# Electric field of a current-carrying wire?

1. May 31, 2010

### Jerbearrrrrr

Suppose there's a square wire loop in the x-y plane centred on the origin. There's a current I flowing in the loop for t>0 (nothing for t<0)
What's the electric field due to the entire loop at (0,0,z) as a function of time?

I have no idea how to approach this.

2. May 31, 2010

### stevenb

Personally, I would approach this using Fourier Analysis. You are looking for the time domain response to a step function. Maxwell's equations tell you the time evolution of the system. Note that the system is linear because no media are specified.

3. May 31, 2010

### jamesmo

Alright, the first thing to try to do is to break it down into a couple of simpler problems, then use superposition. Can you first write down the electric field from a single wire at a distance of z' from the center?

4. May 31, 2010

### jamesmo

Alright, the first thing to try to do is to break it down into a couple of simpler problems, then use superposition. Can you first write down the electric field from a single wire at a distance of z' from the center?

If you are completely lost, start with writing down the magnetic fields at those points at t<0 (I know that one is easy but do it anyway to understand the point.) and t> 0. Once you have done that, the previous poster had suggested using a Fourier series. That is a good idea. Break down j ( a step function) into a Fourier series. so the dB/dt is better defined as a continuous function.

5. May 31, 2010

### Jerbearrrrrr

This is in the context of SR, so we need to introduce retarded times.
I don't remember the problem exactly, but the integral came out as something doable I think. I used:
$$A^a=\frac{\mu_0}{4\pi}\int d^3 x' \frac{ j^a (t',x')}{|x-x'|}$$
t'=retarded time:
$$t ^2 -'t^2 = x^2 - x'^2$$
Parameterize the wire, say, y from -l/2 to +l/2 for a segment length l.
$$A^a = C \int dy \frac {\theta(t-\sqrt(l^2/4 +y^2+z^2) I^a}{\sqrt{z^2 +l^2/4 + y^2}}$$
Integrate by parts to get a delta function and you get something funny with delta functions but it's okay.
Assume I can work through the integrals and find B (i'll get some paper at some point). Dunno if that's right, but the vector calculus doesn't worry me as much as the justifying.

It says justify your answer carefully - are we toying with assumptions or anything. Or do you think it means be careful with the signs when adding up the contributions?

(this isn't homework, so don't be afraid to like 'spoil the ending', it's part of an exam that's already passed x.x but anyway)
Oh balls, just realized I crapped up the integral. Oh well. I had like 20 minutes and I wasn't in electrodynamics mode (The paper had everything from set theory to complex analysis) haha. sigh.

Last edited: May 31, 2010
6. Jun 1, 2010

### stevenb

It seems to me like you are on the right track here in the analysis with the use of retarded potentials.

The way that I would approach this problem would be to look up the analysis of loop antennas. I believe the square loop can be considered 4 short dipole antennas and superpostion can be used. Usually, antenna derivations are done for sinusoidal steady state conditions, so this is where the Fourier Analysis can come in.