# Velocity-selector (Wien filter)

Using a Wien filter is an easy way to select charged particles with a specific speed.

## Setup:

The velocity-selector is a device consisting of perpendicular electric and magnetic fields. A plate capacitor, producing an electric field, is placed in a magnetic field. At the end of the plate capacitor is an aperture. So only particles moving in the middle of the device can pass the Wien filter.
The particle beam enters the velocity-filter so that the velocity-vector of the particles, the electric field and the magnetic field are pairwise perpendicular to each other.

## Function:

When entering the Wien filter electric and magnetic fields cause two different forces (weight force neglected).
The electic field causes the electric force (Coulomb-force) $F_{el}$: $$F_{el}=q\cdot E$$ The motion of charge in a magnetic field causes a Lorentz force:$$F_{\rm{Lorentz}}=q\cdot v\cdot B$$ Only particles, where the amount of electric force $F_{el}$ and Lorentz force $F_{\rm{Lorentz}}$ are equal and show in contrary directions, are not deflected in the velocity-filter. For those particles apply:$$F_{el}=F_{\rm{Lorentz}}\qquad bzw.\qquad q\cdot E=q\cdot v\cdot B$$ Solveing for $v$ provides the velocity, the particles must have to pass the Wien filter: $$v_{\text{passage}}=\frac{E}{B}$$ By particles with velocity $v_0 < v_{\text{passage}}$ the force $F_{el}$ is bigger than $F_{\rm{Lorentz}}$.
By particles with velocity $v_0 > v_{\text{passage}}$ the Lorentz force $F_{\rm{Lorentz}}$ is bigger than $F_{el}$.

## Limitations:

Neither mass of the particles nor the charge of the particles are important for this velocity filter. All particles with the velocity $v_{\text{passage}}$ pass the Wien filter no matter of their mass and their charge.
Also all uncharged particles pass the filter, no matter of their velocity.