Question

Two particles of equal mass $m$ and charge $q$ are placed at a distance of $16\mathrm{cm}$. They do not experience any force. The value of $\frac{q}{m}$ is:

Answer 1

Equilibrium of gravitational force and electrostatic force

1. The forces that are acting on the particles are gravitational force and electrostatic force.
2. The particles will not experience any force if the two particles are in equilibrium.
3. Hence, the gravitational force is equal to the electrostatic force.

$$\begin{gathered} F_g=F_e \\ \frac{G{m}^2}{d^2}=\frac{1}{4\pi {\epsilon }_0}\times \frac{q^2}{d^2} \end{gathered}$$

Here, $F_g$ is the gravitational force, $F_e$ is the electrostatic force, $m$ is the mass of the particles, $G$ is the gravitational constant, $q$ is the charge, ${\epsilon }_0$ is the constant, and $d$ is the distance between the particles.

4. Calculation of the ratio of charge to the mass of the particles.

$$\begin{gathered} \frac{G{m}^2}{d^2}=\frac{1}{4\pi {\epsilon }_0}\times \frac{q^2}{d^2} \\ G{m}^2=\frac{q^2}{4\pi {\epsilon }_0} \\ \frac{q}{m}=\sqrt{4\pi {\epsilon }_{0}G} \end{gathered}$$

Therefore, the ratio of charge to the mass of the particles is $\sqrt{4\pi {\epsilon }_{0}G}$.