Question

Two short electric dipoles are placed as shown ($r$ is distance between their centres). The energy of electric interaction between these dipoles will be:
($C$ is centre of dipole of moment $P_2$)

• A. $\frac{2k{P}_1{P}_2\cos \theta }{r^3}$
• B. $-\frac{2k{P}_1{P}_2\cos \theta }{r^3}$
• C. $\frac{-2k{P}_1{P}_2\sin \theta }{r^3}$
• D. $\frac{-4k{P}_1{P}_2\cos \theta }{r^3}$

Answer 1

The correct option is B $-\frac{2k{P}_1{P}_2\cos \theta }{r^3}$

The general expression of the energy of electric interaction between two dipoles is

$W=-\frac{k}{r^3}\left[\frac{3\left(\overrightarrow{P_1}.\overrightarrow{r}\right)\left(\overrightarrow{P_2}.\overrightarrow{r}\right)}{r^2}-\overrightarrow{P_1}.\overrightarrow{P_2}\right]$

Here,

$\overrightarrow{P_1}.\overrightarrow{r}=P_{1}r\cos 0=P_{1}r$

$\overrightarrow{P_2}.\overrightarrow{r}=P_{2}r\cos \theta$

$\overrightarrow{P_1}.\overrightarrow{P_2}=P_1{P}_2\cos \theta$

Now,

$\begin{array}{c}W=-\frac{k}{r^3}\left[\frac{3\left(P_{1}r\right)\left(P_{2}r\cos \theta \right)}{r^2}-P_1{P}_2\cos \theta \right]\\ =-\frac{k}{r^3}\left[3P_1{P}_2\cos \theta -P_1{P}_2\cos \theta \right]\\ =-\frac{2k{P}_1{P}_2\cos \theta }{r^3}\end{array}$

Hence, Option $B$ is the correct answer.