Question

Two short electric dipoles are placed as shown. The energy of electric interaction between these dipoles will be:

• 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

$W=-\frac{k}{r^3}[\frac{3(\overrightarrow{p_1}.\overrightarrow{r})(\overrightarrow{p_2}.\overrightarrow{r})}{r^2}-\overrightarrow{p_1}.\overrightarrow{p_2}]$

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