Suppose the charge of a proton and an electron differ slightly. One of them is $-e,$ the other is $(e + \Delta e).$ If the net electrostatic force and gravitational force between two hydrogen atoms placed at a distance $d$ (much greater than atomic size) apart is zero, then $\Delta e$ is of the order of $[$ Given: mass of hydrogen $m_h = 1.67 \times 10^{-27} \, kg]$

Six charges,three positive and three negative of equal magnitude,are placed at the corners of a regular hexagon such that the electric field at the center $O$ is double the electric field obtained when only a charge of the same magnitude is placed at $R$. What are the charges at $P, Q, R, S, T,$ and $U$ respectively?

Five balls numbered $1$ to $5$ are suspended using separate threads. Pairs $(1,2), (2,4)$ and $(4,1)$ show electrostatic attraction while pairs $(2,3)$ and $(4,5)$ show repulsion. Therefore,ball $1$ must be:

Three point charges $q_1, q_2, q_3$ are placed at the vertices of a triangle. If the forces on $q_1$ and $q_2$ are $(2\hat{i} - \hat{j}) \, N$ and $(\hat{i} + 3\hat{j}) \, N$,respectively,then what will be the force on $q_3$?

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?

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.