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)