Question

Two short electric dipoles are placed as shown in figure.


The energy of electric interaction between these dipoles will be :

$[\text{Take}k=\frac{1}{4\pi {\epsilon }_0}]$

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

Answer 1

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

We know that for a very small change in potential $dV$, potential energy can be written as,
$U=-{\overrightarrow{p}}_1\cdot \overrightarrow{E}$

But for a dipole with dipole moment $p_2$ we can write that,

$\overrightarrow{E}=\frac{2k{p}_2\cos \theta }{r^3}\widehat{e_r}+\frac{k{p}_2\sin \theta }{r^3}\widehat{e_{\theta }}$

Since, dipole $p_1$ is along the radial component of electric field
$\Rightarrow U=-\overrightarrow{p}.(\overrightarrow{E_r}+\overrightarrow{E_{\theta }})$

$\Rightarrow U=-\overrightarrow{p}.\overrightarrow{E_r}(\because \overrightarrow{p}\perp \overrightarrow{E_{\theta }})$

$U=-p_1\times \left(\frac{2k{p}_2\cos \theta }{r^3}\right)\cos 0^{\circ }$ (From diagram)

$\Rightarrow U=-\frac{2k{p}_1{p}_2\cos \theta }{r^3}$

Hence, option (b) is the correct answer.