Wien filter

Theory

Any charged particle in an electric field will feel a force proportional to the charge and field strength such that ${\displaystyle \overrightarrow{F}=q\overrightarrow{E}}$, where F is force, q is charge, and E is electric field strength. Similarly, any particle moving in a magnetic field will feel a force proportional to the velocity and charge of the particle. The force felt by any particle is then equal to ${\displaystyle \overrightarrow{F}=q\overrightarrow{v}\times \overrightarrow{B}}$, where F is force, q is the charge on the particle, v is the velocity of the particle, B is the strength of the magnetic field, and ${\displaystyle \times }$ is the cross product. In the case of a velocity selector, the magnetic field is always at 90 degrees to the velocity and the force is simplified to ${\displaystyle F=qvB}$ in the direction described by the cross product.

Setting the two forces to equal magnitude in opposite directions it can be shown that ${\displaystyle \frac{E}{B}=v}$. Which means that any combination of electric (${\displaystyle \overrightarrow{E}}$) and magnetic (${\displaystyle \overrightarrow{B}}$) fields will allow charged particles with only velocity ${\displaystyle \overrightarrow{v}}$ through.