Derive an expression for electric field due to an electric dipole at a point on the axial line.

Explanation

The axial line of a dipole is the line passing through the positive and negative charges of the electric dipole .
Consider a system of charges (`-and+q)` separated by a distance 2a. Let .P. ne any point on an axis where the field intehsity is to be determined. Electric field at `P(E_(B))` due to +q
`E_(B)=(1)/(4piepsilon_(0))(q)/((BP)^(2))` along BP
`=(1)/(4piepsilon_(0))(q)/((r-a)^(2))`
Electric field at `P,E_(B)` due to -q
`E_(A)=(1)/(4piepsilon_(0))(q)/((AP)^(2))` along PA
`=(1)/(4piepsilon_(0))(q)/((r+a)^(2))`
Net field at P is given by
`E_(P)=E_(B)-E_(A)`
`=(1)/(4piepsilon_(0))[(q)/((r-a)^(2))-(q)/((r+a)^(2))]`
Simplifying we get
`E_(P)=(1)/(4piepsilon_(0))(4ar)/((r^(2)-a^(2))^(2))`
`E_(P)=(2qa)/(4piepsilon_(0))(2r)/((r^(2)-a^(2))^(2))`
`(2qa=p,K=(1)/(4piepsilon_(0)))`
or `E_(P)=(2kpr)/((r^(2)-a^(2))^(2))` along BP
As a special case :
If `2altltr, " " E_(P)=(2kp)/(r^(3))` along BP