# Magnetic Vector Potential of Hollow Cylinder

I'm modeling a MIT (Magnetic Induction Tomography) apparatus. In order to extract the Jacobian/stiffness matrices for the conductivity distribution, I need to calculate the magnetic vector potentials, A, of each coil using "pre-curled" Biot-Savart.

The coils were modeled as cylinder, but with the following characteristics: 50-turn homogenized coil of 18 gauge (~1[mm] cross-sectional diameter) copper wire. For this 8-coil system, there needs to be 28 unique measurements, ergo 28 Jacobian matrices given by the following:

So, for a small change in voltage between two coils, there will be a small change in conductivity, where Ii and Ij are the current sources for each coil pair, omega is the angular frequency, sigma is the conductivity, big omega relates to some space/pixel and Ai and Aj are the magnetic vector potentials of coil i and j, respectively.

Instead of discretizing/meshing the coils, I would like to find the magnetic vector potential by either using the surface or line integral equation for A. From there I would then be able to calculate A for the two coils i and j, and then take the integral of their dot product to generate the Jacobians.

My question is how do you calculate the magnetic vector potential of the multi-turn coil? I think it'd be easier to treat it as a homogenized copper cylinder, in which case I think I'd be using this equation:

If all I'm given is the coil current (I), how would I setup the above integral to calculate A for the coil/cylinder using Cartesian coordinates? Any help would be greatly appreciated.

• Nm. I got it. I just extracted values from Comsol in a 128x128 grid, and did all the post-processing (dot products, multiplications by the constant, etc.) in Matlab. I wanted to do this in a purely mathematical, rigorous way, and I might do that at some point as an academic exercise. Jun 10, 2018 at 16:24