[2008EM/EVAL-TEST-01]

$ \newcommand {\pieps}{4 \pi \epsilon_0} $

           IMSc-IV Physics-IV : Electricity and Magnetism Jan-Apr 2008 MM : 20

• Show that the magnitude of the electric field of a charged ring, of radius $R$ and total charge $Q$ uniformly distributed, at a point on the axis at distance $d$ from the center is given by $$ E = \frac{Q}{\pieps}\,\frac{d}{(d^2+R^2)^{3/2}} $$
• Four charges $q_1,q_2,q_3,q_4$ are placed on the corners of a square of side $a$.
  1. Find an expression for the potential at a point distance $d$ just above the center.
  2. Obtain an approximate expression for the potential when $d$ is large.
  3. Compute the value of the potential at the center taking and side of the square to be $a=1.0m$.]
• 1. Using Gauss law show that the electric field due to a uniformly charged solid sphere, having radius $R$ and total charge $Q$, at a point inside the sphere is given by  $$ \vec{E} = \frac{Q}{\pieps}\, \frac{\vec{r}}{R^3} $$
  2. Total charge $Q$ is uniformly distributed over volume of a sphere a radius $R$ and charge $-Q$ also uniformly distributed over volume of a second sphere of the same radius. If the two spheres overlap, show that the electric field in the overlap region is uniform. \hfill[4]