Cylindrical shell in magnetic field [closed]

Question

Exercise: 'Infinitely long cylindrical shell of inner radius a and outer radius b of material of magnetic susceptibility χ is placed in otherwise uniform magnetic $B_0$ perpendicular to cylinder's axis. What is the resulting magnetic field. Hint: in this problem field H can be expressed through scalar potential $H = -\nabla{w}$.'

I found the general formula (Coordinate system: $\textbf{B}_0=|B_0|\hat{\textbf{x}}$, $x=r\cos{\phi}$) for potential w in three possible zones.
The inside of cylinder (r < a): $w(r,\phi) = k_1r\cos{\phi}$
The shell (a < r < b): $w(r,\phi) = (k_2r+\frac{k_3}{r})\cos{\phi}$
The outside cylinder (b < r): $w(r,\phi) = (-\frac{B_0}{\mu_0}r+\frac{k_4}{r})\cos{\phi}$
In order to find coefficients, I need to know boundary conditions. What are boundary conditions are at $r=a$ and $r=b$?

Answer 1

The magnetic boundary conditions at an interface are:

$$\mathbf{B}_{in,\perp} = \mathbf{B}_{out,\perp}$$ and

$$\mathbf{H}_{in,||} = \mathbf{H}_{out,||}$$

Here, $\mathbf{H}= - \nabla w$ and $\mathbf{B}= \mu_0(\mathbf{H} + \mathbf{M}) = \mu_0 [- \nabla w + \chi (- \nabla w)]$.

With these equations, you can set the boundary conditions at the two interfaces for the magnetic scalar potential $w$.