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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.
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}'
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
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.
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.