression for the electromagnetic energy density, u_EM? 3) The expression that I'm getting for u_EM doesn't seem that remarkable, so either I'm doing something wrong, or I'm not sure what kind of analysis you're looking for before we plug in numbers. Perhaps this goes back to my question (1)? (1) If  the field is zero, but the charge is not, you need to say why. z=0 is fine. ( 2) You don't need to derive it, just use it. (3) Well, I found something I thought was a bit remarkable.  Look at how your answer depends on the physical parameters.  Is that what you expected? Sept  13 I'm not sure what Griffiths is getting at when he says "in the quasistatic case" in problem 7.24 part (a). Is he implying that we shouldn't consider the back EMF yet, or are we supposed to take the time average of dPhi/dt, or what? The current in the long wire changes SLOWLY, so you can calculate the B it produces using static methods (Ampere's law, for example, ignoring displacement current.) Sept 13 I'm  not sure if 1) the EMF in the toroid is the same as that through single loop [my gut feeling -- but that seems like the loops would be in parallel & that doesn't make sense since they're directly connected] or 2) the EMF in the toroid is N * (the EMF in one loop) [which I guess makes more sense if I think of the loops in series, but if the EMF is the potential difference around one loop, & they add all the way the toroid, how would you know where to start or end?]. And from there I'm not sure if the resistor is in series with the coils, or in parallel, or if that matters since it's the only resistor in the circuit. The emf is the integral around the whole circuit, which includes all N coils.  It doesn't matter where you start and end, but in a real toroid there would be a place where the wire connects out to somewhere else (like the resistor), so you might want to start there. Remember, that EMF and potential difference are NOT the same thing!  Only static fields contribute to potential difference. In series. Sept 13 Do we need to worry that the resistor in the circuit is throwing off the rotational symmetry of the toroid about the z-axis? Or can we assume that the resistor can be placed arbitrarily at any point in the circuit and argue that it doesn't matter where? Imagine that the resistor is connected to the toroid by very fine wire.  It does not change the geometry of the toroid at all. Sept 13 I'm thinking that if you had a flat, symmetrical "figure 8" of wire, you'd just add the flux through each loop to get the total flux Phi = Phi1 + Phi2, and therefore you could find the total EMF = -dPhi/dt. If you took that figure 8 and folded it so that you had 2 parallel loops, would the EMF be cut in half, or would it stay the same? I keep waffling on whether you consider the flux through each loop (EMF stays the same), or the flux through area that the SYSTEM presents (EMF cut in half). This is a very good question, as it really gets at the idea.  By folding the figure 8, you don't change the flux!  That is really the whole issue, right there. Notice that the integral of E dot dl doesn't change when you fold the loop, and so phi, and hence d phi/dt can't change either. Sept 13 For Griffiths 7.20 must I derive the magnetic field a distance z  above a current loop, and that for a magnetic dipole, or may I just  quote the results from Ch 5 of the textbook? You may quote the result.  But be sure that you COULD derive it if asked to.