Question

Find the resultant electric field due to an electric dipole of dipole moment, 2aq, (2a being the separation between the charges ±± q) at a point distant 'x' on its equator.

Answer 1

Consider a dipole consisting of two electric charges +q and −q between a small distance AB = 2a with centre O. 
Now, let us find the electric field intensity at point P.

The magnitudes of the electric field at point P due to the two charges +q and −q are given by

`E_(+q) = q/(4πε_0)  1/(x^2+a^2) .....(1)`

and ` E_(-q) = q/(4πε_0)  1/(x^2+a^2)   ...... (2)`

`∴ E_(+q) = E_(-q) `

The directions of E_+q and E_−q are as shown in the figure. The components normal to the dipole axis along PE and PF cancel away. The components along the dipole axis add up.

Therefore, the resultant electric field at point P is given as

`E = - (E_(+q) + E_(-q) cosΘ)` (The negative sign shows that the field is opposite to the dipole moment of the dipole.)

`E=(-2qa)/(4πε_0 (x^2+a^2)^(3/2))   ......(3)`

When point P lies at a large distance (x >> a) from the dipole, the above expression reduces to 

`E = (-2qa)/(4πε_0 x^3 )   ....... (4)`

∴ `p =1 xx 2a`

∴ `E = (-p)/(4πε_0 x^3 )` (x >> a)

The direction of this electric field is along PR (or BA).