Uniform circular motion in a magnetic field

When $\vec{v}\perp\vec{B}$ and $\vec{B}$ is uniform, the Lorentz force is always perpendicular to the velocity: its magnitude is constant but its direction changes continuously. This is exactly the behaviour of a centripetal force, so the trajectory is a circle traced at constant speed.

Setting the magnitude of the Lorentz force $|qvB|$ equal to the centripetal force $mv^2/r$:

$qvB = \frac{mv^2}{r} \quad\Longrightarrow\quad \ev{r = \frac{mv}{|q|B}}$

The radius of this circle is called the Larmor radius. It grows with the particle’s momentum $mv$ and decreases as the field increases.

Positive charge in an inward field: the Lorentz force always points towards the centre and the trajectory is a circle of radius $r = mv/(|q|B)$.

Observations on the Larmor radius

• If $B$ increases, $r$ decreases: a stronger field curves the path more.
• If $B = 0$, $r\to\infty$: straight-line motion, no force.
• If $m$ increases, $r$ increases: different masses separate — this is the principle of the mass spectrometer.

Going further — Mass spectrometer

The fact that $r$ depends on $m$ (for the same $v$, $q$, $B$) makes it possible to distinguish isotopes: the same element with two different mass numbers, for example ${}^{235}\text{U}$ and ${}^{238}\text{U}$, produces two circles of slightly different radius in the same magnetic field. This is the basis of the mass spectrometer, a fundamental instrument in analytical chemistry and in the study of cosmic rays.

Try it — interactive simulation

Move the sliders to change the field, speed, pitch angle, mass and charge sign, and watch how the radius responds while the period stays the same.

A charged particle in a uniform magnetic field: a circle when the velocity is perpendicular to the field, a helix when it has a parallel component.

Links

Topics: Magnetismo Concepts: Forza di Lorentz · Moto circolare uniforme · Forza centripeta

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