Derive an expression for electric field intensity at any point due to short dipole ?

Explanation

Consider a short electric dipole AB having dipole moment `vecp`. Let P be the point at a distance r from the centre O. Let the line OP of the dipole makes an angle `theta` with the direction of dipole moment `vecP`.
Resolve `vecP` into two components.
(i) `p cos theta` along OP
(ii) `p sin theta` perpendicular to OQ
Now, Point P is on the axial line w.r.t.p `cos theta`
`:." "E_(1)=(2 p cos theta)/(4 pi varepsilon_(0)r^(3))` along PD
Point P is on the equatorial line `w.r.t. p sin theta`
`:." E_(2)=(p sin theta)/(4 pi varepsilon_(0)r^(3))` along PC
Since `E_(1) and E_(2)` are perpendicular to each other, so the resultant electric field is given by,
`E=sqrt(E_(1)^(2)+E_(2)^(2)+2E_(1)E_(2)cos theta)`
`But" "theta=90^(@)`
`:." "E=sqrt(E_(1)^(2)+E_(2)^(2)+2E_(1)E_(2)cos 90^(@))`
`E=sqrt(E_(1)^(2)+E_(2)^(2)+O)`
`E=sqrt(E_(1)^(2)+E_(2)^(2))`
`E=sqrt(((2 p cos theta)/(4pi varepsilon_(0)r^(3)))^(2)+((p sin theta)/(4 pi varepsilon_(0)r^(3)))^(3))`
`E=p/(4 pi varepsilon_(0)r^(3))sqrt(4 cos^(2) theta+sin^(2)theta)`
`E=p/(4 pi varepsilon_(0)r^(3))sqrt(3 cos^(2)theta+cos^(2)theta+sin^(2)theta)`
since`" "sin^(2)theta+cos^(2)theta=1`
`:." "E=p/(4 pi varepsilon_(0)r^(3))sqrt(3 cos^(2)theta+1)`