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

Options

A. `q_(1)m_(2) =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)`

Explanation

To solve the problem, we need to analyze the forces acting on the two positively charged spheres suspended from a common point. Here’s a step-by-step solution:

### Step 1: Understand the Setup
We have two spheres of masses \( m_1 \) and \( m_2 \) that are positively charged with charges \( q_1 \) and \( q_2 \) respectively. They are suspended from the ceiling by identical massless strings of length \( l \). Both strings make the same angle \( \theta \) with the vertical.

### Step 2: Analyze Forces on Each Sphere
For each sphere, there are three forces acting on it:
1. The gravitational force acting downward: \( F_g = m_i g \) (where \( i \) is either 1 or 2).
2. The tension \( T \) in the string acting along the string.
3. The electrostatic repulsive force \( F \) between the two charged spheres.

### Step 3: Set Up the Equilibrium Conditions
In equilibrium, the net force acting on each sphere must be zero. This gives us two equations based on the forces acting on each sphere.

For sphere 1:
- The vertical component of the tension balances the weight:
\[ T \cos \theta = m_1 g \]
- The horizontal component of the tension balances the electrostatic force:
\[ T \sin \theta = F \]

For sphere 2:
- The vertical component of the tension balances the weight:
\[ T \cos \theta = m_2 g \]
- The horizontal component of the tension balances the electrostatic force:
\[ T \sin \theta = F' \]

### Step 4: Relate the Forces
Since both spheres are at the same angle \( \theta \) and are in equilibrium, we can equate the horizontal components of the forces:
\[ T \sin \theta = F \quad \text{(for sphere 1)} \]
\[ T \sin \theta = F' \quad \text{(for sphere 2)} \]

### Step 5: Equate the Tension Forces
From the vertical force balance equations, we have:
\[ T = \frac{m_1 g}{\cos \theta} \quad \text{(for sphere 1)} \]
\[ T = \frac{m_2 g}{\cos \theta} \quad \text{(for sphere 2)} \]

Setting these equal gives:
\[ \frac{m_1 g}{\cos \theta} = \frac{m_2 g}{\cos \theta} \]
Since \( g \) and \( \cos \theta \) are common, we can simplify this to:
\[ m_1 = m_2 \]

### Conclusion
Thus, we conclude that the masses of the two spheres are equal:
\[ m_1 = m_2 \]