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## Homework Statement

An infinitely long cylinder of radius R carries a "frozen-in" magnetization, parallel to the axis, ## \vec M = ks \hat z ##. There are no free currents. Find the magnetic field inside and outside the cylinder by two different methods:

(a) Locate all the bound currents and calculate the field they produce.

(b) Use Ampere's law (in the form of 6.20) to find ## \vec H ##, and then get ## \vec B ## from 6.18.

## Homework Equations

(6.20) $$ \oint \vec H \cdot d \vec l = I_{f, enc} $$

(6.18) $$ \vec H = \frac{1}{\mu_{0}} \vec B - \vec M $$

(*) $$ \vec J_{B} = \nabla \times \vec M $$

(**) $$ \vec K_{B} = \vec M \times \hat n $$

## The Attempt at a Solution

Let s be the distance from the z axis. Then using the form of the curl for cylindrical coordinates,

$$ \vec J = [ \frac {\partial } {\partial s} (ks) ] \hat \phi = -k \hat \phi, $$

and

$$ \vec K = ks \hat \phi. $$

Since bound currents are circulating on the surface, the object behaves somewhat like a solenoid, so the magnetic field due to the surface current is

$$ \vec B = \mu_{0} nI, I = KL \rightarrow \vec B = \mu_{0} n(kR)(2\pi R) $$

I am not very sure about how to handle the volume current, but I think it should act like concentric solenoids, so I integrated from 0 to R, with I = JA

$$ - k \mu_{0} n \int_0^R \pi s^2 \, ds = -\frac{1}{3} k \mu_{0} n R^3 $$

And like a solenoid the magnetic field outside is zero.

But when I do part (b), 6.20 and 6.18 imply that ## \vec B = \mu_{0} \vec M , ## which is clearly not the same as what I got for part (a). Where did I go wrong?