Two identical pendulums A an dB are suspended from the same point. Both are givne 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 the two identical pendulums A and B that are suspended from the same point and given positive charges, we need to analyze the forces acting on each pendulum and their equilibrium conditions.

### Step-by-Step Solution:

1. **Understanding the Setup**:
- Two identical pendulums A and B are suspended from the same point.
- Pendulum A has a charge \( Q \) (greater charge) and pendulum B has a charge \( q \) (smaller charge).
- Both pendulums make angles \( \theta_1 \) and \( \theta_2 \) with the vertical, respectively.

2. **Electrostatic Force**:
- Since both charges are positive, they will repel each other due to electrostatic force.
- The electrostatic force \( F \) between them can be expressed using Coulomb's law:
\[ F = k \frac{Qq}{d^2} \]
where \( k \) is Coulomb's constant and \( d \) is the distance between the two charges.

3. **Forces Acting on the Pendulums**:
- For pendulum A:
- The tension \( T_1 \) acts along the string.
- The weight \( mg \) acts downward.
- The electrostatic force \( F \) acts horizontally away from B.
- For pendulum B:
- The tension \( T_2 \) acts along the string.
- The weight \( mg \) acts downward.
- The electrostatic force \( F \) acts horizontally away from A.

4. **Equilibrium Conditions**:
- For pendulum A:
- Vertical component:
\[ T_1 \cos \theta_1 = mg \]
- Horizontal component:
\[ T_1 \sin \theta_1 = F \]
- For pendulum B:
- Vertical component:
\[ T_2 \cos \theta_2 = mg \]
- Horizontal component:
\[ T_2 \sin \theta_2 = F \]

5. **Relating the Tensions and Angles**:
- From the vertical components, we have:
\[ T_1 \cos \theta_1 = T_2 \cos \theta_2 \]
- From the horizontal components, we have:
\[ T_1 \sin \theta_1 = T_2 \sin \theta_2 \]

6. **Dividing the Equations**:
- Dividing the horizontal and vertical equations gives:
\[ \frac{T_1 \sin \theta_1}{T_1 \cos \theta_1} = \frac{T_2 \sin \theta_2}{T_2 \cos \theta_2} \]
- This simplifies to:
\[ \tan \theta_1 = \tan \theta_2 \]
- Therefore, we conclude that:
\[ \theta_1 = \theta_2 \]

7. **Conclusion**:
- Since the angles \( \theta_1 \) and \( \theta_2 \) are equal, it implies that the tensions \( T_1 \) and \( T_2 \) must also be equal, given that the masses of the pendulums are identical.
- Thus, the answer to the question is that \( \theta_1 = \theta_2 \).