Find the potential energy of a charge `q_(0)` placed at the centre of regular hexagon of side a, if charge q is placed at each vertex of regular hexagon ?

Answer

To find the potential energy of a charge \( q_0 \) placed at the center of a regular hexagon with charge \( q \) at each vertex, we can follow these steps:

### Step 1: Understand the Geometry of the Hexagon
A regular hexagon has six equal sides. If the side length is \( a \), the distance from the center of the hexagon to each vertex (where charge \( q \) is placed) can be determined. For a regular hexagon, this distance is equal to \( a \).

### Step 2: Identify the Charges and Their Positions
We have:
- Charge \( q_0 \) at the center of the hexagon.
- Charge \( q \) at each of the six vertices.

### Step 3: Use the Formula for Electric Potential Energy
The potential energy \( U \) due to a point charge is given by the formula:
\[ U = k \frac{q_1 q_2}{r} \]
where:
- \( k \) is Coulomb's constant,
- \( q_1 \) and \( q_2 \) are the magnitudes of the charges,
- \( r \) is the distance between the charges.

### Step 4: Calculate the Potential Energy Contribution from Each Vertex Charge
Since all vertex charges are at the same distance \( r = a \) from the charge \( q_0 \), the potential energy contributed by one charge \( q \) at a vertex is:
\[ U_{single} = k \frac{q_0 q}{a} \]

### Step 5: Calculate Total Potential Energy from All Vertex Charges
Since there are six vertex charges, the total potential energy \( U_{total} \) is:
\[ U_{total} = 6 \times U_{single} = 6 \left( k \frac{q_0 q}{a} \right) \]
Thus, we can write:
\[ U_{total} = \frac{6k q_0 q}{a} \]

### Final Answer
The potential energy of the charge \( q_0 \) placed at the center of the regular hexagon is:
\[ U = \frac{6k q_0 q}{a} \]

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