A regular hexagon of side `a` has a charge Q at each vertex. Potential at the centres of hexagon is (k= `(1/4piepsilon_0)`)

Options

A. Zero
B. `kQ/a`
C. `12(kQ)/a `
D. `6(kQ)/a `

Answer

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

### Step-by-Step Solution:

1. **Understanding the Geometry of the Hexagon**:
A regular hexagon has six equal sides and six vertices. The distance from each vertex to the center of the hexagon is the same.

2. **Finding the Distance from the Center to a Vertex**:
In a regular hexagon with side length \( a \), the distance \( r \) from the center of the hexagon to any vertex can be calculated. For a regular hexagon, this distance is given by:
\[ r = \frac{a}{\sqrt{3}} \cdot 2 = \frac{a}{\sqrt{3}} \cdot \sqrt{3} = a \]
(This can be derived from the properties of the hexagon and the equilateral triangles formed within it.)

3. **Calculating the Electric Potential Due to One Charge**:
The electric potential \( V \) due to a point charge \( Q \) at a distance \( r \) is given by:
\[ V = k \frac{Q}{r} \]
where \( k = \frac{1}{4\pi \epsilon_0} \).

4. **Calculating the Total Potential at the Center**:
Since there are six charges, each contributing equally to the potential at the center, the total potential \( V_{total} \) at the center is the sum of the potentials due to each charge:
\[ V_{total} = V_1 + V_2 + V_3 + V_4 + V_5 + V_6 \]
Since each charge is at the same distance \( r \) from the center:
\[ V_{total} = 6 \cdot k \frac{Q}{r} \]
Substituting \( r = a \):
\[ V_{total} = 6 \cdot k \frac{Q}{a} \]

5. **Final Expression for the Potential**:
Thus, the potential at the center of the hexagon is:
\[ V = \frac{6kQ}{a} \]

### Final Answer:
The potential at the center of the hexagon is \( \frac{6kQ}{a} \).

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