Derive an expression for the electric field at any point along the axial line of an electrical dipole.

Explanation

The electric field at a point P on axial line of dipole due to +q charge B.
` E_B =(1)/(4 pi in _0) .(q)/((r-a)^(2)) ` along BP
and electric field due to -q charge at A is
` " " E_A =(1)/(4 pi in _0) .(q)/( (r+a) ^(2)) ` along PB
` therefore ` Net electric field at point P is given by
` E= E_B -E_A =(q)/(4 pi in_0) [(1)/((r-a)^(2) )-( 1)/((r+a)^(2)) ]= (q. 4ar)/(4 pi in _0(r^(2) -a^(2) )^(2)) `
`rArr " " E= (2pr)/(4 pi in _0(r^(2) -a^(2))^(2)) , ` where p=q ,2a=dipole moment of given dipole.
For a short dipole or if the point P is situated far away, then a `ltlt ` r and hence,
` " "E =(1)/(4piin _0) .(2p)/(r^(3)) `along ABP
Vectorially ` oversetto E= (1)/(4 pi in _0) .(oversetto 2p)/(r^(3)) `