(i) Derive the expression for the specific for the electric potential due to an electric dipole at a point on its axial line.
(ii) Depiet the equivpotential surfaces due to an electric dipole.

Explanation

(i) Expression : Potential at P due to charge `+q` and -q.
`V_(-q) = (-q)/(4piin_(0)r_(1))` and `V_(+q) = q/(4piin_(0) r_(2))`
Potential at P due to dipole `V = V_(-q) + V_(+q) = (q)/(4piin_(0)) [1/(r_(2)) - 1/(r_(1))] "…….." (i)`

From the figure, `r_(1)^(2) = r^(2) +d + 2ar cos theta`
and, `r_(2)^(2) = r^(2) + a^(2) + 2ar cos(180^(@) - theta) = r^(2) + a^(2) - 2ar cos theta`
`rArr r_(1)^(2)= r^(2)[1+(a^(2))/(r^(2)) +(2a)/(r ) costheta] = r^(2)[1+(2a)/(r) costheta]` as `r gt gt a, (a^(2))/(r^(2)) = 0`
`rArr r_(1) = r[1+(2a)/(r) cos theta]^(1/2)`
So `1/(r_(1)) = 1/r[1+(2a)/(r) cos theta]^(-1/2)`
and `1/(r_(2)) = = 1/r [1-(2a)/(r) cos theta]^(-1//2)`
Using it for `(i)`
`V = (q)/(4piin_(0)r)[1+a/r cos theta - (1- a/r cos theta)]`
[Applying Binomial Theorem and neglecting highest power of `(2a)/(r)` ]
Hence, `V = (q)/(4piin_(0)r) [1+q/r cos theta - 1+ a/r cos theta]`
`= (qxx2a cos theta)/(4piin_(0)r^(2))`
`rArrV = (p cos theta)/(4piin_(0)r^(2)) , {p = q xx 2a = " dipole moment"}`
Vaxial `= (pcos theta)/(4piin_(0)r^(2))` ( `:' theta = 0^(@)` on axial line) `= (p)/(4pi in_(0)r^(2))`
(ii) `V_("net") = V_(A) + V_(B) = (-kq)/((r^(2)+a^(2))) + (-kq)/((r^(2)+a^(2)) = 0`