Consider a solid sphere of radius 5 cm made of a perfect electric conductor. If one million electrons are added to this sphere, these electrons will be distributed.

Consider a function $$\overrightarrow f=\frac1{r^2}\widehat r$$ where r is the distance from the origin and $$\widehat r$$ is the unit vector in the radial direction. The divergence of the function over a sphere of radius R, which includes the origin, is

Match the following.

Two semi-infinite dielectric regions are separated by a plane boundary at y = 0. The dielectric constants of region 1 (y< 0) and region 2 (y>0) are 2 and 5, respectively. Region 1 has uniform electric field $$\overrightarrow E=3{\widehat a}_x\;+\;4{\widehat a}_y\;+\;2{\widehat a}_z$$, where $${\widehat a}_x$$, $${\widehat a}_y$$, and $${\widehat a}_Z$$ are unit vectors along the x, y and z axes, respectively. The electric field in region 2 is