(a) Derive and expression for the electric field at any point on the equatorial line of an electric dipole.
(b) Two identical point charges, q each kept 2m apart in air. A third point charge Q of unknown magnitude and sign is placed on the line joining the charges such that the system remains in equilibrium. Find the position and nature of Q.

Explanation

### Solution:

#### (a) Derivation of Electric Field at Any Point on the Equatorial Line of an Electric Dipole

1. **Understanding the Dipole**:
An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance '2a'. The dipole moment \( p \) is given by:
\[
p = q \cdot 2a
...