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Elliptical Solenoid

  • #1

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!
 
Last edited:

Answers and Replies

  • #2
Dr Transport
Science Advisor
Gold Member
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  • #3
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?
 
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  • #4
Ben Niehoff
Science Advisor
Gold Member
1,879
160
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.
 

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