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?

 

[paranormal]

I would first clarify the question and make sure I understand the problem correctly. The problem seems to be asking for the surface current K and the vector potential A_{\phi} for an insulated coil on a sphere that generates a uniform field inside the sphere and a dipole field outside. The parameters given are the radius of the sphere, the conductivity and permeability of the material, and the desired field inside the sphere.

To solve this problem, I would start by using the given information to write down the relevant equations and then manipulate them to find the desired quantities. The equation B=\nabla X A can be used to find the surface current K, which can then be used to find the vector potential A_{\phi}.

I would also check if the given solution A_{\phi} satisfies the boundary conditions for both inside and outside the sphere. If it does not, then further calculations or adjustments may be needed.

If I am unable to solve the problem using the given information, I would consult relevant textbooks or literature to find similar examples and use them as a guide to solving the problem. Additionally, I would seek help from colleagues or professors who may have more expertise in this particular area of physics. Overall, solving this problem would require a combination of understanding the concepts, using relevant equations, and applying problem-solving skills.