Field at the Centre of a Regular Polygon with Charges at Corners
In a regular polygon with n sides, each corner is equidistant from the centre, located at a distance r. Identical charges of magnitude q are placed at each corner of the polygon. The field at the centre can be calculated as follows:

Calculating the Field at the Centre
- Due to the symmetry of the polygon, the electric fields produced by the charges at each corner will cancel each other out in pairs.
- Since the charges are identical in magnitude and the distances are the same, the magnitude of the electric field produced by each pair of charges will be the same.
- By considering the components of the electric fields in the x and y directions, it can be shown that the net electric field at the centre of the polygon is zero.

Explanation
- The electric field at a point is a vector quantity that depends on the distance and direction from the point where the charge is located.
- In this case, the electric fields produced by the charges at the corners of the polygon will have components in different directions, resulting in their cancellation at the centre.
- This cancellation occurs due to the symmetrical arrangement of charges around the centre of the polygon.
- Therefore, the net electric field at the centre of the regular polygon with charges at the corners is zero.
In conclusion, the field at the centre of a regular polygon with charges at the corners is zero due to the cancellation of electric fields produced by the symmetrical arrangement of charges.