The value of electric potential at any point due to any electric dipole is $(k = \frac{1}{4 π ε_{0}})$

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$k \cdot \frac{\vec{p} \times \vec{r}}{r^{2}}$

$k \cdot \frac{\vec{p} \times \vec{r}}{r^{3}}$

$k \cdot \frac{\vec{p} . \vec{r}}{r^{3}}$

answer is D.

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Detailed Solution

Given data: Electric potential at any point due to any electric dipole

Formula used: Electric potential $p : v = v_{1} + v_{2}$ ; dipole moment , $p = q \times 2 l$

Detailed solution:

Let P,Q be the dipole will be a distance of 2L between then consider a point X at a distance ‘r’ from the midpoint of PQ.

$Q X = R X = O X - O R = r - L \cos θ$

$P X = S X = O X + O S = r + L \cos θ$ (as dipoles are nearer )

Electric potential at X :

$v = v_{1} + v_{2} v = \frac{1}{4 π ξ} \frac{- q}{P X} + \frac{1}{4 π ξ} \frac{q}{Q X} v = \frac{k q 2 / \cos (θ)}{r^{2} - l^{2} \cos^{2} θ}$

We know dipole moment

$p = q \times 2 l v = \frac{k p \cos (θ)}{r^{2}} = \frac{k p r \cos (θ)}{r^{3}} v = k \frac{\vec{p} \cdot \vec{r}}{r^{3}}$

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