If `3` charges are placed at the vertices of equilateral triangle of charge 'q' each. What is the net potential energy, if the side of equilateral `Delta` is `l cm` ?

Options

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

Answer

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

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

### 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, \( k = \frac{1}{4 \pi \epsilon_0} \).

In our case, since all charges are equal and placed at the vertices of the triangle, the potential energy between any two charges (say \( q \) and \( q \)) separated by distance \( l \) is:
\[ U_{12} = k \frac{q \cdot q}{l} = k \frac{q^2}{l} \]

### Step 3: Calculate the Total Potential Energy
Since there are three pairs of charges in the triangle (1-2, 2-3, and 3-1), the total potential energy \( U_{net} \) is the sum of the potential energies of these pairs:
\[ U_{net} = U_{12} + U_{23} + U_{31} \]
Since each pair has the same potential energy:
\[ U_{net} = 3 \cdot U_{12} = 3 \cdot k \frac{q^2}{l} \]

### Step 4: Substitute the Value of \( k \)
Now, substituting the value of \( k \):
\[ U_{net} = 3 \cdot \left( \frac{1}{4 \pi \epsilon_0} \right) \frac{q^2}{l} \]
Thus, we can write:
\[ U_{net} = \frac{3q^2}{4 \pi \epsilon_0 l} \]

### Final Result
The net potential energy of the system of three charges is:
\[ U_{net} = \frac{3q^2}{4 \pi \epsilon_0 l} \]