(a) Derive an expression for the electric field at any point on the equatorial line of an electric dipole.
(b) Two identical point charges, each, are kept
apart in air. A third point charge
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
.
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(a) Let us consider the two charges
and 
, be seperated by a distance
On the equatorial line of the dipole let any point
.at a distance 
from the line joining the two charges.
Now,
The electric field at
due to the charge is : for 
, the electric field :
For
, the electric field :
Adding horizontal components, such that vertical will cancel out.
Hence,
Let us suppose,
be the unit vector in the direction of the dipole moment.
If,
Then,
(b) According to the Coulomb's law, All the charges must be in equilibrium and the force depends upon the distance and hence, the position of
must be in the mddle of the two charges.
Now, the nature of
must have opposite sign to that of 
, to balance the forces.