Three charges $+5 q, Q$ and $-2 q$ are kept along a straight line in the same order such that $+5 q$ and $-2 q$ charges are at a distance of $\frac{2 r}{3}$ and $\frac{r}{3}$ from the charge $Q$ respectively. If the net force on the charge $-2 q$ is zero,then $Q$ is

The ratio of electrostatic and gravitational forces acting between an electron and a proton separated by a distance $5 \times 10^{-11} \, m$ will be (Charge on electron $= 1.6 \times 10^{-19} \, C$,mass of electron $= 9.1 \times 10^{-31} \, kg$,mass of proton $= 1.6 \times 10^{-27} \, kg$,$G = 6.7 \times 10^{-11} \, N m^2/kg^2$).

$A$ proton and an anti-proton come close to each other in vacuum such that the distance between them is $10 \, cm$. Consider the potential energy to be zero at infinity. The velocity at this distance will be ........... $\, m/s$.

Two identical spheres each of radius $R$ are kept at a center-to-center spacing of $4R$ as shown in the figure. They are charged,and the electrostatic force of interaction between them is first calculated assuming them to be point-like charges at their centers $(F_c)$,and the force is also measured experimentally $(F_m)$. ($F_c$ and $F_m$ denote the magnitude of the force.)

An oil drop of $12$ excess electrons is held stationary under a constant electric field of $2.55 \times 10^{4} \; N \, C^{-1}$ (Millikan's oil drop experiment). The density of the oil is $1.26 \; g \, cm^{-3}$. Estimate the radius of the drop. $(g = 9.81 \; m \, s^{-2}; e = 1.60 \times 10^{-19} \; C)$

Three charges $-q_1$,$+q_2$,and $-q_3$ are placed as shown in the figure. The $x$-component of the force on $-q_1$ is proportional to