- #1
RawrSpoon
- 18
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Homework Statement
Consider the charging capacitor in problem 7.34
(A fat wire, radius a, carries a constant current I, uniformly distributed over its cross section. A narrow gap of wire, of width w, w<<a, forms a parallel-plate capacitor)
a) Find the electric and magnetic fields in the gap, as functions of the distance s from the axis and the time t (Assume the charge is zero at t=0)
b) Find the energy density uem and the Poynting vector S in the gap.
c) Determine the total energy in the gap, as a function of time. Calculate the total power flowing into the gap by integrating the Poynting vector over the appropriate surface. Check that the power input is equal to the rate of increase of energy in the gap.
Homework Equations
I've solved a and b, the electric field is
[tex]E= \frac{It}{\pi a^2 \epsilon_0}\hat{z}[/tex]
the magnetic field is
[tex]B= \frac{\mu_0 I s}{\pi a^2}\hat{\phi}[/tex]
the energy density is
[tex]u_{em}= \frac{I^2 \mu_0}{8 \pi^2 a^4}[4c^2t^2+s^2][/tex]
the Poynting vector is
[tex]S= -\frac{I^2 t s}{2 \epsilon_0 \pi^2 a^4 }\hat{s}[/tex]
However, I don't know how to solve c. How would I find the total energy?
The Attempt at a Solution
First I figured I could solve this via an addition of work done for each magnetic and electric fields
[tex]W_{total}=\frac{\epsilon_0}{2}\int E^2 d\tau + \frac{1}{2 \mu_0}\int B^2 d\tau[/tex]
However, this didn't give me a solution in the solutions manual.
I then figured that since the energy density uem is the energy per unit volume, I could integrate via
[tex]U_{em}=\int u_{em} d\tau[/tex]
Alternatively, as the Poynting vector is the energy per unit time, per unit area, I figure maybe I could integrate twice, once as a surface integral and the other with respect to time?
I'm kind of lost, because while I've done just fine without the solutions manual, my attempt at the work haven't been fruitful. Additionally, the solutions manual states the solution is
[tex]U_{em}=\frac{\mu_0 w I^2 b^2}{2 \pi a^4}[(ct)^2+\frac{b^2}{8}][/tex]
in which I have no idea what w is supposed to represent.
I'd greatly appreciate it if anyone could point me in the right direction!