Two positively charged spheres of masses `m_(1), " and " m_(2)` are suspended from a common point at the ceiling by identical insulating massless strings of length l. Charged on the two spheres are `q_(1) " and " q_(2)`, respectively. At equilibrium both strings make the same angle `theta` with the vertical. Then

Options

A. `q_(1) m_(1) = q_(2) m_(2)`
B. `m_(1) = m_(2)`
C. `m_(1) = m_(2) sin theta`
D. `q_(2) m_(1)=q_(1)m_(2)`

Answer

To solve the problem, we will analyze the forces acting on each of the charged spheres and use the equilibrium conditions to derive the required relationships.

### Step-by-Step Solution:

1. **Identify Forces Acting on Each Sphere**:
Each sphere experiences three forces:
- The gravitational force acting downward: \( F_g = m_i g \) (where \( i = 1, 2 \) for spheres 1 and 2 respectively).
- The tension in the string acting along the string: \( T_i \).
- The electrostatic force of repulsion between the two charged spheres: \( F_e = \frac{k q_1 q_2}{r^2} \), where \( r \) is the distance between the two spheres.

2. **Geometry of the Setup**:
Since both strings make the same angle \( \theta \) with the vertical, we can express the horizontal distance between the two spheres in terms of \( L \) and \( \theta \):
\[ r = 2L \sin(\theta) \]

3. **Equilibrium Conditions**:
For each sphere to be in equilibrium, the net force in both the vertical and horizontal directions must be zero.

- **Vertical Forces**:
For sphere 1:
\[ T_1 \cos(\theta) = m_1 g \]
For sphere 2:
\[ T_2 \cos(\theta) = m_2 g \]

- **Horizontal Forces**:
The horizontal component of the tension must balance the electrostatic force:
For sphere 1:
\[ T_1 \sin(\theta) = F_e = \frac{k q_1 q_2}{(2L \sin(\theta))^2} \]
For sphere 2 (similar equation):
\[ T_2 \sin(\theta) = F_e = \frac{k q_1 q_2}{(2L \sin(\theta))^2} \]

4. **Express Tensions**:
From the vertical force equations, we can express the tensions:
\[ T_1 = \frac{m_1 g}{\cos(\theta)} \]
\[ T_2 = \frac{m_2 g}{\cos(\theta)} \]

5. **Substituting Tensions into Horizontal Force Equations**:
Substitute \( T_1 \) and \( T_2 \) into the horizontal force equations:
\[ \frac{m_1 g}{\cos(\theta)} \sin(\theta) = \frac{k q_1 q_2}{(2L \sin(\theta))^2} \]
\[ \frac{m_2 g}{\cos(\theta)} \sin(\theta) = \frac{k q_1 q_2}{(2L \sin(\theta))^2} \]

6. **Equating Forces**:
Since both equations equal \( \frac{k q_1 q_2}{(2L \sin(\theta))^2} \), we can set them equal to each other:
\[ \frac{m_1 g \sin(\theta)}{\cos(\theta)} = \frac{m_2 g \sin(\theta)}{\cos(\theta)} \]

7. **Final Relationship**:
This leads to the conclusion:
\[ m_1 = m_2 \]