We have to show that the components of due to a dipole are given, at distant points, by

where x and z are coordinates of point P in the figure.

Solution:             Click For PDF Version

We note from the diagram that the distance of the point P from the charge +q is

and its distance from the charge -q is

X-component of the electric field due to the dipole at the point P will be

As the point P is distant from the dipole, we can assume that

We use the approximation that

and we get

where we have substituted , for the dipole moment.

The Z-component of the electric field is

As the point P is distant from the dipole, we can assume that

We use the approximation that

and we have

On simplifying the algebraic expression, we easily get