The time period of revolution of a charge `q_1` and of mass m moving in a circular path of radius r due to Coulomb force of attraction due to another charge `q_2` at its centre is

Options

A. `sqrt((16piepsilon_0mr^3)/(q_1q_2))`
B. `sqrt((8pi^2epsilon_0mr^3)/(q_1q_2))`
C. `sqrt((epsilon_0mr^3)/(16q_1q_2))`
D. `sqrt((16pi^3epsilon_0mr^3)/(q_1q_2))`

Answer

To find the time period of revolution of a charge \( q_1 \) of mass \( m \) moving in a circular path of radius \( r \) due to the Coulomb force of attraction from another charge \( q_2 \) at its center, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Forces Acting on the Charge**:
The charge \( q_1 \) is moving in a circular path due to the electrostatic force exerted by the charge \( q_2 \). This force can be expressed using Coulomb's law:
\[ F = \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} \]

2. **Centripetal Force Requirement**:
For \( q_1 \) to move in a circular path, the centripetal force required for circular motion must equal the electrostatic force. The centripetal force \( F_c \) is given by:
\[ F_c = \frac{m v^2}{r} \]
where \( v \) is the velocity of the charge \( q_1 \).

3. **Set the Forces Equal**:
Setting the electrostatic force equal to the centripetal force gives us:
\[ \frac{1}{4 \pi \epsilon_0} \frac{q_1 q_2}{r^2} = \frac{m v^2}{r} \]

4. **Solve for Velocity \( v \)**:
Rearranging the equation to solve for \( v^2 \):
\[ v^2 = \frac{q_1 q_2}{4 \pi \epsilon_0 m r} \]
Taking the square root gives:
\[ v = \sqrt{\frac{q_1 q_2}{4 \pi \epsilon_0 m r}} \]

5. **Calculate the Time Period \( T \)**:
The time period \( T \) of one complete revolution is given by the distance traveled in one revolution divided by the velocity:
\[ T = \frac{2 \pi r}{v} \]
Substituting the expression for \( v \):
\[ T = \frac{2 \pi r}{\sqrt{\frac{q_1 q_2}{4 \pi \epsilon_0 m r}}} \]

6. **Simplify the Expression**:
To simplify, we can multiply the numerator and denominator by \( \sqrt{4 \pi \epsilon_0 m r} \):
\[ T = \frac{2 \pi r \sqrt{4 \pi \epsilon_0 m r}}{\sqrt{q_1 q_2}} \]
This simplifies to:
\[ T = 2 \pi \sqrt{\frac{4 \pi \epsilon_0 m r^3}{q_1 q_2}} \]

7. **Final Result**:
Thus, the final expression for the time period \( T \) is:
\[ T = 2 \pi \sqrt{\frac{4 \pi \epsilon_0 m r^3}{q_1 q_2}} \]