A moving charge in $\vec{E}$ and $\vec{B}$ fields

Once we have found electric and magnetic fields, we can control the motion of charged particles using them. This is how a TV, a mass spectrometer, a cyclotron and and many other devices work. The key equation is the Lorentz force law:

$$\vec{F}=q(\vec{E}+\vec{v}\times \vec{B})$$ (3)

This gives the force on a particle of charge q moving at velocity $\vec{v}$ and being acted upon by an electric field $\vec{E}$ and a magnetic field $% \vec{B}$. To find the motion of the particle, just use Newton's law $\vec{F}% =m\vec{a}=m\frac{d^{2}\vec{x}}{dt^{2}}$, where $\vec{x}$ is the position of the particle. Qualitatively, the charge is accelerated along the direction of the electric field, while it ``spirals'' around the direction of the magnetic field. Very complex motion can occur when these two basic motions are combined, especially if the electric and magnetic fields change in space and time.