Charges `+q -4q` and `+2q` are arranged at the corners of an equilatoral triangle of side `0.15m`.If the `q=1 muC` their mutual potential energy is

Options

A. `0.4J`
B. `0.5J`
C. `0.6J`
D. `0.8J`

Explanation

To find the mutual potential energy of the charges arranged at the corners of an equilateral triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Charges and Their Values:**
- Let the charges be:
- \( q_1 = +q = +1 \mu C = +1 \times 10^{-6} C \)
- \( q_2 = -4q = -4 \times 1 \mu C = -4 \times 10^{-6} C \)
- \( q_3 = +2q = +2 \times 1 \mu C = +2 \times 10^{-6} C \)

2. **Determine the Side Length of the Triangle:**
- The side length \( L \) of the equilateral triangle is given as \( 0.15 m \).

3. **Use the Formula for Potential Energy Between Two Point Charges:**
- The potential energy \( U \) between two point charges \( q_i \) and \( q_j \) separated by a distance \( r \) is given by:
\[ U_{ij} = k \frac{q_i q_j}{r} \]
- Where \( k = 9 \times 10^9 \, N m^2/C^2 \) is Coulomb's constant.

4. **Calculate the Potential Energy for Each Pair of Charges:**
- **For \( q_1 \) and \( q_2 \):**
\[ U_{12} = k \frac{q_1 q_2}{L} = 9 \times 10^9 \frac{(1 \times 10^{-6})(-4 \times 10^{-6})}{0.15} \]
\[ U_{12} = 9 \times 10^9 \frac{-4 \times 10^{-12}}{0.15} = -240 \times 10^{-3} = -0.24 \, J \]

- **For \( q_1 \) and \( q_3 \):**
\[ U_{13} = k \frac{q_1 q_3}{L} = 9 \times 10^9 \frac{(1 \times 10^{-6})(2 \times 10^{-6})}{0.15} \]
\[ U_{13} = 9 \times 10^9 \frac{2 \times 10^{-12}}{0.15} = 120 \times 10^{-3} = 0.12 \, J \]

- **For \( q_2 \) and \( q_3 \):**
\[ U_{23} = k \frac{q_2 q_3}{L} = 9 \times 10^9 \frac{(-4 \times 10^{-6})(2 \times 10^{-6})}{0.15} \]
\[ U_{23} = 9 \times 10^9 \frac{-8 \times 10^{-12}}{0.15} = -480 \times 10^{-3} = -0.48 \, J \]

5. **Calculate the Total Mutual Potential Energy:**
- The total potential energy \( U \) is the sum of the potential energies of each pair:
\[ U = U_{12} + U_{13} + U_{23} \]
\[ U = (-0.24) + (0.12) + (-0.48) = -0.60 \, J \]

### Final Answer:
The mutual potential energy of the three charges is \( U = -0.60 \, J \).