I am having a lot of difficulty finding an approach to solving the following question:
A dyon is a particle with both electric and magnetic charge; in suitable units
$$\mathbf{E} = \frac{-Q}{4\pi\varepsilon_0 r^2 } \mathbf{e_r}$$
$$\mathbf{B} = \frac{-P}{4\pi\mu_0 r^2 } \mathbf{e_r}$$
A particle of mass $m$ and charge $e$ moves in the field of the dyon. Using the equation of motion for a charged particle
$$\mathbf{F}= e (\mathbf{E} + \mathbf{v} \times \mathbf{B})$$
and the expression for velocity in spherical polars
$$\dot{\mathbf{r}} = \dot{r} \mathbf{e_r} + r \dot\theta \mathbf{e_\theta} + r \dot\varphi \sin\theta \,\mathbf{e_\varphi}$$
find the vector equation of motion in the spherical polar basis $\{\mathbf{e_r}, \mathbf{e_\theta}, \mathbf{e_\varphi} \}$
I have attempted to solve the problem by first finding the cross product of $\mathbf{v} \times\mathbf{B}$
Since $\mathbf{v}= \dot{\mathbf{r}}$
Then just following through with the given equation of motion by adding vectors $\mathbf{E}+(\mathbf{v} \times \mathbf{B})$
Then finally multiplying everything through with the scalar $e$.
However I am not completely sure that this is the correct way to go about this problem, all help is very much greatly appreciated.