Five charges, q each are placed at the corners of a regular pentagon of side a. (Refer the adjoining figure)
(a) (i) What will be the electric field at O, if the centre of the pentagon?
(ii) What will be the electric field at O if the charge from one of the corners (say A) is removed?
(iii) What will be the electric field at O if the charge q at A is replaced by -q?
(b) How would your answer to (a) be affected if pentagon is replaced by n-sided regular polygon with charge q at each of its corners?

Explanation

(a) (i) In fig, the electric field at 0, the center of pentagon would be zero.
(ii) When charge q from corner A is removed , electric field at 0 would be
`E_(1) = (q xx 1)/(4pi in_(0) r^(2))` along OA.
(iii) If charge q at A is replaced by `-q`, it is equivanlent to adding charge `-2q` at A. Therefore, electric field at 0 would be
`E_(2) = (2q)/(4pi in_(0) r^(2))` along OA.
(b) When pentagon is replaced by n sides regular polygon with charge q at each of its corners, the electric field at 0 would continue be to zero.