Question

A long, cylindrical wire of radius b carries a current i distributed uniformly over its cross section. Find the magnitude of the magnetic field at a point inside the wire at a distance a from the axis.  

Answer 1

Given:
Magnitude of current = i 
Radius of the wire = b 

For a point at a distance a from the axis,
Current enclosed,

\[i' = \frac{i}{\pi b^2} \times \pi a^2\]

By Ampere's circuital law,

\[\oint B . dl = \mu_0 i'\]

For the given conditions,

\[B \times 2\pi a =    \mu_0 \frac{i}{\pi b^2} \times \pi a^2 \] 

\[ \Rightarrow B = \frac{\mu_0 ia}{2\pi b^2}\]