Two small spheres of masses `M_(1)`and `M_(2)` are suspended by weightless insulating threads of lengths `L_(1)` and `L_(2)`. The spheres carry charges `Q_(1)` and `Q_(2)` respectively. The spheres are suspended such that they are in level with one another and the threads are inclined to the vertical at angles `theta_(1)` and `theta_(2)`respectively . Which one of the following conditions is essential for `theta_(1) = theta_(2)` ?

Options

A. `M_(1) != M_(2)` but `Q_(1) = Q_(2)`
B. `M_(1) = M_(2)`
C. `Q_(1) = Q_(2)`
D. `L_(1) = L_(2)`

Answer

To solve the problem step by step, we will analyze the forces acting on the two charged spheres and derive the condition for the angles of inclination to be equal.

### Step 1: Draw the Diagram
Draw a diagram showing two small spheres (mass \(M_1\) and \(M_2\)) suspended by weightless insulating threads of lengths \(L_1\) and \(L_2\). Indicate the angles of inclination \(\theta_1\) and \(\theta_2\) with respect to the vertical.

### Step 2: Identify Forces Acting on Each Sphere
For each sphere, identify the forces acting on it:
- The gravitational force acting downwards: \(F_{g1} = M_1 g\) for sphere 1 and \(F_{g2} = M_2 g\) for sphere 2.
- The electrostatic force \(F\) acting between the two charged spheres due to their charges \(Q_1\) and \(Q_2\).

### Step 3: Write the Equations of Motion
For sphere 1:
- The vertical component of the tension \(T_1\) balances the weight:
\[ T_1 \cos(\theta_1) = M_1 g \]
- The horizontal component of the tension provides the electrostatic force:
\[ T_1 \sin(\theta_1) = F \]

For sphere 2:
- The vertical component of the tension \(T_2\) balances the weight:
\[ T_2 \cos(\theta_2) = M_2 g \]
- The horizontal component of the tension provides the electrostatic force:
\[ T_2 \sin(\theta_2) = F \]

### Step 4: Relate the Forces
From the equations derived, we can express the tangent of the angles:
\[ \tan(\theta_1) = \frac{F}{M_1 g} \]
\[ \tan(\theta_2) = \frac{F}{M_2 g} \]

### Step 5: Set the Angles Equal
Since we want \(\theta_1 = \theta_2\), we can set the two tangent equations equal to each other:
\[ \frac{F}{M_1 g} = \frac{F}{M_2 g} \]

### Step 6: Simplify the Equation
Assuming \(F\) is not zero, we can cancel \(F\) from both sides:
\[ \frac{1}{M_1} = \frac{1}{M_2} \]
This implies:
\[ M_1 = M_2 \]

### Conclusion
The essential condition for \(\theta_1 = \theta_2\) is that the masses of the two spheres must be equal:
\[ M_1 = M_2 \]