An electric dipole has a fixed dipole moment $\stackrel{\rightharpoonup }{p}$ , which makes angle $\theta$ with respect to $x-$axis. When subjected to an electric field ${\overrightarrow{E}}_1=E\hat{\mathrm{i}}$ , it experiences a torque ${\stackrel{\rightharpoonup }{T}}_1=\tau \hat{\mathrm{k}}$. When subjected to another electric field $E_2=\sqrt{3}{E}_1\hat{\mathrm{j}}$ it experiences a torque $T_2=-T_1$. The angle $\theta$ is

(i) Three equal charges $q_0$ each are placed at three corners of an equilateral triangle of side$a$. Find out force acting on one of the charge due to other two charges?

(ii) In the above question, if one of the charge is replaced by negative charge then find out force acting on it due to other charges?

(iii) Repeat the part (i) if magnitude of each charge is doubled and side of triangle is reduced to half.

A solid sphere of radius $R$ is uniformly charged with charge density $\mathrm{\rho }$ in its volume. A spherical cavity of radius $\frac{R}{2}$ is made in the sphere as shown in the figure. Find the electric potential at the centre of the sphere.

A uniform surface charge of density $\sigma$ is given to a quarter of a disc extending up to infinity in the first quadrant of $x-y$ plane. The centre of the disc is at the origin $O$. Find the $z-$component of the electric field at the point $(0, 0, z)$ and the potential difference between the points $(0, 0, d) \& (0, 0, 2d).$

A thin non-conducting ring of radius $R$ has a linear charge density $\mathrm{\lambda }={\mathrm{\lambda }}_{0 }\mathrm{cos\phi }$, where ${\mathrm{\lambda }}_0$ is a constant, $\phi$ is the azimuthal angle. Find the magnitude of the electric field strength,

(a) at the centre of the ring,

(b) on the axis of the ring as a function of the distance $x$ from its centre. Investigate the obtained function at $x\gg R$.