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Why do ions with different masses follow different curved paths in a magnetic field?

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Figure 1: An electron bombardment ion source in cross section. An electron beam is drawn from the filament and accelerated across the region in which the ions are formed and toward the electron trap. An electric field produced by the repeller forces the ion beam from the source through the exit slit. — mass spectrometry Figure 1: *An electron bombardment ion source in cross section.* An electron beam is drawn from the filament and accelerated across the region in which the ions are formed and toward the electron trap. An electric field produced by the repeller forces the ion beam from the source through the exit slit.

In a mass spectrometer, ions with different masses follow different curved paths because their trajectories depend on their mass-to-charge ratio when they possess the same kinetic energy.1

When ions of charge $z$ are accelerated through a voltage drop $V$ , they acquire an identical kinetic energy, $z V = \frac{1}{2} m v^{2}$ . Because each ion has a different mass $m$ , each must travel at a unique velocity $v$ to maintain this energy. When these ions enter a magnetic field $B$ at a right angle, the magnetic force acts as a centripetal force, causing them to move in a circular path with a radius $r$ .

The relationship between these variables is defined by the equation:
$\frac{m}{z} = \frac{B^{2} r^{2}}{2 V}$

Because the magnetic field and the accelerating voltage are constant, the radius of curvature $r$ becomes a function of the mass-to-charge ratio ( $m / z$ ). Consequently, ions with different masses are deflected by different amounts, allowing the spectrometer to separate them into distinct components.1