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A positive charge $q$ is placed in a spherical cavity made in a positively charged sphere. The centres of sphere and cavity are displaced by a small distance $\vec l $ . Force on charge $q$ is :
in the direction parallel to vector $\vec l $
in radial direction
in a direction which depends on the magnitude of charge density in sphere
direction can not be determined.
Solution
Electric field inside solid sphere is given as$:$ $E=\frac{\rho \vec{r}}{3 \epsilon_{o}}$
where $\rho$ is charge density and $\vec{r}$ is position vector of point $P$ inside sphere $w.r.t.$ it's center. Using superposition principle for resulting electric field$:$
$E=E_{\text {sphere}}-E_{\text {cavity}}$
$=\frac{\rho r_{p /\vec{sphere}}}{3 \epsilon_{o}}-\frac{\rho r_{p / \vec {carity}}}{3 \epsilon_{o}}$
$=\frac{\rho\left(r_{p / \vec {sphere}}-r_{p / \vec {cavity}}\right)}{3 \epsilon_{\mathrm{o}}}$
$=\frac{\rho \vec{l}}{3 \epsilon_{o}}$
Thus the force will be parallel to the direction of displacement $\vec{l}$.
