a = − E p [ . (a)] where ˆ p is the unit vector along the dipole axis (from – q to q ). Also ˆ ( a E p [ . (b)] The total field at P is ˆ ( ( a a ⎡ ⎤ ⎢ ⎥ ⎣ ⎦ E E E p ˆ ( o a r a p ( .
) For r >> a ˆ q a E p ( r >> a ) ( . ) (ii) For points on the equatorial plane The magnitudes of the electric fields due to the two charges + q and – q are given by E a [ . (a)] – E a [ . (b)] and are equal.
The directions of E + q and E – q are as shown in Fig. . (b). Clearly, the components normal to the dipole axis cancel away.
The components along the dipole axis add up. The total electric field is opposite to ˆ p . We have E = – ( E + q + E – q ) cos θ ˆ p / ˆ ( o q a a = − p ( . ) At large distances ( r >> a ), this reduces to ˆ ( o q a a = − >> E p ( .
) From Eqs. ( . ) and ( . ), it is clear that the dipole field at large distances does not involve q and a separately; it depends on the product qa .
This suggests the definition of dipole moment. The dipole moment vector p of an electric dipole is defined by p = q × a ˆ p ( . ) that is, it is a vector whose magnitude is charge q times the separation a (between the pair of charges q , – q ) and the direction is along the line from – q to q . In terms of p , the electric field of a dipole at large distances takes simple forms: At a point on the dipole axis o r p E (r >> a) ( .
) At a point on the equatorial plane o