Model of a Bainbridge Mass Spectrometer

A Wien filter (velocity-selector) allows to isolate particles with a specific velocity of a particle beam. To investigate the mass of the particles the experimental setup must be extended. A simple setup to examine the mass of particles is the Bainbridge Mass Spectrometer.

Setup:

In the extended setup the magnetic field does not end on the aperture. So, after passing the velocity-selector, stay in the magnetic field. There the Lorentz force is the only force that affects the motion of the particles. So the particles start moving in a circle. In addition to the setup of a Wien filter there is a detector plate on the back of the aperture. This plate detects where the particles impinge.
Setup of a Bainbridge Mass Spekctrometer

Function:

If a charged particle moves with velocity v in the presence of a magnetic field B, then it will experience a Lorentz force perpendicular to the direction on motion:$$F_{\rm{Lorentz}}=q\cdot v\cdot B$$ This force causes that the particles start moving in a circle because the Lorentz force acts as centripetal force on the particles. The radius of the circular motion is:$$F_{\rm{Lorentz}}=F_{\rm{Zentripetal}}\Rightarrow q\cdot v\cdot B=m\cdot \frac{v^2}{r} \Rightarrow r=\frac{m\cdot v}{q\cdot B}$$ Because of the velocity selctor all particles have the velocity $v=v_{\text{passage}}$.
With $v_{\text{passage}}=\frac{E}{B}$ follows$$r=\frac{m\cdot v_{\text{passage}}}{q\cdot B}= \frac{m\cdot E}{q\cdot B^2}$$ So particles with equal charge and different mass move on circular paths with different radius. So, they collide with the detector plate on different places. With the distance $d=2\cdot r$ between the impact point and the aperture of the velocity selector, the mass of the particles can be calculated with$$m=\frac{q\cdot B\cdot d}{2\cdot v_{\text{passage}}}=\frac{q\cdot B^2\cdot d}{2\cdot E}$$

Restrictions:

With a Bainbridge Mass Spectrometer only charged particles can be examined. So uncharged particles must be charged at first.
Furthermore, the charge must be known to calculate the mass of the particles. Otherwise, only the specific charge q/m can be calculated.
Particles with the same charge-mass-ratio impact impact on the detector plate in the same distance d from the aperture.