Six point charges are placed at the vertices of a hexagon of side 1 m as shown in figure. Net electric field at the centre of the hexagon is

Options

A. Zero
B. `(6q)/(4pi epsilon_(0))`
C. `(q)/(pi epsilon_(0))`
D. `(q)/(4pi epsilon_(0))`

Answer

To find the net electric field at the center of a hexagon with point charges placed at its vertices, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Configuration**:
We have six point charges located at the vertices of a regular hexagon. Let's assume three of these charges are positive (+q) and three are negative (-q).

2. **Determine the Distance to the Center**:
The distance from each vertex of the hexagon to the center (O) can be calculated. For a hexagon with a side length of 1 m, the distance (r) from the center to a vertex is given by:
\[ r = \frac{s}{\sqrt{3}} = \frac{1}{\sqrt{3}} \text{ m} \]
where \(s\) is the side length of the hexagon.

3. **Calculate Electric Field Due to One Charge**:
The electric field (E) due to a single point charge at a distance r is given by:
\[ E = \frac{k \cdot |q|}{r^2} \]
where \(k\) is Coulomb's constant. Substituting \(r\):
\[ E = \frac{k \cdot |q|}{\left(\frac{1}{\sqrt{3}}\right)^2} = 3k \cdot |q| \]

4. **Consider Symmetry**:
Due to the symmetry of the hexagon, the electric fields due to the positive charges will point away from the charges, while the fields due to the negative charges will point towards the charges.

5. **Calculate the Net Electric Field**:
The electric fields from opposite charges will cancel each other out in pairs. Thus, the net electric field at the center will be the vector sum of the fields from the three positive charges and the three negative charges.

Since the configuration is symmetrical, the net electric field can be simplified:
\[ E_{\text{net}} = 0 \]
This is because the contributions from the positive and negative charges will cancel each other out perfectly.

6. **Conclusion**:
Therefore, the net electric field at the center of the hexagon is zero.

### Final Answer:
The net electric field at the center of the hexagon is **0 N/C**.