Physics 7701: Problem Set #8

Recent changes to this page:

• 25-Oct-2013 --- original version.
• 27-Oct-2013 --- added more comments on capacitor force problem (#2).
• 29-Oct-2013 --- added more comments on 4b.

1. Simple capacitors. Use the definition of capacitance for two conductors as stated in the problem.
   • The general procedure is to assume charges +Q and -Q on the two conductors, use Gauss's law (based on the symmetry of the problem) to calculate the electric field, use the electric field to calculate the potential difference V (which can be calculated by a line integral in the most convenient direction and should be linearly dependent on Q), and then use Q=CV to find C.
   • Remember that capacitance should depend on geometric factors only.
   • Check your results. Units of course, but also limiting or special cases. For example, for the spheres, if you let b and a both get very large with fixed d = b - a, what system (i.e., what type of capacitor) does this become like? Does your result got to the correct formula? How is this limit reached for the case of the cylinder?
2. Force between conductors. [Note: My solution is based on finding the force from a derivative of the energy with respect to the separation of the plates. There are alternative methods that seem to work even more directly for this problem (but maybe not so easy to generalize). E.g., calculate the force on the charge of one plate due to the electric field from the other plate only. This gets the same answer as the energy derivative method.] For this system we can find the electric field from Gauss's law (state the approximations you make, e.g., what do you assume based on large A and small d?) and then integrate to find the energy as a function of the parameters in the problem. We can express this energy in terms of either the charge Q or the potential difference d.
   1. A small amount of work must be done to separate the plates a small amount Δd. This is the force (recall ΔW = FΔd). You are told to keep the charge fixed, so what version of the energy formula should you use? What sign should the answer be? (That is, do you expect an attractive [negative] or repulsive [positive] force?) Check that your answer scales as expected when you imagine scaling (e.g., doubling) the parameters such as the charge and the area.
   2. Now repeat except that the potential difference must be held fixed when you calculate the derivative that determines the force. Again, what sign do you expect? There is a subtlety here: the force is between charges, but at fixed potential charges move on and off the plates (we need a battery or equivalent to maintain the potential). So you need to calculate the change in energy not only due to the electric field in the capacitor but due to the energy change &Delta Q V (which you can do using the relation between Q and V). (The best discussion of this is in Zangwill 5.6.)
3. Two electrostatic theorems.
   1. The idea in applying Green's theorem is to choose f and g to get the ingredients you want. Obviously you want to choose one of them (say f) to be φ(x), but what can you choose for g so that the integral gives you φ(0)? How does the fact that there are no charges in the volume help you? Does one of the surface terms give you the average we want? How can you use the freedom to add a constant to g to ensure that the other term is zero on the surface?
   2. These types of theorems are usually proven by contradiction: assume the opposite result is true and derive consequences that contradict the assumption. To use part (a), you have to consider a spherical volume inside of R with your origin at the center. If you assume that the origin is a maximum or minimum, what can you conclude from part (a) that contradicts this assertion?
4. A variation of Coulomb's law. The first step in this problem is to figure out which formulas for electrostatics still hold when Coulomb's law for the potential is no longer simply proportional to 1/r. If you get stuck using the general function f, try it with the known case of 1/r and some other specific scalar function.
   1. What is the formula for the potential in terms of the charge density given the potential for a point charge?
      • Recall how to represent the charge density for a plane (check the units).
      • Carry out any integrals you can do, making any changes of variables to simplify the integrations (these changes are safest to do in Cartesian coordinates, based on my own experience :). What is the most useful coordinate system (Cartesian, cylindrical, or spherical) for your final result?
      • The answer for the scalar potential will be be in the form of a single integral. What variables should the potential depend on?
   2. There should be z dependence in the integral from part (a). What you would like is to move this dependence from the integrand to the limits of the integral. Then when you calculate the electric field from the potential you won't have an integral in your final answer. What variable change will do this?
      • You need to be able to take the derivative with respect to a variable that appears in the limits of a definite integral. (You can derive the result by using the definition of a derivative as the limit of [f(x+h)-f(x)]/h as h goes to zero. You will have the difference of two integrals, which you can interpret as a small area that is easy to approximate in the small h limit --- draw a picture.) If a function of that variable appears instead, use the chain rule of differentiation.
5. Variational principle for capacitance.
   1. How is the integral in the equation related to the energy stored in the volume? What is this same energy in terms of the capacitances and coefficients of induction (see lecture notes page 166) for the particular values of the potentials here?
   2. To show that the functional C[Ψ] is minimized when Ψ=Φ (the solution for the potential), write Ψ = Φ + δΦ. What is δΦ on the surfaces? Expand and use Green's identities to simplify. If you can show the difference with C is an integral with a positive definite integrand, you are done!
6. Electrostatic Green functions. Getting the expression for the difference is just a matter of plugging into Green's theorem and doing the integrals over delta functions that arise.
   1. Just use the definition of Dirichlet boundary conditions on the surface.
   2. Follow the instructions.
   3. Use the formula for Φ(x) as an integral over the Green function and the charge density with the shifted definition from (b) and show that the additional terms add to zero (use Gauss's law).