Write the expression for electric field at a point on the axis of a short electric dipole.

Explanation

Let `+1C` be the test charge at P, at a distance r from the centre of an electric dipole.
Test charge +1C experiences a force of repulsion along BP due to +q at and atrraction due to -q along PA.
Since +q is nearer than -q from P, the resultant force points along BP.
Force experienced by a unit +ve charge is called electric field.
Net `vecE=vecE_(+q)-vecE_(-q)=q/(4 pi epsilo_(0))=[1/((r-a)^(2))-1/((r+a)^(2))]hatp`
where `hatp` represents the unit vector of dipole moment point from -q to +q.
`:.vecE=(q/(4 pi epsilon_(0)))((2(2ar))/((r^(2)-a^(2))^(2)))hatp`
i.e. `vecE=((2pr)/(4 pi epsilon_(0)))(1/((r^(2)-a^(2))^(2)))hatp`
`vecE=(1/(4 pi epsilon_(0)))(2bar(p)r)/((r^(2)-a^(2))^(2))`
where `p=2aq` and `vecp=phatp`