Lorentz Force

The Lorentz force is the summation of both the Electric Force and the Magnetic Force that act on an object.

Definition §

Lorentz Force

$${\vec{F}}_{net}={\vec{F}}_E+{\vec{F}}_B=q({\vec{E}}_{ext}+\vec{v}\times {\vec{B}}_{ext})$$

Note

• $q$ = Charge of particle (in $C$)
• ${\vec{E}}_{ext},{\vec{B}}_{ext}$ = External Electric Field and Magnetic Field, respectively (in $N/C$ and $T$)
• $\vec{v}$ = Velocity of particle (in $m/s$)
  For the magnetic force to contribute, $\vec{v}\nparallel \vec{B}$

Implications §

In the case where the external magnetic and electric fields can be controlled, and are equal in magnitude, i.e.

$$\Vert \vec{E}\Vert =\Vert \vec{B}\Vert$$

Then the only way for the particle to not experience any force (and therefore change direction) is if:

$$\vec{v}=\frac{\Vert E\Vert }{\Vert B\Vert }$$

This allows us to filter particles of certain velocities. Such a principle was used in by J.J Thomson in his experiment.