Question

Inside a long straight uniform wire 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 sire. If the magnetic induction inside the cavity is $\frac{1}{X}{\mu }_0\overrightarrow{j}\times \overrightarrow{l}$. Find X?

Consider, in particular the case $I=0$.

Answer 1

Consider a cross section of the cylinder perpendicular to its axis. The cylindrical cavity is shown at a distance $l$ from $O$ the axis point $P$ inside the cavity. The cavity means an absence of current which is equivalent to an opposite current density $j$ in the cavity plus the true current through the cross section of the wire.
By superposition theorem.
$\overrightarrow{B_P}=\frac{1}{2}{\mu }_0\overrightarrow{j}\times (\overrightarrow{OP}-\overrightarrow{CP})=\frac{1}{2}{\mu }_0\overrightarrow{j}\times \overrightarrow{l}$
It is uniform field.
In case $l$ vanishes, point $C$ coincides with the point $O$, then there is no magnetic field in the cavity.