An equation for any charge distribution in a constant external electric field. Is it correct? Is the equation (torque, $\tau = p \times E$) correct for any kind of charge distribution? (E is an external constant electric field.) If so, why? Will somebody prove it to me? I'm only sure of its validation for a simple electric dipole. 
 A: Consider a system of stationary point charges $q_i$, with position vectors $\mathbf{r}_{i}$ in an external electric field, assuming a value $\mathbf{E}_{i}$ at the position of the $i$th charge. The net torque on this system, about the origin of coordinates is:
$$\mathbf{\tau} =  \sum_{i} \mathbf{r}_{i} \times (q_i \mathbf{E}_{i})$$
or
$$\mathbf{\tau} = \sum_{i} q_i\mathbf{r}_{i} \times \mathbf{E}_{i}$$
In general, as the dipole moment of an arbitrary charge distribution is defined as $\mathbf{p} = \sum_{i} q_i \mathbf{r}_{i}$, we see that torque is not always expressible as the cross product of the dipole moment and the electric field.
However, in the special case that the electric field is uniform, or the charges are close enough together that the variation in the electric field can be neglected, the electric field $\mathbf{E}$ can be factored out of the sum (as it contributes the same value to every term), and one gets
$$\mathbf{\tau} = (\sum_{i} q_i \mathbf{r}_{i}) \times \mathbf{E} = \mathbf{p} \times \mathbf{E}$$
