Question

Inside a long straight uniform with of round cross-section there is a long round cylindrical cavity whose axis is parallel to the axis of the wire and displaced from the latter by a distance $l$. A direct current of density $j$ flows along the wire. Find the magnetic induction inside the cavity. Consider, in particular, the case $I=0$.

Answer 1

We can think of the given current which will be assumed uniform, as arising due to a negative current, flowing in the cavity, superimposed on the true current, everywhere including the cavity. Then, by superposition,
$\overrightarrow{B}=\frac{1}{2}{\mu }_0\overrightarrow{j}\times (A\overrightarrow{P}-B\overrightarrow{P})=\frac{1}{2}{\mu }_0\overrightarrow{j}\times \overrightarrow{l}$
If $\overrightarrow{l}$ vanishes so that the cavity is concentric with the conductor, there is no magnetic field in the cavity.