# The formula of the force exerted on an electric dipole by non-uniform electric field

When an electric dipole of moment $\mathbf{P}$ is located in a non-uniform electric field $\mathbf{E}$, there is an net force exerted on it.

However, the formula of the force in some books is read $\mathbf{F}=\nabla(\mathbf{P}·\mathbf{E})$, while in other books, it is $\mathbf{F}=(\mathbf{P}·\nabla)\mathbf{E}$. Obviously, the two formula are not the same. So, which one is true?

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Both formulas are equivalent, if you are in the electrostatic approximation and your dipole vector does not depend on the position $\mathbf{r}$.

Let's consider the expression $\mathbf{F}=\nabla_{\mathbf{r}}(\mathbf{p} \cdot \mathbf{E})$ which can be easily obtained from the potential energy function

$U=-\mathbf{p} \cdot \mathbf{E}$

and its relation with the force $\mathbf{F}=\nabla_\mathbf{r} U$. Now, recall the vector identity

$\nabla_\mathbf{r}(\mathbf{a}\cdot \mathbf{b})= (\mathbf{a} \cdot \nabla_\mathbf{r}) \mathbf{b}+(\mathbf{b} \cdot \nabla_\mathbf{r}) \mathbf{a} + \mathbf{a} \times (\nabla_\mathbf{r} \times \mathbf{b})+ \mathbf{b} \times (\nabla_\mathbf{r} \times \mathbf{a})$

for $\mathbf{a}=\mathbf{a}(\mathbf{r})$ and $\mathbf{b}=\mathbf{b}(\mathbf{r})$ two arbitrary vectors. For $\mathbf{p}=\mathbf{a} \neq \mathbf{p}(\mathbf{r})$ [independent of the position] and $\mathbf{b}=\mathbf{E}(\mathbf{r}$) we have

$\nabla_\mathbf{r}(\mathbf{p}\cdot \mathbf{E})= (\mathbf{p} \cdot \nabla_\mathbf{r}) \mathbf{E}+(\mathbf{E} \cdot \nabla_\mathbf{r}) \mathbf{p} + \mathbf{p} \times (\nabla_\mathbf{r} \times \mathbf{E})+ \mathbf{E} \times (\nabla_\mathbf{r} \times \mathbf{p})$

As the dipole vector does not depend on the position we can drop the second and the fourth terms. In the electrostatic approximation, Faraday's law reads $\partial_t \mathbf{B}=\mathbf{0}\Leftrightarrow \nabla_\mathbf{r} \times \mathbf{E}(\mathbf{r})=\mathbf{0}$ [this is known as ''Carn's law''] so that the electric field is irrotational and the curl vanishes. Then we can drop the third term and

$\nabla_\mathbf{r}(\mathbf{p}\cdot \mathbf{E})= (\mathbf{p} \cdot \nabla_\mathbf{r}) \mathbf{E}$

Thanks for your answer! But I think the potential energy function $U=-\mathbf{p} \cdot \mathbf{E}$ is only valid in the uniform electric field. How can you prove it for a dipole in the non-uniform field? –  hlew Nov 18 '12 at 9:10
Perhaps I was not clear... I am not saying that it has to be constant but rather that the variations of the electric field are smooth. What I have in mind is that you can make a valid Taylor expansion of $\mathbf{E}(\mathbf{r}_\pm)$ around the center of the dipole $\mathbf{r}_0$. Indeed you can derive the expression for the force using this method, too. –  DaniH Nov 18 '12 at 15:31