Magnetic field in asymmetric hollow cylinder

[bobred]

Homework Statement

Cylinder of radius $a$ and a cylindrical hole $b < a$ is displaced a distance $d$ in x-direction. Current density $\textbf{J}=J_z\textbf{e}_z$. Show that a uniform magnetic field inside the hole is

$\textbf{B}=\frac{\mu_0}{2}J_zd\textbf{e}_y$

Homework Equations

Using previous result of whole cylinder

$\textbf{B}=\frac{\mu_0}{2}J_z(-y\textbf{e}_x + x\textbf{e}_y)$

and that the cylindrical hole can be modeled as a charge density of $-J_z\textbf{e}_z$.

The Attempt at a Solution

I tried superposition of fields so

$\textbf{B}_1=\frac{\mu_0}{2}J_z(-y\textbf{e}_x + x\textbf{e}_y)$

$\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x + d)\textbf{e}_y)$

$\textbf{B}= \textbf{B}_1 + \textbf{B}_2$ to which I get

$\textbf{B}=-\frac{\mu_0}{2}J_zd\textbf{e}_y$

Any ideas, is this the right approach?

 

[bobred]

Sorted, silly mistake

$\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x + d)\textbf{e}_y)$

should be

$\textbf{B}_2=-\frac{\mu_0}{2}J_z(-y\textbf{e}_x + (x - d)\textbf{e}_y)$