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## Homework Statement

A coaxial cable has a linear insulating material of magnetic susceptibility [tex]\chi_m[/tex] separating the conductors. A current I flows down the inner conductor and returns along the outer one. Find the magnetic field in the region between the tubes. As a check, calculate the magnetization and the bound currents and confirm that they generate the correct field

## Homework Equations

## The Attempt at a Solution

I don't have agreement between the two methods. I think my result for the first part is probably right, so it must be the second part that's wrong.

Firstly, I have

[tex]\oint \textbf{H} \cdot d \textbf{L} = I[/tex]

[tex]\Rightarrow 2 \pi r H = I[/tex]

[tex]\Rightarrow H = \frac{I}{2 \pi r} [/tex]

[tex]\Rightarrow B = \frac{\mu I}{2 \pi r} [/tex]

[tex]=(\chi_m + 1) \frac{\mu_0 I}{2 \pi r} [/tex]

And for the second part I have

[tex]\textbf{M}=\chi_m \textbf{H}[/tex]

[tex]=\chi_m \frac{I}{2 \pi r} \hat{\textbf{\phi}}[/tex]

[tex]\textbf{j_{bound}} =\nabla \cdot \textbf{M}[/tex]

[tex]=0[/tex]

[tex]\textbf{k_{bound}} =\textbf{M}\times \hat{\textbf{n}} [/tex]

[tex]=\textbf{M}\times \hat{\textbf{r}}[/tex]

[tex]=-\chi_m \frac{I}{2 \pi r} \hat{\textbf{z}}[/tex]

finally

[tex]\oint \textbf{B} \cdot d \textbf{L} = \mu_0 I (1+\textbf{k}_{bound})[/tex]

[tex]\Rightarrow B =\mu_0 \frac{I}{2 \pi r} (1-\frac{\chi_m}{2 \pi r} )[/tex]

Now clearly the bound current should be in the same direction as the current I, so I have messed up the cross product(EDIT: No, wait. It should be in the opposite direction to I, so I had it right the first time...not sure what I've done wrong then). The main thing though is the division of chi by 2 pi r, as I can't see how to get rid of that. Any help here would be appreciated, thanks :) (EDIT2: I just realised that I've neglected a factor of 1/r in computing the cross product, so there should be r^2 in the denominator of k_bound instead of r, but that doesn't solve my problem.)

P.S Sorry about the weird tex, I'm not sure what I've done. It's meant to read j_bound=div(M) and k_bound=M X n. The rest looks right.

(EDIT2: no wonder I'm having so much trouble with this problem, I've made so many mistakes. I just realised that the bound current is A/m, and I've been treating it as though it is just A. I also realised that I could r=r_1 for the surface current because we are only dealing with a current that resides at r_1. I now have this

[tex]B=I/2\pi r(1-\chi_m/r_1)[/tex]

It's looking much better, but still not right. I'm still hoping someone can help me with this.

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