Derive the formula for the electric potential due to an electric dipole at a point from it.

Explanation

A dipole is placed with origin at its midpoint. Its charge -q and +q is separated by a distance 2a. The magnitude of a dipole moment of the dipole is p= 2aq and its direction is from -q to +q.
Let point P away from the midpoint of dipole and `r_(1)` and `r_(2)` are the distance of P from +q and q respectively which is shown in the figure .

The potential due to the charge +q at point P is
`V_(1) (k(+q))/(r_(1))` and
The potential due to the charge -q,
`V_(2)=(k(-q))/(r_(2))= (kq)/(r_(2))`
According to superposition principle, the total potential at point P
`V=V_(1)+V_(2)`
`=(kq)/(r_(1))-(kq)/(r_(2))`
`=kq [(1)/(r_(1))-(1)/(r_(2))]`
Draw qN `bot` OP In `DeltaqON angleqON = theta`
`:. ON = r- r_(1)`
and `cos theta= (ON)/(Oq) implies ON = Q q cos theta`
`:. r- r_(1) = a cos theta implies r_(1) = r - a cos theta `
`=r(1-(a_(r1))/(r) cos theta)`
`implies r_(1)^(2)= r^(2) (1-(a)/(r)cos theta)^(2)`
`=r^(2)(1-""(2acostheta)/(r)+(a^(2))/(r^(2))cos^(2)theta)`
`=r^(2)(1-(2acostheta)/(r))`
Putting `rgtgt a implies (a^(2))/(r^(2))=0`
and draw OM `bot` (-qp=P) then in `Delta- ` qOM P(-q) M `= theta` and (-q) M `= r_(2) -r `
`:. cos theta((-q)M)/(O-(-q))implies (-q)M =m O (-q) cos theta`
`:. r_(2) -r = acostheta`
`:> r_(2) = r +a cos theta`
`:. r_(2)= r(1+(a)/(r) cos theta)`
Similarly `r_(2)^(2) =r^(2) (1+(2acostheta)/(r))`
Now from equation (4 )
`:. (1)/(r_(1))=(1)/(r)(1-(2acostheta)/(r))^(-1//2)`
`=(1)/(r)(1+(acostheta)/(r))`
only two terms be taken from expansion and,
`(1)/(r_(2))=(1)/(r)(1-(acostheta)/(r))`
Now from equation (5)
only two terms be taken from expansion .
`implies` Putting the values of equation (6) and (7) in equation (1) ,
`V=(kq)/(r)[(1+(acostheta)/(r))-(1-(acostheta)/(r))]`
`=(kq)/(r)[(2acostheta)/(r)]`
`=(k(p)costheta)/(r^(2)) [ because 2aq=p]`
`=(kvecP.vecr)/(r^(2))[becausep costheta= vecP.hatr]`
`:. V = (p.hatr)/(4piin_(0)r^(2))or (pcostheta)/(4piin_(0)r^(2))or (kpcostheta)/(r^(2))`
Hence potential due to a point charge decreases according `(1)/(r)` and potential due to a dipole decreases according to `(1)/(r^(2))` distance . The molecular dipoles are very small and such an approximation is very well applicable to them.