Off-axis magnetic field due to a current loop in cartesian coordinates

[johnpatitucci]

Hi there,


a few days ago I derived the probably well-know expression for the magnetic field of a current loop including elliptic integrals of the first and second kind (it can be seen here http://plasmalab.pbworks.com/f/bfield.pdf" ). As I'd like to rotate and shift the position of the current loop I tried to transform the expressions for the fields $$B_r, B_z$$ into cartesian coordinates but failed because i also need to transform the elliptic integrals and I don't know how to do that.

Do you think I have to derive the whole thing again (starting with the current density's in x-,y- and z-direction but now strictly in cartesian coordinates which is very tedious) or is there a way to get the wanted cartesian expression from the one's in cylindrcal coordinates which I have already written down.

Thanks for your comments !

 

[marcusl]

I would instead express the field in spherical coordinates (see Jackson's Classical Electrodynamics, or everywhere on the web). Then use the spherical harmonic addition theorem for rotations, and spherical harmonic translation theorem for translations, to position your loop.

You can find these theorems in Steinborn and Ruedenberg, Rotation and Translation of...Spherical Harmonics, Advances in Quantum Chemistry, v. 7 (1979), and certainly elsewhere.

 

[johnpatitucci]

Thanks @marcusl. That is a pretty good idea and I going to try it now.


Yesterday, I found a paper (you can see it here: http://ntrs.nasa.gov/archive/nasa/casi.ntrs.nasa.gov/20010038494_2001057024.pdf" ) where the current loop expressions are written down in spherical as well as cartesian coordinates. On the first page the authors claim to have derived the cartesian result from the spherical expression but I don't know how. Anybody got a clue how one could do that ?