A non-relativistic charged particle files through the electric field of a cyclindrical capacitor and gets into a unifrom transverse magnetic field with induction `B`(fig). In the capacitor the particle moves along the are of a circle, in the magnetic field, along a semi-circle of radius `r`. The potential differnce applied to the capacitor is equal to `V`, the radii of the electrodes are equal to `a` and `b`, with `a lt b`. Find the velocity of the particle and its specific charge `q//m`.

Explanation

Inside the capacitor, the electric field follows `a (1)/(r)` law, and so the potential can be written as
`varphi = (V In r//a)/(In b//a), E = (-V)/(In b//a) (1)/(r)`,
Here `r` is the distance from the axis of the capacitor.
Also, `(mv^(2))/(r) = (qV)/(In b//a r) (1)/(r)` or `mv^(2) = (qV)/(In b//a)`
On the other hand
`mv = q B r` in the magentic field.
Thus, `v = (V )/(B r In b//a)` and `(q)/(m) = (v)/(Br) = (V)/(B^(2) r^(2) In (b//a))`