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# Lorentz Force

The Lorentz force equation describes the force on a particle of charge $q$ with velocity $v$ moving in an electric field $E$ and magnetic field $B$:

${\mathbf {F}}=q{\mathbf {E}}+q{\mathbf {v}}\times {\mathbf {B}}$

The electric force is $q{\mathbf {E}}$, such that the magnitude of the force is proportional to charge and the field strength. The force acts in the direction of the electric field.

The magnetic force is $q{\mathbf {v}}\times {\mathbf {B}}$, where the cross product term $\times$ indicates that the magnetic field only operates on the part of the velocity of the charge that is not aligned with the magnetic field. This means that a particle in a magnetic field will curve around the field. Convention for the direction of curvature follows the right-hand rule. Place a flat hand with fingers in direction of velocity, bend fingers in direction of the magnetic field, and your thumb will point in the direction of the magnetic force on a positive charge.

Units are $F$ in $N = kg \cdot m/s^2$, $E$ in $N/C$, $B$ in $T = N/(m \cdot A)$, $v$ in $m/s$ and $q$ in $C = A \cdot s$

Use the 2-D demo below to see how changing the various parts of the Lorentz equation affects a charged particle's motion.

Purple represents negative charges, and orange represents positive charges.

0.00 N/C or V/m

(-) (+) +0.00 T
m/s
m/s
(-) (+) C/kg
(-) (+) C/kg

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A velocity selector uses perpendicular E and B fields to allow through only particles with a particular velocity, perpendicular to both fields. In the demo set E to "up/down" and some value. Set B to another value. You should find that only particles with velocity v = E/B move in a straight line. All other values curve away.

### Basics of mass spectrometry

Three main pieces are an electric field to accelerate charged particles (from an ionization chamber) up to speed, magnetic and electric fields perpendicular to the velocity and each other to select a narrow range of velocities, and a magnetic field to curve particles based on mass/charge ratio.

Relevant equations:
velocity selector (E and B create equal and opposite forces): $v = E/B$
radius of curvature (derived from $F = ma = qvB$): $r = \frac{m\,v}{q\,B} = \frac{m\,E}{q\,B1\,B2}$

(-) (+) 1.00

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