Three identical spheres each of radius a and mass m are placed such that their centers lie on the vertices of an equilateral triangle of side length 3a. The spheres are non conducting and each one carries a charge Q uniformly distributed over its surface. All the three spheres are simultaneously released and after releasing they move under the influence of their mutual electrostatic forces only. Based on this information, answer the following questions :
The total electrostatic potential energy stored in the system at the time of releasing is :

Options

A. `(KQ^(2))/(2a)`
B. `(3KQ^(2))/(2a)`
C. `(5KQ^(2))/(2a)`
D. `(15KQ^(2))/(2a)`

Explanation

To find the total electrostatic potential energy stored in the system of three identical spheres, we will break down the problem step by step.

### Step 1: Understand the Configuration
We have three identical spheres, each with a charge \( Q \), radius \( a \), and mass \( m \). The centers of these spheres form the vertices of an equilateral triangle with a side length of \( 3a \).

### Step 2: Calculate the Self-Energy of Each Sphere
The self-energy \( U_{\text{self}} \) of a charged sphere can be calculated using the formula:
\[ U_{\text{self}} = \frac{kQ^2}{2a} \]
where \( k \) is Coulomb's constant.

Since there are three identical spheres, the total self-energy of the system is:
\[ U_{\text{self, total}} = 3 \times U_{\text{self}} = 3 \times \frac{kQ^2}{2a} = \frac{3kQ^2}{2a} \]

### Step 3: Calculate the Interaction Energy Between the Spheres
Next, we need to calculate the interaction energy between each pair of spheres. The distance between the centers of any two spheres is \( 3a \).

The interaction energy \( U_{\text{interaction}} \) between two charges \( Q \) separated by a distance \( r \) is given by:
\[ U_{\text{interaction}} = \frac{kQ^2}{r} \]
For our case, since the distance \( r = 3a \), the interaction energy between any two spheres is:
\[ U_{\text{interaction}} = \frac{kQ^2}{3a} \]

Since there are three pairs of spheres (A-B, A-C, and B-C), the total interaction energy is:
\[ U_{\text{interaction, total}} = 3 \times \frac{kQ^2}{3a} = \frac{kQ^2}{a} \]

### Step 4: Calculate the Total Electrostatic Potential Energy
The total electrostatic potential energy \( U_{\text{total}} \) of the system is the sum of the total self-energy and the total interaction energy:
\[ U_{\text{total}} = U_{\text{self, total}} + U_{\text{interaction, total}} \]
Substituting the values we calculated:
\[ U_{\text{total}} = \frac{3kQ^2}{2a} + \frac{kQ^2}{a} \]

To combine these terms, we can express \( \frac{kQ^2}{a} \) as \( \frac{2kQ^2}{2a} \):
\[ U_{\text{total}} = \frac{3kQ^2}{2a} + \frac{2kQ^2}{2a} = \frac{5kQ^2}{2a} \]

### Final Answer
Thus, the total electrostatic potential energy stored in the system at the time of releasing is:
\[ \boxed{\frac{5kQ^2}{2a}} \]