Physics 7701: Problem Set #8
Here are some hints, suggestions, and comments on the assignment.
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.
- 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?
- 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
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.
- 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.
- 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.)
- Two electrostatic theorems.
- 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?
- 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?
- 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.
- 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?
- 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.
- Variational principle for capacitance.
- 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?
- 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!
- 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
- Just use the definition of Dirichlet boundary conditions on the surface.
- Follow the instructions.
- 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).
Physics 7701: Assignment #8 hints.
Last modified: 09:43 am, October 29, 2013.