Determine torque on a dipole and quadrupole (in external E-Field)

[LeoJakob]

Homework Statement

Determine the torque on a dipole $\vec{M}_{d}$ and on a quadrupole $\vec{M}_{q}$ in an external electric field.
Hint 1: Develope Taylor series of the electric field $\vec{E}(\vec{r})$ around $\vec{r}=0$ up to and including the first order, using that in
Hint 2: $\operatorname{rot} \vec{E}=0$

Relevant Equations

$$
\vec{M} = \int \rho(\vec{r}) \vec{r} \times \vec{E}(\vec{r}) d^{3} \vec r .
$$

For the dipole moment I calculated

$$\begin{aligned}
M &= \int \rho(\mathbf{r}) \mathbf{r} \times \mathbf{E}(\mathbf{r}) d^{3} \mathbf{r} \\
\mathbf{E}(\mathbf{r}) &\approx \mathbf{E}(\mathbf{0}) + \sum_{i=1}^{3} \nabla E_{i}(\mathbf{0}) \cdot \mathbf{r} \\
\mathbf{M}_{D} &= \mathbf{p} \times \mathbf{E} \\
&= \left( \int \rho(\mathbf{r}) \mathbf{r} d^{3} \mathbf{r} \right) \times \mathbf{E}(\mathbf{r}) \\
&= \int \rho(\mathbf{r}) \mathbf{r} \times \left[ \mathbf{E}(\mathbf{0}) + \sum_{i=1}^{3} \left( \nabla E_{i}(\mathbf{0}) \cdot \mathbf{r} \right) \mathbf{e}_i \right] d^{3} \mathbf{r} \\
&= \int \rho(\mathbf{r}) \left( \mathbf{r} \times \mathbf{E}(\mathbf{0}) + \sum_{i=1}^{3} \mathbf{r} \times \left[ \left( \nabla E_{i}(\mathbf{0}) \cdot \mathbf{r} \right) \mathbf{e}_i \right] \right) d^{3} \mathbf{r}
\end{aligned}$$

I don't know how to simplify this equation any further. Are there ways to simplify this equation?

How do I calculate the torque on the quadrupole?

 

[ergospherical]

Try looking at the components, e.g.$$M_i = \int \rho(\mathbf{r}) \epsilon_{ijk} x_j E_k (\mathbf{r}) d^3 \mathbf{r}$$Then expand the field$$E_k (\mathbf{r}) = \left[ E_k (\mathbf{r}') + x_l \frac{\partial}{\partial x_l'} E_k(\mathbf{r}') + \dots \right]_{\mathbf{r}' = \mathbf{0}}$$You will be able to identify the dipole moment $p_i = \int \rho(\mathbf{r}) x_i d^3 \mathbf{r}$.