Choose the correct alternative (a), (b), (c) or (d) for each of the questions given below :
A point charge .q. is kept at each of the vertices of an equilateral triangle having each side .a.. Potential energy is

Options

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

Answer

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

### Step 1: Understand the Configuration
We have three point charges, each of charge \( q \), located at the vertices of an equilateral triangle with each side of length \( a \).

### Step 2: Calculate the Potential Energy Between Each Pair of Charges
The potential energy \( U \) between two point charges \( q_1 \) and \( q_2 \) separated by a distance \( r \) is given by the formula:
\[ U = k \frac{q_1 q_2}{r} \]
where \( k \) is Coulomb's constant, \( k = \frac{1}{4 \pi \epsilon_0} \).

### Step 3: Identify the Pairs of Charges
In our case, we have three charges:
- Charge at vertex 1: \( q_1 = q \)
- Charge at vertex 2: \( q_2 = q \)
- Charge at vertex 3: \( q_3 = q \)

The pairs of charges are:
1. \( (q_1, q_2) \)
2. \( (q_2, q_3) \)
3. \( (q_3, q_1) \)

### Step 4: Calculate the Potential Energy for Each Pair
Since the distance between each pair of charges is \( a \), we can calculate the potential energy for each pair:
1. For \( (q_1, q_2) \):
\[ U_{12} = k \frac{q \cdot q}{a} = k \frac{q^2}{a} \]
2. For \( (q_2, q_3) \):
\[ U_{23} = k \frac{q \cdot q}{a} = k \frac{q^2}{a} \]
3. For \( (q_3, q_1) \):
\[ U_{31} = k \frac{q \cdot q}{a} = k \frac{q^2}{a} \]

### Step 5: Sum the Potential Energies
The total potential energy \( U_{total} \) of the system is the sum of the potential energies of all pairs:
\[ U_{total} = U_{12} + U_{23} + U_{31} \]
Substituting the values:
\[ U_{total} = k \frac{q^2}{a} + k \frac{q^2}{a} + k \frac{q^2}{a} = 3 k \frac{q^2}{a} \]

### Step 6: Final Expression
Thus, the total potential energy of the system is:
\[ U_{total} = \frac{3 k q^2}{a} \]

### Step 7: Identify the Correct Option
Given that \( k = \frac{1}{4 \pi \epsilon_0} \), we can express the potential energy as:
\[ U_{total} = \frac{3 q^2}{4 \pi \epsilon_0 a} \]
Among the options provided, the correct answer corresponds to this expression.