Jackson's 5.35: Insulated Coil on a Sphere

[shaun_chou]

Homework Statement

The question is Jackson's 5.35. Insulated coil on a sphere (radius a) to generate uniform field B0 in the z-direction inside the sphere and dipole field outside the sphere. Conductivity is $$\sigma$$ and permeability is $$\mu$$. 1st question is to find the surface current K and prove that vector potential $$A_{\phi}=\frac{B_0a^2}{2}\frac{r_{<}}{r_{>}^2}\sin\theta$$ in which $$r_{<}$$ and $$r_{>}$$ is the smaller (bigger) of r and a.

Homework Equations

$$\boldface{B}=\nabla X\boldface{A}$$

The Attempt at a Solution

I can only find A using $$B=-1/2 rXA$$ inside the sphere but I didn't know the magnetic field outside. Or I have to use the example in the text to integrate the vector potential of a circular current loop from the bottom to top of the sphere?

 

[gabbagabbahey]

I think you need to first determine the current density, and then integrate it (using $\textbf{A}=\frac{\mu_0}{4\pi}\int \frac{\textbf{K}}{|\textbf{x}-\textbf{x}'|}da'$ ) to find the vector potential directly.

Alternatively, if may be sufficient to simply take the divergence and curl of the vector potential you are given and show that it produces the correct field and an appropriate gauge. However, I suspect Jackson intends you to use the first method.

 

[shaun_chou]

I'm sorry for the stupid question but I really don't know how to get K with the current condition. I'm going nuts with Jackson...sad!

 

[gabbagabbahey]

Let's start with the exact form of $\textbf{B}$...can you deduce what that must be?