Find Magnetic Field of an Elliptical Solenoid | Biot-Savart Law

[anothersivil]

Homework Statement

I have to find an expression for the magnetic field of an elliptical solenoid. This is an actual solenoid sitting on the desk next to me, but even an infinite/very long solenoid approximation would be a wonderful start.

Homework Equations

Biot-Savart law

The Attempt at a Solution

Therein lies the problem... I've not the slightest clue where to start. I just started working in a lab at my school and the grad student here left me this problem while he's away on vacation. I've only seen Biot-Savart from my Intro to Electromagnetism class, so I'm not to savvy with it yet. A poke in the right direction would be greatly appreciated!

 

[Dr Transport]

How about starting with elliptical coordinates and setting up Biot-Savart...

http://en.wikipedia.org/wiki/Elliptic_coordinates will get you started.

 

[anothersivil]

I tried looking up elliptical cylindrical coordinates and when I saw all of the sinh's and cosh's I got kinda scared... but I did come across something else. I found that for an elliptical loop of current, the magnetic field is:

u[0] = 4*pi*10^(-7)
I = current
a = semi-major axis
b = semi-minor axis
k = sqrt(1-a^2/b^2)

B = (u[0]*I)/(pi*a)*E(k), where E(k) is a complete elliptical integral of the second kind.

E(k) = int( sqrt(1-k^2*sin^2(theta)), theta = 0..pi/2).

Given the specs that I have, k=0.826, I found an integral table (CRC standard mathematical tables, 18th edition) and found that E(k) = 1.372.

Plugging in my other numbers (this is assuming 1 amp of current), I got the field to be:

B = 1.065 gauss.

This seems to be somewhat correct since the current for a circular loop of about the same size is pretty small too. Now I need to find a way to incorporate the number of turns (in this case, 1010 turns) into my equation so that I can get an idea of how strong the elliptical solenoid will be. Any ideas?

 

[Ben Niehoff]

I'm fairly sure that the magnetic field inside an ideal (infinite) solenoid is independent of the shape of its cross section. What matters are the cross-sectional area and the sheet current density.