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Mathematics for Physics

35 Cyclotron Motion

A circular or spiraling path of a charged particle in a uniform magnetic field that is used to describe orbital motion when the magnetic force stays perpendicular to the velocity.

The cyclotron radius. A uniform magnetic field does no work and bends the velocity into a circle in the plane perpendicular to \(\mathbf{B}\). This principle is used to compute the cyclotron radius from the perpendicular speed.

The cyclotron radius is

\[ R = \dfrac{mv_{\perp}}{|q|B} \]

where

  • \(R\) is the cyclotron radius.
  • \(m\) is the mass of the particle.
  • \(v_{\perp}\) is the speed perpendicular to \(\mathbf{B}\).
  • \(q\) is the charge.
  • \(B\) is the magnitude of the magnetic field.

The cyclotron frequency. The angular frequency of that circular motion is independent of speed. This principle is used to time the orbits in a cyclotron.

The cyclotron frequency is

\[ \omega = \dfrac{|q|B}{m} \]

where

  • \(\omega\) is the cyclotron angular frequency.
  • \(q\) is the charge.
  • \(B\) is the magnitude of the magnetic field.
  • \(m\) is the mass of the particle.

Helical motion along \(\mathbf{B}\). A velocity component parallel to \(\mathbf{B}\) is unchanged, so the trajectory is a helix. This principle is used to describe charged-particle motion along field lines.

The pitch of the helix is set by

\[ v_{\parallel} = \mathbf{v}\cdot\hat{\mathbf{B}} \]

where

  • \(v_{\parallel}\) is the speed along the field.
  • \(\mathbf{v}\) is the velocity.
  • \(\hat{\mathbf{B}}\) is the unit vector along \(\mathbf{B}\).