Obtain the equation of electric field by dipole at a point on axis of dipole.

Explanation

Let the point P be at distance r from the centre of the dipole on the side of the charge q as shown in figure, then electric field by - q charge.

The electric field by `-q` charge
`vecE_(-q) =-q/(4pi epsilon_(0)(r+a)^(2))hatp =(-kq)/(r+a)^(2)hatp`
(`therefore` distance from `-q` to `P = r+a`).........(1)
Electric field by `+q` charge.
`vecE_(+q) = -q/(4pi epsilon_(0)(r-a)^(2))hatp = - (kq)/(r-a)^(2) hatp`
(`therefore` distance from `+q` to P `=r -a)`.......(2)
The unit field at P is
`vecE =vecE_(+q) + vecE_(-q) = kq[1/(r-a)^(2) -1/(r+a)^(2)] hatp_(n)`
(`therefore` From equation (1) and (2) )
`therefore vecE = kq(4ar)/(r^(2)-a^(2))^(2)hatp`..........(3)
For `r gt gt a, r^(2) -a^(2) = r^(2)`, using in equation (3)
`vecE = (k(4qa))/(r^(3))hatp =(2kp)/r^(3) (therefore p = q(2a))`..........(4)