As shown in the figure, a point charge $q_{1} = +1 \times 10^{-8} \ C$ is placed at the origin in the $x-y$ plane and another point charge $q_{2} = +3 \times 10^{-6} \ C$ is placed at the coordinate $(10, 0)$. In that case, which of the following graph$(s)$ shows most correctly the electric field vector $E_{x}$ in the $x$-direction?

The dimension of $\frac{1}{2} \varepsilon_0 E^2$ (where $\varepsilon_0$ is the permittivity of free space and $E$ is the electric field) is:

$A$ particle of mass $m$ and charge $q$ is kept at the top of a fixed frictionless sphere. $A$ uniform horizontal electric field $E$ is switched on. The particle loses contact with the sphere when the line joining the center of the sphere and the particle makes an angle $45^{\circ}$ with the vertical. The ratio $\frac{qE}{mg}$ is:

In $1959$,Lyttleton and Bondi suggested that the expansion of the Universe could be explained if matter carried a net charge. Suppose that the Universe is made up of hydrogen atoms with a number density $N$,which is maintained constant. Let the charge on the proton be: $e_p = -(1 + y)e$,where $e$ is the electronic charge.
$(a)$ Find the critical value of $y$ such that expansion may start.
$(b)$ Show that the velocity of expansion is proportional to the distance from the centre.

$A$ negatively charged plate has a surface charge density of $2 \times 10^{-6} \, C/m^2$. Find the minimum initial distance of an electron moving toward the plate such that it does not strike the plate,given that its initial kinetic energy is $200 \, eV$.

The figure shows a nonconducting ring which has positive and negative charge non-uniformly distributed on it such that the total charge is zero. Which of the following statements is true?