Charge `q_(2)` of mass m revolves around a stationary charge `q_(1)` in a circulare orbit of radius r. The orbital periodic time of `q_(2)` would be

Options

A. `[(4 pi^(2) mr^(3))/(k q_(1)q_(2))]^(1//2)`
B. `[(k q_(1) q_(2))/(4 pi^(2) mr^(3))]^(1//2)`
C. `[(4 pi^(2)mr^(4))/(k q_(1) q_(2))]^(1//2)`
D. `[(4 pi^(2) mr^(2))/(k q_(1) q_(2))]^(1//2)`

Answer

To find the orbital periodic time of charge \( q_2 \) revolving around a stationary charge \( q_1 \), we can follow these steps:

### Step 1: Identify the forces acting on the charge \( q_2 \)
The charge \( q_2 \) experiences an electrostatic force due to the charge \( q_1 \), which acts as the centripetal force required for circular motion.

### Step 2: Write the expression for the electrostatic force
The electrostatic force \( F \) between two point charges is given by Coulomb's law:
\[ F = \frac{k \cdot |q_1 \cdot q_2|}{r^2} \]
where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between them.

### Step 3: Set the electrostatic force equal to the centripetal force
For circular motion, the centripetal force \( F_c \) required to keep the charge \( q_2 \) in orbit is given by:
\[ F_c = m \cdot \omega^2 \cdot r \]
where \( m \) is the mass of the charge \( q_2 \) and \( \omega \) is the angular velocity.

Setting the electrostatic force equal to the centripetal force, we have:
\[ \frac{k \cdot |q_1 \cdot q_2|}{r^2} = m \cdot \omega^2 \cdot r \]

### Step 4: Rearrange the equation to solve for \( \omega^2 \)
Rearranging the equation gives:
\[ \omega^2 = \frac{k \cdot |q_1 \cdot q_2|}{m \cdot r^3} \]

### Step 5: Find \( \omega \)
Taking the square root of both sides, we find:
\[ \omega = \sqrt{\frac{k \cdot |q_1 \cdot q_2|}{m \cdot r^3}} \]

### Step 6: Calculate the time period \( T \)
The time period \( T \) of one complete revolution is related to the angular velocity \( \omega \) by the formula:
\[ T = \frac{2\pi}{\omega} \]
Substituting the expression for \( \omega \):
\[ T = \frac{2\pi}{\sqrt{\frac{k \cdot |q_1 \cdot q_2|}{m \cdot r^3}}} \]

### Step 7: Simplify the expression for \( T \)
This can be simplified as follows:
\[ T = 2\pi \sqrt{\frac{m \cdot r^3}{k \cdot |q_1 \cdot q_2|}} \]

### Final Expression
Thus, the orbital periodic time \( T \) of charge \( q_2 \) is:
\[ T = 2\pi \sqrt{\frac{m \cdot r^3}{k \cdot |q_1 \cdot q_2|}} \]