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

Explanation

Consider a point .P. on the axis oif an electric dipole at a distance .r. from its mid point as show in the figure . The magnitude of dipole moment of the dipole .P. (directed from -q to +q) is given by P `= q xx2a rarr` (1)
Electric field at P due to q is
`E= (1)/(4piepsilon_(0))(q)/(BP)=(1)/(4piepsilon_(0))(q)/(r.a)` along BP produced
Resultant electric field at P is
`E =E_(2)-E_(1)[ E_(2) gt E_(1) "because " (r-a) lt (r+a) ]`
`= (q)/(piepsilon) (.(1)/((r " "a))-(1)/((r " "a)))`
`E =(.)/(piepsilon)(((r+a)^(2)-(r-a)^(2))/((r-a)^(2)(r+a)^(2)))[(r+a)^(2)-(r-a)^(2)=4ar]`
`E=(q)/(4piepsilon_(0))(4ar)/((r^(2)-a^(2))^(2))`
`E=(1)/(4piepsilon_(0))(2rp)/((r^(2)-a^(2))^(2))[(because` From (1), P=2aq]
For `r gt gt a E = (1)/(4piepsilon_(0))((2P)/(r^(3))) hatP`