Three charges `+q,+q and -q` are located at the corners of an equilaterial triangle of side a. Calculate potential energy of the system.

Answer

To calculate the potential energy of the system of three charges located at the corners of an equilateral triangle, we can follow these steps:

### Step 1: Identify the Charges and Their Positions
We have three charges:
- Charge \( Q_1 = +q \) at point A
- Charge \( Q_2 = +q \) at point B
- Charge \( Q_3 = -q \) at point C

The charges are positioned at the corners of an equilateral triangle with side 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.

#### Pair 1: Between Charges A and B
For charges \( Q_1 \) and \( Q_2 \):
\[ U_{AB} = k \frac{(+q)(+q)}{a} = k \frac{q^2}{a} \]

#### Pair 2: Between Charges B and C
For charges \( Q_2 \) and \( Q_3 \):
\[ U_{BC} = k \frac{(+q)(-q)}{a} = -k \frac{q^2}{a} \]

#### Pair 3: Between Charges C and A
For charges \( Q_3 \) and \( Q_1 \):
\[ U_{CA} = k \frac{(-q)(+q)}{a} = -k \frac{q^2}{a} \]

### Step 3: Sum the Potential Energies
Now, we can find the total potential energy \( U_{total} \) of the system by summing the potential energies of all pairs:
\[ U_{total} = U_{AB} + U_{BC} + U_{CA} \]
Substituting the values we calculated:
\[ U_{total} = k \frac{q^2}{a} - k \frac{q^2}{a} - k \frac{q^2}{a} \]
\[ U_{total} = k \frac{q^2}{a} - 2k \frac{q^2}{a} \]
\[ U_{total} = -k \frac{q^2}{a} \]

### Final Result
The total potential energy of the system is:
\[ U_{total} = -k \frac{q^2}{a} \]

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