A point charge .q. is kept at each of the vertices of an equilateral triangle having each side .a.. Total electrostatic potential energy of the system is:

Options

A. `((1)/(4pi epsi_(0))) (3q^(2))/(a^(2))`
B. `((1)/(4pi epsi_(0))) (3q)/(a)`
C. `((1)/(4pi epsi_(0))) (3q^(2))/(a)`
D. `((1)/(4pi epsi_(0))) (3q)/(a^(2))`

Explanation

To find the total electrostatic potential energy of a system of point charges located at the vertices of an equilateral triangle, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Charges and Geometry**:
- We have three point charges, each of magnitude \( Q \), located at the vertices of an equilateral triangle with each side of length \( a \).

2. **Calculate the Potential Energy for Each Pair of Charges**:
- The electrostatic potential energy \( U \) between two point charges \( Q_1 \) and \( Q_2 \) separated by a distance \( r \) is given by the formula:
\[ U = k \frac{Q_1 Q_2}{r} \]
- Here, \( k \) is Coulomb's constant, which can be expressed as \( k = \frac{1}{4 \pi \epsilon_0} \).

3. **Determine the Number of Pairs**:
- In our case, we need to consider the potential energy for all pairs of charges. Since there are three charges, the pairs are:
- Pair 1: Charge at vertex 1 and Charge at vertex 2
- Pair 2: Charge at vertex 2 and Charge at vertex 3
- Pair 3: Charge at vertex 3 and Charge at vertex 1
- Thus, we have a total of 3 pairs.

4. **Calculate the Potential Energy for Each Pair**:
- For each pair, the distance \( r \) is equal to \( a \):
\[ U_{pair} = k \frac{Q \cdot Q}{a} = k \frac{Q^2}{a} \]
- Since there are 3 pairs, the total potential energy \( U_{total} \) is:
\[ U_{total} = 3 \cdot U_{pair} = 3 \cdot k \frac{Q^2}{a} \]

5. **Substituting the Value of \( k \)**:
- Now, substituting \( k = \frac{1}{4 \pi \epsilon_0} \):
\[ U_{total} = 3 \cdot \frac{1}{4 \pi \epsilon_0} \cdot \frac{Q^2}{a} \]
- This simplifies to:
\[ U_{total} = \frac{3Q^2}{4 \pi \epsilon_0 a} \]

### Final Answer:
The total electrostatic potential energy of the system is:
\[ U_{total} = \frac{3Q^2}{4 \pi \epsilon_0 a} \]