Two identical pendulums A and B are suspended from the same point. Both are given positive charge, with A having more charge than B. They diverge and reach equilibrium with the suspension of A and B making angles `theta_(1) and theta_(2)` with the vertical respectively.

Options

A. `theta_(1) gt theta_(2)`
B. `theta_(1) lt theta_(2)`
C. `theta_(1) = theta_(2)`
D. The tension in A is greater than that in B

Answer

To solve the problem of two identical pendulums A and B that are given positive charges and reach equilibrium at angles θ₁ and θ₂ with the vertical, we can analyze the forces acting on each pendulum and their equilibrium conditions.

### Step-by-Step Solution:

1. **Identify the Forces Acting on Each Pendulum:**
- Each pendulum experiences three main forces:
- The gravitational force (weight) acting downwards: \( mg \)
- The tension in the string acting along the string: \( T \)
- The electrostatic force due to the repulsion between the two charged pendulums.

2. **Analyze the Electrostatic Force:**
- Since both pendulums are positively charged, they will repel each other. Let the charge on pendulum A be \( q_A \) and the charge on pendulum B be \( q_B \), with \( q_A > q_B \).
- The electrostatic force \( F \) between the two charges can be expressed using Coulomb's Law:
\[ F = k \frac{q_A q_B}{r^2} \]
where \( k \) is Coulomb's constant and \( r \) is the distance between the two charges.

3. **Set Up the Equilibrium Conditions:**
- In equilibrium, the forces acting on each pendulum must balance. For pendulum A, we can resolve the forces into components:
- Vertical component: \( T \cos(\theta_1) = mg \)
- Horizontal component: \( T \sin(\theta_1) = F \)
- For pendulum B, similarly:
- Vertical component: \( T \cos(\theta_2) = mg \)
- Horizontal component: \( T \sin(\theta_2) = F' \) (where \( F' \) is the electrostatic force acting on B).

4. **Relate the Angles and Forces:**
- Since both pendulums are identical and experience the same gravitational force, the tension \( T \) in the strings will be the same when they are at equilibrium.
- Therefore, from the vertical force balance, we have:
\[ T \cos(\theta_1) = mg \quad \text{and} \quad T \cos(\theta_2) = mg \]
- This implies that \( \cos(\theta_1) = \cos(\theta_2) \), leading to \( \theta_1 = \theta_2 \) if the angles are small.

5. **Conclusion:**
- Since pendulum A has a greater charge than pendulum B, it will experience a greater electrostatic force, causing it to diverge more than pendulum B. However, the equilibrium conditions imply that the angles will be equal due to the identical nature of the pendulums and the balancing of forces.
- Therefore, we conclude that:
\[ \theta_1 = \theta_2 \]

### Final Answer:
The angles \( \theta_1 \) and \( \theta_2 \) made by pendulums A and B with the vertical are equal, i.e., \( \theta_1 = \theta_2 \).