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Question

(B) The general expression for the potential energy of interaction between two short electric dipoles $\vec{p}_1$ and $\vec{p}_2$ separated by a dista

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$\frac{-2kP_1P_2\cos 	heta}{r^3}$

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(B) The general expression for the potential energy of interaction between two short electric dipoles $\vec{p}_1$ and $\vec{p}_2$ separated by a distance vector $\vec{r}$ is given by:$W = -\frac{k}{r^3} \left[ \frac{3(\vec{p}_1 \cdot \vec{r})(\vec{p}_2 \cdot \vec{r})}{r^2} - \vec{p}_1 \cdot \vec{p}_2 ight]$From the given configuration:$1$. The vector $\vec{r}$ is along the direction of $\vec{p}_1$,so $\vec{p}_1 \cdot \vec{r} = P_1 r \cos(0) = P_1 r$.$2$. The angle between $\vec{p}_2$ and $\vec{r}$ is $	heta$,so $\vec{p}_2 \cdot \vec{r} = P_2 r \cos 	heta$.$3$. The angle between $\vec{p}_1$ and $\vec{p}_2$ is $	heta$,so $\vec{p}_1 \cdot \vec{p}_2 = P_1 P_2 \cos 	heta$.Substituting these values into the expression:$W = -\frac{k}{r^3} \left[ \frac{3(P_1 r)(P_2 r \cos 	heta)}{r^2} - P_1 P_2 \cos 	heta ight]$$W = -\frac{k}{r^3} [3 P_1 P_2 \cos 	heta - P_1 P_2 \cos 	heta]$$W = -\frac{k}{r^3} [2 P_1 P_2 \cos 	heta]$$W = -\frac{2kP_1P_2\cos 	heta}{r^3}$

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

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