Question

Electric field strength due to dipole of moment p at a distant point r along the axis of the dipole is

Answer 1

Electric field strength in point A

$$E=\frac{q}{4\pi \epsilon_0}\left(\frac{1}{r_2^2}-\frac{1}{r_1^2}\right)$$

We may apply the mean value theorem

$$E=\frac{q}{4\pi \epsilon_0}\frac{d}{dr} \left(\frac{1}{r^2} \right) \bigg|_{r=\xi}(r_2-r_1)=\frac{1}{4\pi \epsilon_0}\frac{2p}{\xi^3}$$

where

$$\xi \in [r_2,r_1]$$

Dipole is a point particle, hence

$$\xi=r$$

where r is a distance between dipole and point A.

So the electric field strength due to dipole of moment p at a distant point r along the axis of the dipole is

$$E=\frac{1}{4\pi \epsilon_0}\frac{2p}{r^3}$$

or in vector form

$$\vec E=\frac{1}{4\pi \epsilon_0}\frac{2\vec p}{r^3}$$