Homework Help: Total energy and power of electromagnetic fields

1. Apr 20, 2015

RawrSpoon

1. The problem statement, all variables and given/known data
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.

2. Relevant equations
I've solved a and b, the electric field is
$$E= \frac{It}{\pi a^2 \epsilon_0}\hat{z}$$
the magnetic field is
$$B= \frac{\mu_0 I s}{\pi a^2}\hat{\phi}$$
the energy density is
$$u_{em}= \frac{I^2 \mu_0}{8 \pi^2 a^4}[4c^2t^2+s^2]$$
the Poynting vector is
$$S= -\frac{I^2 t s}{2 \epsilon_0 \pi^2 a^4 }\hat{s}$$

However, I don't know how to solve c. How would I find the total energy?

3. The attempt at a solution
First I figured I could solve this via an addition of work done for each magnetic and electric fields
$$W_{total}=\frac{\epsilon_0}{2}\int E^2 d\tau + \frac{1}{2 \mu_0}\int B^2 d\tau$$
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
$$U_{em}=\int u_{em} d\tau$$

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
$$U_{em}=\frac{\mu_0 w I^2 b^2}{2 \pi a^4}[(ct)^2+\frac{b^2}{8}]$$
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!

2. Apr 20, 2015

RawrSpoon

Nevermind, I solved it after all! I used my original method by finding the sum of the two components of work and realized that the volume element in cylindrical coordinates is $$s ds d\phi dz$$ so I solved the integrals that way, which gave me a variation of the solution. I had completely forgotten that w was the length of the gap, and when evaluating the integrals I realized that the z component is from 0 to w, which is why the answer made sense.

The power was simple after that.

Nevertheless, hopefully this post will help anyone stuck on this problem anyway :P