# Force on a charge from an induced dipole

1. ### Yitzach

61
1. The problem statement, all variables and given/known data
A point charge q is situated a large distance r from a neutral atom of polarizability $$\alpha$$. Find the force of attraction between them.

2. Relevant equations
$$\vec{E}_{mono}(r)=\frac{q}{4\pi\epsilon_0r^2}\hat{r}$$
$$\vec{E}_{dip}(r,\theta)=\frac{p}{4\pi\epsilon_0r^3}(2\cos\theta\hat{r}+\sin\theta\hat{\theta})$$
$$\vec{p}=\alpha\vec{E}$$

3. The attempt at a solution
$$\vec{E}_{mono}(r)=\frac{q}{4\pi\epsilon_0r^2}\hat{r}$$
$$\vec{p}=\alpha\vec{E}$$
$$\vec{p}=\frac{\alpha q}{4\pi\epsilon_0r^2}\hat{r}$$
$$\vec{E}_{dip}(r,\theta)=\frac{p}{4\pi\epsilon_0r^3}(2\cos\theta\hat{r}+\sin\theta\hat{\theta})$$
$$\vec{E}_{dip}(r,\pi)=\frac{\alpha q}{16\pi^2\epsilon^2_0r^5}(-2\hat{r})$$
$$\vec{E}_{dip}(r,\pi)=-\frac{\alpha q}{8\pi^2\epsilon^2_0r^5}\hat{r}$$
$$\vec{F}=q\vec{E}$$
$$\vec{F}=-\frac{\alpha q^2}{8\pi^2\epsilon^2_0r^5}\hat{r}$$

The result I got was unexpected because that is a repulsive force.
Do I need to go about a longer way or did I mess it up somewhere?

Last edited: Dec 6, 2009