Derive an expression for field intensity at any point on axial line of dipole.
Or
Derive an expression for electric field intensity due to electric dipole at any point on the axis of the dipole. What is the magnitude and direction of electric field intensity, when the distance of the point from the centre of the dipole is very large as compared to the length of the dipole.

Explanation

Let us take an electric dipole consisting of two point charges - and + q separated by a small distance 2a. We wish to calculate electric field intensity at point P, on the axial line of dipole.
Let`" "OP = r, AP= r + a and BP=r-a`
If `E_(1)` be the electric field intensity at P, due to charge - q placed at point A, then

`abs(vecE_(1))=1/(4pivarepsilon_(0))q/((AP)^(2))`
`implies" "abs(vecE_(1))=1/(4pivarepsilon_(0))q/((r+a)^(2))`
The direction of `(vecE_(1))` is along `vec(PA)`. if `vecE_(2)` be the electric field intensity at P, due to change `+-q` placed at the point B, then
`absvec(E_(2))=1/(4pivarepsilon_(0))q/((BP)^(2))=1/(4pivarepsilon_(0))q/((r-a)^(2))`
The direction of `vec(E_(2))` is along `vec(PX)`
As `abs(vecE_(2)) > abs (vecE_(1))`, the resultant at p is given by : `[:. "Point B is nearer to p than A"]`
`absvecE=abs(vecE_(2))-abs(vecE_(1))`
`=1/(4 pivarepsilon_(0))q/((r-a)^(2))-1/(4pivarepsilon_(0))q/((r+a)^(2))[1/((r-a)^(2))-1/((r+a)^(2))]`
`=q/(4pivarepsilon_(0))[((r+a)^(2)-(r-a)^(2))/((r^(2)-a^(2))^(2))]=q/(4 pivarepsilon_(0))(4ar)/((r^(2)-a^(2))^(2))=(q xx 2a xx 2r)/(4pivarepsilon_(0)(r^(2)-a^(2))^(2))`
`:." "abs(vecE)=(abs(vecp)2r)/(4pivarepsilon_(0)(r^(2)-a^(2))^(2))" "[:. q xx 2a =abs(vecp)]`
When the distance of the point from the center of the dipolr is very large as compare to langth of dipole
`i.e.,2a"<<"rimpliesr^(2)-a^(2)~~r^(2)`
Then,`" "abs(vecE)=(abs(vecp)2r)/(4pivarepsilon_(0)r^(4))impliesabs(vecE)=1/(4pivarepsilon_(0))(2abs(vecp))/r_(3)`
Cleraly,`abs(vecE)prop1/r^(3)`. The direction of `vecE` is along `vec(PX)`.