Two equal charges $q$ are placed at $x = -a$ and $x = a$ on the $x$-axis. $A$ particle of mass $m$ and charge $q_0 = q/2$ is placed at the origin. If the charge $q_0$ is given a small displacement $(y << a)$ along the $y$-axis,the net force acting on the particle is proportional to .......

Coulomb's law for electrostatic force between two point charges and Newton's law for gravitational force between two stationary point masses,both have inverse-square dependence on the distance between the charges and masses respectively.
$(a)$ Compare the strength of these forces by determining the ratio of their magnitudes $(i)$ for an electron and a proton and $(ii)$ for two protons.
$(b)$ Estimate the accelerations of electron and proton due to the electrical force of their mutual attraction when they are $1 \mathring A \left( = 10^{-10} \, m \right)$ apart? $\left( m_{p} = 1.67 \times 10^{-27} \, kg, m_{e} = 9.11 \times 10^{-31} \, kg \right)$

Two small spheres of masses $M_1$ and $M_2$ are suspended by weightless insulating threads of lengths $L_1$ and $L_2$ respectively. The charges on the spheres are $Q_1$ and $Q_2$ respectively. The spheres are suspended such that they lie in a horizontal line and the threads make angles $\theta_1$ and $\theta_2$ with the vertical as shown in the figure. Which of the following conditions is necessary for $\theta_1 = \theta_2$?

Five charges, each $+Q$, are placed at five vertices of a regular hexagon of side length $L$. What is the magnitude of the force on a charge $-Q$ placed at the center of the hexagon?

$(a)$ In a quark model of elementary particles,a neutron is made of one up quark [ charge $\frac{2}{3}e$ ] and two down quarks [ charges $-\frac{1}{3}e$ ]. Assume that they have a triangle configuration with side length of the order of ${10^{ - 15}} \ m$. Calculate the electrostatic potential energy of the neutron and compare it with its mass $939 \ MeV$. $(b)$ Repeat the above exercise for a proton which is made of two up and one down quark.

In one model of the electron,the electron of mass $m_e$ is thought to be a uniformly charged shell of radius $R$ and total charge $e$,whose electrostatic energy $E$ is equivalent to its mass $m_e$ via Einstein's mass-energy relation $E = m_e c^2$. In this model,$R$ is approximately ($m_e = 9.1 \times 10^{-31} \, kg$,$c = 3 \times 10^8 \, ms^{-1}$,$1 / 4 \pi \varepsilon_0 = 9 \times 10^9 \, Nm^2C^{-2}$,magnitude of the electron charge $e = 1.6 \times 10^{-19} \, C$).