by using Biot Savart Law, the calculation is very terrible. since it involve the cubic inverse distance. However, using vector potential is a way to make an analytic solution. first in cylindrical coordinate, the current density is:

the vector potential, without loss of generality, we set the field point on x-z plane and the vector potential is pointing direction.

Oh, shit, i am tired on typing equation by Latex. anyway,

the finial result can be obtained in J.D. Jackson’s book. and it take the form of Elliptic Integral.

where K(m) is the complete elliptic integral of first kind, and E(m) is the complete elliptic integral of second kind.

and

and using the derivative properties of elliptic integral, we can have the magnetic field in analytic form. The coil radius is , at x-y plan. The observation point is

Here is the PDF for the detail calculation. Field of Single Coil. Note that, in the pdf, the B-field is smaller by factor of 2. The denominator should be . As we check the magnetic field on the z-axis, the above formula will reduce to

where

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elmanuelito

Sep 15, 2013@ 19:06:46Having a quick look at your pdf file, it seems you ended up with 4 pi in the denominator of the B field while it should probably be 2 pi. Please double check what I’m saying, but I believe the formula should give on the axis:

$B_z(\rho=0, z=0 )=\frac{ \mu_0 I }{ 2 a }$

The formula you wrote gives half of this value since K(0)=E(0)=pi/2.

The scaling error could come maybe from the derivative of the elliptic function and the k, k^2 convention, but I didn’t have time to look at it.

Also it seems you results have opposite sign than usual, but that’s probably just a matter of convention.

Thanks for a nice document, I hope you can shed light on this problem and maybe prove me wrong.

Cheers!

GoLuckyRyan

Nov 07, 2014@ 12:39:24Thank you very much for the comment! True, the derivative of the K and E is wrong, there is no 1/2 after the derivative.

For the sign problem. I cannot find the origin. Somehow, the plotting is correct. I guess it could be :

1) I corrected it when plotting.

2) That could be in the Curl(A), but I did not check.

Thank you very much! and sorry for late reply.

GoLuckyRyan

Sep 16, 2015@ 15:42:19The sign error was found on the line F(pi)-F(0).