A charged particle moving in an electromagnetic field exhibits a "drift" in addition to its gyromotion and any acceleration due to a component of the electric field parallel to the magnetic field. This drift motion has velocity \((\boldsymbol{E}\times\boldsymbol{B})/B^2\) and is therefore known as the \(\boldsymbol{E}\times\boldsymbol{B}\) drift. In the simple case of constant, crossed magnetic and electric fields, the Lorentz equation of motion, \(m\ddot{\boldsymbol{r}} = q(\boldsymbol{E} + \dot{\boldsymbol{r}} \times \boldsymbol{B})\) can be solved analytically to give the particle's trajectory:
where the fields are \(\boldsymbol{B} = (0,0,B_z)\) and \(\boldsymbol{E} = (0,E_y,0)\); the initial velocity is \(\boldsymbol{v} = (0,v_\perp,0)\); the gyrofrequency is \(\Omega = qB_z/m\); and \(m\) and \(q\) are the particle's mass and charge respectively.
The Jupyter Notebook here depicts this solution and is also available on my github page.
