A hollow sphere of radius carries a total charge $Q$ distributed uniformly over its surface. A small area $dA$ of the sphere is cut off. Find the electric field at the center due to the remaining sphere.

$\left(\mathrm{i}\right)$ Two infinitely long line charges each of linear charge density $\lambda$ are placed at an angle $\theta$ as shown in figure. Find out electric field intensity at a point $P$, which is at a distance $x$ from point $O$ along angle bisector of line charges. 

$\left(\mathrm{ii}\right)$ Repeat the above question if the line charge densities are $\mathrm{\lambda }$ and  $-\mathrm{\lambda }$as shown in figure.

If three infinite charged sheets of uniform surface charge densities $\mathrm{\sigma }, 2\mathrm{\sigma } \mathrm{and} –4\mathrm{\sigma }$ are placed as shown in figure, then find out electric field intensities at points $\mathrm{A}, \mathrm{B}, \mathrm{C} \mathrm{and} \mathrm{D}.$

Find out electric field intensity due to uniformly charged solid non-conducting sphere of volume charge density $\rho$ and radius $R$ at following points:

(i) at a distance $r$ from the surface of a sphere(inside),

(ii) at a distance $r$, from the surface of a sphere(outside),

Find out electric field intensity if the sphere is a hollow non-conducting sphere of volume charge density $\rho$ and radius $R$ at the following points, 

(i) at a distance $r$ from the surface of a sphere(inside),

(ii) at a distance $r$, from the surface of a sphere(outside),

A thread carrying a uniform charge $\lambda$ per unit length has the configurations shown in the figures $a$ and $b$. Assuming the curvature radius $R$ to be considerably less than the length of the thread, find the magnitude of the electric field strength at the point $O$.

A point charge $20 \mathrm{\mu C}$ is shifted from infinity to a point $P$ in an electric field with zero acceleration. If the potential of that point is $1000 \mathrm{volt}$, then

$\left(\mathrm{i}\right)$ Find out work done by an external agent against the electric field?

$\left(\mathrm{ii}\right)$ What is the work done by the electric field?

$\left(\mathrm{iii}\right)$ If the kinetic energy of the charged particle is found to increase by $10 \mathrm{mJ}$ when it is brought from infinity to point $P$, then what is the total work done by an external agent?

$\left(\mathrm{iv}\right)$ What is the work done by the electric field in the part $\left(\mathrm{iii}\right)$.

$\left(\mathrm{v}\right)$ If a point charge $30 \mathrm{\mu C}$ is released at rest at a point $P$, then find out its kinetic energy at a large distance?