An infinitely long positively charged straight thread has a linear charge density $\lambda \text{ Cm}^{-1}$. An electron revolves along a circular path having its axis along the length of the wire. The graph that correctly represents the variation of the kinetic energy of the electron as a function of the radius $r$ of the circular path from the wire is:

$A$ small bob of mass $100 \ mg$ and charge $+10 \ \mu C$ is connected to an insulating string of length $1 \ m$. It is brought near to an infinitely long nonconducting sheet of charge density $\sigma$ as shown in the figure. If the string subtends an angle of $45^{\circ}$ with the sheet at equilibrium,the charge density of the sheet will be (Given,$\varepsilon_0 = 8.85 \times 10^{-12} \ F/m$ and acceleration due to gravity,$g = 10 \ m/s^2$): (in $nC/m^2$)

In a cuboid of dimension $2 L \times 2 L \times L$,a charge $q$ is placed at the centre of the surface ' $S$ ' having an area of $4 L^2$. The flux through the opposite surface to ' $S$ ' is given by

$A$ non-conducting solid sphere of radius $R$ has a uniform volume charge density $\rho$. The electric potential at the center of the sphere is related to the potential at the surface and outside the sphere due to this uniform charge distribution.
Statement-$1$: When a charge $q$ is moved from the surface to the center of the sphere,the change in its potential energy is $q\rho R^2 / 6\varepsilon_0$.
Statement-$2$: The electric field at a distance $r$ $(r < R)$ from the center of the sphere is $\rho r / 3\varepsilon_0$.

Two infinitely long parallel wires having linear charge densities $\lambda_1$ and $\lambda_2$ respectively are placed at a distance of $R$ meters. The force per unit length on either wire will be $(K = \frac{1}{4\pi\varepsilon_0})$.

$A$ charged ball $B$ hangs from a silk thread $S$,which makes an angle $\theta$ with a large charged conducting sheet $P$,as shown in the figure. The surface charge density $\sigma$ of the sheet is proportional to