What Is the Magnetic Vector Potential of a Finite Length Coil?

[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 length Coil.

Homework Statement

The equation for a magnetic vector potential is given by.

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

Homework 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

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

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.

Please help me out.

 

[Goddar]

Hi.
Typically, direct integrations of this kind are very difficult and you're going to have a hard time if you don't make any simplifications. For example, the field here is likely to get very complicated near the coil so i would certainly not expect a solution in closed form... Instead i would work with a solenoid, for which the magnetic field is easy to determine and is a good approximation as long as you don't get too close to the coil, then find a suitable vector potential from that.
(Incidentally, in your attempt of a solution the denominator is wrong...)