Two point charges are located on the x - axis : `q_1 = -e` at `x = 0` and `q_2 = + e` at `x = a`. A third charge `q_3 = + e` is brought to `x = 2a`.
The total potential energy of the system of three charges.

Options

A. `(e^2)/(4 pi epsilon_0 a)`
B. `(e^2)/(8 pi epsilon_0 a)`
C. `(-e^2)/(8 pi epsilon_0 a)`
D. `(-e^2)/(4 pi epsilon_0 a)`

Explanation

To find the total potential energy of the system of three charges, we will follow these steps:

### Step 1: Identify the Charges and Their Positions
We have three charges:
- \( q_1 = -e \) located at \( x = 0 \)
- \( q_2 = +e \) located at \( x = a \)
- \( q_3 = +e \) located at \( x = 2a \)

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

### Step 3: Calculate the Potential Energy Between Each Pair of Charges
1. **Potential Energy between \( q_1 \) and \( q_2 \)**:
- Distance \( r_{12} = a \)
- \( U_{12} = k \frac{(-e)(e)}{a} = -\frac{k e^2}{a} \)

2. **Potential Energy between \( q_2 \) and \( q_3 \)**:
- Distance \( r_{23} = a \)
- \( U_{23} = k \frac{(e)(e)}{a} = \frac{k e^2}{a} \)

3. **Potential Energy between \( q_1 \) and \( q_3 \)**:
- Distance \( r_{13} = 2a \)
- \( U_{13} = k \frac{(-e)(e)}{2a} = -\frac{k e^2}{2a} \)

### Step 4: Sum the Potential Energies
Now, we can find the total potential energy \( U_{total} \) of the system by summing the potential energies calculated above:
\[ U_{total} = U_{12} + U_{23} + U_{13} \]
Substituting the values we calculated:
\[ U_{total} = \left(-\frac{k e^2}{a}\right) + \left(\frac{k e^2}{a}\right) + \left(-\frac{k e^2}{2a}\right) \]

### Step 5: Simplify the Expression
Notice that \( -\frac{k e^2}{a} \) and \( \frac{k e^2}{a} \) cancel each other out:
\[ U_{total} = 0 - \frac{k e^2}{2a} = -\frac{k e^2}{2a} \]

### Step 6: Substitute the Value of \( k \)
Now substitute \( k = \frac{1}{4 \pi \epsilon_0} \):
\[ U_{total} = -\frac{1}{2a} \cdot \frac{e^2}{4 \pi \epsilon_0} = -\frac{e^2}{8 \pi \epsilon_0 a} \]

### Final Answer
Thus, the total potential energy of the system of three charges is:
\[ U_{total} = -\frac{e^2}{8 \pi \epsilon_0 a} \]