# Magnetic Vector Potential of Coil

1. Feb 22, 2014

### VVS

Hi

Basically I want to examine the effect of a magnetic vector potential created by a coil on the spin of an electron in a Coulomb potential.
The Hamiltonian of a charged particle in a Vector Potential is well known.
But I have a problem in calculating the Magnetic Vector Potential of a finite lenght Coil.

1. The problem statement, all variables and given/known data

The equation for a magnetic vector potential is given by.

$\vec{A}(\vec{r},t)=\frac{\mu_{0}}{4\pi}\int_{\Re^{3}}\frac{\vec{J}(\vec{r}',t)}{\left|\vec{r}-\vec{r}'\right|}d^{3}\vec{r}'$

2. Relevant equations
The vector equation in cylindrical coordinates for a coil is

$\vec{r}'=\hat{i}\rho_{0} cos(\vartheta)+\hat{j}\rho_{0} sin(\vartheta)+\hat{k}\frac{\vartheta}{2\pi}$

Therefore the equation for the Current Density is

$\vec{J}(\vec{r}',t)=(-\hat{i}\rho_{0} sin(\vartheta)+\hat{j}\rho_{0} cos(\vartheta)+\hat{k}\frac{1}{2\pi})\delta (\rho-\rho_{0})$

The position of any point in space in cylindrical coordinates is given by

$\vec{r}=\hat{i}\rho cos(\vartheta)+\hat{j}\rho sin(\vartheta)+\hat{k}z$

3. The attempt at a solution
One can write the Volume integral in cylindrical coordinates.

$\vec{A}(\vec{r},t)=\frac{\mu_{0}}{4\pi}\int_{0}^{∞}\int_{0}^{2\pi}\int_{-h/2}^{h/2}\frac{(-\hat{i}\rho_{0} sin(\vartheta)+\hat{j}\rho_{0} cos(\vartheta)+\hat{k}\frac{1}{2\pi})\delta (\rho-\rho_{0})}{(\rho-\rho_{0})^2+(z-\frac{1}{2\pi})^2}\rho dz d\vartheta d\rho$

And performing the integral you finally end up with only the k component.
Which must be wrong because I know that the Magnetic Vector Potential is finite outside of the coil.