A positive charge is distributed in a spherical region with charge density $\mathrm{\rho }= {\mathrm{\rho }}_0 \mathrm{r}$ for $\mathrm{r} \le \mathrm{R}$ (where ${\mathrm{\rho }}_0$ is a positive constant and $\mathrm{r}$ is the distance from centre). Find out electric potential and electric field at following locations.

(a) At a distance $\mathrm{r}$ from centre inside the sphere.

(b) At a distance r from centre outside the sphere.

A ball of radius $R$ carries a positive charge whose volume density depends only on the separation $r$ from the ball’s centre as $\rho ={\rho }_0 \left(\frac{1 – r}{R}\right)$, where ${\rho }_0$ is a constant. Assuming the permittivity of the ball and the environment to be equal to unity, find :

(i) The magnitude of the electric field strength as a function of the distance r both inside and outside the ball;

(ii) The maximum intensity $E_{\max }$ and the corresponding distance $r_{\mathrm{m}}$.

Consider an equilateral triangle $ABC$ of side $2a$ in the plane of the paper as shown. The centroid of the triangle is $O$. Equal charges $\left(Q\right)$ are fixed at the vertices $A, B$ and $C$. In what follows consider all motion and situations to be confined the plane of the paper.

(a) A test charge $\left(q\right)$, of same sign as $Q$, is placed on the median $AD$ at a point at a distance $\delta$ below $O$. Obtain the force $\left(F\right)$ felt by the test charge.
(b) Assuming $\delta \ll a$ discuss the motion of the test charge when it is released.
(c) Obtain the force $\left(F_D\right)$ on this test charge if it is placed at the point $D$ as shown in the figure.
(d) In the figure below mark the approximate locations of the equilibrium point(s) for this system. Justify your answer.
(e) Is the equilibrium at $O$ stable or unstable if we displace the test charge in the direction of $OP$? The line $PQ$ is parallel to the base $BC$. Justify your answer.
(f) Consider a rectangle $ABCD$. Equal charges are fixed at the vertices $A, B, C$ and $D. O$ is the centroid. In the figure below mark the approximate locations of all the neutral points of the system for a test charge with same sign as the charges on the vertices. Dotted lines are drawn for the reference.
(g) How many neutral points are possible for a system in which $N$ charges are placed at the $N$ vertices of a regular $N$ sided polygon ?.

An electrostatic field line leaves at angle $\alpha$ from point charge $q_1$ and connects with point charge $–q_2$ at angle $\beta$ (see figure).

 Then find the relationship between $\alpha$ and $\beta$ .

The figure shows two infinitely large conducting plates $A$ and $B$. If the electric field at $C$ due to charge densities ${\mathrm{\sigma }}_1, {\mathrm{\sigma }}_2, {\mathrm{\sigma }}_3$ and ${\sigma }_4$ is $E\hat{\mathrm{i}}$, find ${\mathrm{\sigma }}_2$ and ${\mathrm{\sigma }}_{3 }$in terms of $E$. State whether this much information is sufficient to find ${\mathrm{\sigma }}_1$ and ${\mathrm{\sigma }}_4$ in terms of $E$. Derive a relation between${\mathrm{\sigma }}_1$ and ${\mathrm{\sigma }}_4.$ 

In a neutral conducting hollow sphere of inner and outer radii $5 \mathrm{cm}$ and $10 \mathrm{cm}$ respectively, a point charge $Q=1 \mathrm{C}$ is placed at point $A$, that is $3 \mathrm{cm}$ from the centre $C$ of the hollow sphere. An external uniform electric field of magnitude $20 \mathrm{N} {\mathrm{C}}^{-1}$ is also applied. Net electric force on this charge is $15 \mathrm{N}$, away from the centre of the sphere as shown. Then find:

(a) Force due to external electric field on the outer surface of the shell.

(b) Net force on the shell.

(c) Net force on a point charge due to the shell.

The figure shows a system of three concentric metal shells $A, B$ and $C$ with radii $a, 2a$ and $3a$ respectively. Shell $B$ is earthed and shell $C$ is given a charge $Q$. Now if shell $C$ is connected to shell $A$, then find the final charge on the shell $B$.