Electrostatics: work moving point dipole

1. Oct 7, 2009

Bapelsin

1. The problem statement, all variables and given/known data

Two electrical dipoles with dipole moments $$\vec{p}_0=p_0\hat{y}$$ and $$\vec{p}_1=p_1\hat{y}$$ are located in the xy-plane. $$\vec{p}_0$$ i located at the origin and $$\vec{p}_1$$ is initially in (x,y)=(a,0). What work is required to move $$\vec{p}_1$$ (with unhanged directon) to the point (x,y)=(0,a)?

2. Relevant equations

Dipole potential: $$\phi_D(\vec{r})=\frac{\vec{p}\cdot\hat{r}}{4\pi\epsilon_0r^2}$$
Work: $$W=Q\phi$$

3. The attempt at a solution

Taking the difference of the potential in the two cases:

$$\left(\phi_{D, p_o}+\frac{p_1\hat{y}\cdot a\hat{x}}{4\pi\epsilon_0a^2}\right) - \left(\phi_{D,p_0}+\frac{p_1\hat{y}\cdot a\hat{y}}{4\pi\epsilon_0a^2}\right)=-\frac{p_1}{4\pi\epsilon_0a}$$

Here comes the step that I'm not sure about. The work is the potential times the point charge - how is it when we have a point dipole? Attempt:

$$W=Q\phi = -\frac{p_0}{a}\cdot\frac{p_1}{4\pi\epsilon_0a}=-\frac{p_0p_1}{4\pi\epsilon_0a^2}$$

Does this make sense? Any help appreciated!

Thanks!

2. Oct 7, 2009

kuruman

Start with the expression for the potential energy of one dipole in the external electric field of the other dipole

$$U=-\vec{p}_{1}\cdot\vec{E}_{2}$$

where

$$E_{2}=\frac{1}{4 \pi\epsilon_{0}}\large(\frac{(3\vec{p}_{2} \cdot \vec{r})\vec{r}}{r^{5}}-\frac{\vec{p}_2}{r^3} \large)$$