A long cylinder of uniform cross section and radius R is carrying a current i along its length and current density is uniform cross section and radius r in the cylinder parallel to its length. The axis of the cylinderical cavity is separated by a distance d from the axis of the cylinder. Find the magnetic field at the axis of cylinder.

Explanation

Current density `j=i/(pi(R^@-r^2))`
Current through the smaller cross section
`i'=jpir^2` (in absence of cavity)
Magnetic field due to this current at a point on the axis of bigger
cylinder
`B_1=(mu_0i')/(2pid)=(mu_0)/(2pid)xxjpir^2=(mu_0)xxi/(pi(R^@-r^2))`
Let `B_2` be the magnetic field due to remaining portion. Then
`B_1+B_2=0 or B_2==-B_1` or `|B_2|=|B_1|`
`:. B_2=(mu_0i)/(2pid)xxr^2/(R^2-r^2)`