Define electric dipole moment. Is it a scalar or a vector ? Derive the expression for the electric field of a dipole at a point on the equatorial line of the dipole.

Explanation

For definition of dipole moment, see point Number 29 under the heading "Chapter At a Glance" Dipole moment is a vector. Let us calculate the electrostatic field at a point P on the equatorial line at a distance .r. form mid- point O of an electric dipole AB.
Obviously, `" "|oversetto (E_A) | =|oversetto (E_B)| =(1)/(4 pi in _0) .(q)/( (a^(2) + r^(2))) `
Resultant field at point P is `oversetto E =oversetto (E_A) +oversetto (E_B) `
Let us resolve ` oversetto (E_A) and oversetto (E_B) ` along and perpendicular to the dipole axis. We find that components `E_A sin theta and E_B sin theta ` nullify each other and hence
` |oversetto E| =(oversetto (E_A) +oversetto (E_B)) cos theta =2 .(1)/(4 pi in _0) .(q)/((a^(2) + r^(2)) ) .(a)/(sqrt(a^(2) +r^(2)) ) ` where p=q.2a =dipole moment of electric dipole
This direction of ` oversetto E ` is opposite to that of ` oversetto p i.e. , oversetto E =-(oversetto p)/( 4 pi in _0( r^(2) +a^(2)) ^(3//2)) `
If ` r gt gt a , `then teh above relation may be modified as
` " " oversetto E =-( oversetto p)/( 4 pi in _0 r^(3)) `
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