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## Homework Statement

A spherical capacitor with inner and outer radii a and b, contains a dielectric material with small conductivity [itex] \sigma [/itex] between its spheres.Find the vector potential and magnetic field of this configuration.

## Homework Equations

## The Attempt at a Solution

[itex]

\nabla^2\phi=0 \Rightarrow \frac{1}{r^2}(r^2\frac{d\phi}{dr})=0 \Rightarrow \phi=-\frac{p}{r}+q \\

I=\frac{\Delta \phi}{R}=\frac{p(\frac 1 a -\frac 1 b)}{ \int_a^b \frac{dr}{4\pi \sigma r^2 }}\Rightarrow I=4\pi \sigma p \Rightarrow \vec{J}=-\frac{\sigma p}{r^2}\hat{r}

[/itex]

[itex]

\vec{A}= - \frac{\mu_0}{4\pi} \int_a^b \int_0^\pi \int_0^{2\pi} \frac{\sigma p \hat{r}}{r^{'2}}\frac{r^{'2} \sin\theta d\varphi d\theta dr}{\sqrt{ r^2+r^{'2}-2r r^{'} \cos\gamma }}=

- \frac{\sigma p \mu_0}{4\pi r} \int_a^b \int_0^\pi \int_0^{2\pi} \hat{r} \frac{\sin\theta d\varphi^{'} d\theta^{'} dr^{'}}{\sqrt{ 1+(\frac{r^{'}}{r})^2-2 \frac{r^{'}}{r} \cos\gamma }}=

- \frac{\sigma p \mu_0}{4\pi r} \int_a^b \int_0^\pi \int_0^{2\pi} \hat{r} \sin\theta (\sum_{n=0}^\infty P_n(\cos\gamma) (\frac{r^{'}}{r})^n) d\varphi^{'} d\theta^{'} dr^{'}

[/itex]

And then,I expanded the Legendre functions using the following identity:

[itex]

P_n(\cos\gamma)=\frac{4\pi}{2n+1}\sum_{m=-n}^n Y^{m*}_n(\theta^{'},\varphi^{'})Y^m_n(\theta,\varphi)

[/itex]

After some calculations,I arrived at:

[itex]

\vec{A}=\frac{\sigma \mu_0 p}{12 r^2}(b^2-a^2)\sin\theta e^{i\varphi}(\hat{x}+\hat{y})

[/itex]

As you can see,its complex!

Another issue is about the symmetry of the problem. The spherical symmetry of the configurations suggests that the magnetic field should be radial but a radial magnetic field violates Gauss's law in magnetism([itex] \oint \vec{B}\cdot \vec{dS}=0 [/itex]). On the other hand,any other form for the magnetic field, doesn't respect the symmetry of the configuration.

I even thought that maybe the magnetic field is zero but as you can see easily,the vector potential isn't the gradient of a scalar field.

Please help!

Thanks