Show mathematically that the electric field due to a short dipole at a distance 'd' along its axis is twice the value of field at the same distance along the equatorial line.

Explanation

We know that electric field at a distance .d. along axis of an electric dipole is
` " " E_("axial") = (1)/( 4 pi in _0) .(2pd)/((d^(2) -a^(2)) ^(2)) `
and for short dipole `(a lt lt d), ` we have
` " "E_("axial ") =(1)/(4 pi in _0) .(2p)/(d^(3)) `
Again electric field at a distance .d. along the equatorial line of dipole is
` " " E_("equatorial ") = ( 1)/(4 pi in _0) .(p)/( (d^(2) +a^(2)) ^(3//2)) `
and in case of a short dipole
` " "E_("equatorial") =(1)/(4 pi in _0) .(p)/(d^(3)) `
Comparing (i) and (ii) we, have
` (E_("axial") )/(E_("equatorial")) = 2 or E_("axial") =2xx E_("equatorial")) `