Derive an expression for-electric field intensity at any point on equatorial line neutral line of dipole.
Or
Two point charges - q and + q are placed at a distance '2a' apart. Calculate the electric field intensity at a point A situated at a distance r along the perpendicular bisector of the line joining the charges.

Explanation

Let us take an electric dipole which consists of two point charges - q and + q separated by very small distance 2a. We wish to calculate the electric field intensity at point P, on equatorial line of dipole.
Electric field intensity at P due to - q charge placed at point A is given by
`abs(vecE_(1))=1/(4pivarepsilon_(0))q/((AP)^(2))`
`implies" "abs(vecE_(1))=1/(4pivarepsilon_(0))q/((a^(2)+r^(2)))" ".......(i)`
The direction of E, is along PA.
Resolve `E_(1)` into two components `E_(1) cos theta` along `PR and E_(1) sin theta` along PO. Electric field intensity at P, due to +q charge placed at B is given by :
`abs(vecE_(2))=1/(4pivarepsilon_(0))q/((BP)^(2))=1/(4pivarepsilon_(0))q/((a^(2)+r^(2)))" ".........(ii)`
The direction of `E_(2)` is along PD.
Resolve `E_(2)` into two components. i.e., `E_(2)sintheta` along PF and `E_(2), cos theta` along PR. The components `E_(1) sin theta` along PO and `E_(2), sin theta` along PF are equal and opposite and hence they cancel each other.
Hence, net electric field intensitv at P is given by
`E = E_(1) cos theta + E_(2) cos theta`
Substitute the values of `abs(vecE_(1)) and abs(vecE_(2))` from equations `(i) and (ii)`, we have
`E = 1/(4pivarepsilon_(0))q/((a^(2)+r^(2)))costheta+1/(4pivarepsilon_(0))q/((a^(2)+r^(2)))costheta`
`implies" "E=2[1/(4pivarepsilon_(0)).q/((a^(2)+r^(2)))costheta]" ".......(iii)`
`implies "From" Delta AOP,costheta=(OA)/(AP)impliescostheta=a/((a^(2)+r^(2))^(1/2))`
Putting the value of `cos theta` in equation (iii), we get
`E=2[1/(4pivarepsilon_(0)).(q2a)/((a^(2)+r^(2))).a/((a^(2)+r^(2))^(1//2))]`
`implies" "E=1/(4pivarepsilon_(0))(q2a)/((a^(2)+r^(2))^(3//2))`
`implies" "E=1/(4pivarepsilon).p/((a^(2)+r^(2))^(3//2))" "[:.p=qxx2a]`