Two identical charges are placed at the two corners of an equilateral triangle. The potential energy of the system is U. The work done in bringing an identical charge from infinity to the third vertex is

Options

A. U
B. 2U
C. 3U
D. 4U

Answer

To solve the problem of finding the work done in bringing an identical charge from infinity to the third vertex of an equilateral triangle where two identical charges are already placed at the other two vertices, we can follow these steps:

### Step-by-Step Solution

1. **Understanding the Initial Configuration**:
- We have two identical charges \( q \) placed at two corners of an equilateral triangle with side length \( a \).
- The potential energy of this system is given as \( U \).

2. **Calculating the Initial Potential Energy**:
- The potential energy \( U \) of the two charges can be calculated using the formula:
\[ U = k \frac{q^2}{a} \]
- Here, \( k \) is Coulomb's constant.

3. **Introducing the Third Charge**:
- Now, we bring an identical charge \( q \) from infinity to the third vertex of the triangle.
- This charge will interact with the two existing charges.

4. **Calculating the Final Potential Energy**:
- When the third charge is brought to the third vertex, the potential energy of the new system (with three charges) can be calculated.
- The potential energy \( U' \) of the system with three charges is the sum of the potential energies of each pair:
\[ U' = U_{12} + U_{23} + U_{31} \]
- Each pair contributes \( k \frac{q^2}{a} \) to the potential energy:
\[ U' = k \frac{q^2}{a} + k \frac{q^2}{a} + k \frac{q^2}{a} = 3k \frac{q^2}{a} \]
- Since \( U = k \frac{q^2}{a} \), we can express \( U' \) as:
\[ U' = 3U \]

5. **Calculating the Work Done**:
- The work done \( W \) by the external agent in bringing the charge from infinity to the third vertex is equal to the change in potential energy:
\[ W = U' - U \]
- Substituting the values we found:
\[ W = 3U - U = 2U \]

### Final Answer
The work done in bringing the identical charge from infinity to the third vertex is \( 2U \).

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