Larmor Radius

Intuition

The Larmor radius is the radius of the circular motion a charged particle makes when it gyrates around a magnetic field line. It measures how tightly a particle is bound to the field.

Formal Definition

\[\\rho_s = \\frac{m_s v_{\\perp}}{|q_s| B}\]

where \(v_{\\perp}\) is the velocity component perpendicular to \(\\mathbf{B}\).

Mathematical Formulation

For a thermal species with \(v_{\\perp} \\sim v_{th} = \\sqrt{2 k_B T_s/m_s}\):

\[\\rho_{th} = \\frac{\\sqrt{2 m_s k_B T_s}}{|q_s| B}\]

Derivation

Lorentz force provides centripetal acceleration:

\[|q| v_\\perp B = \\frac{m v_\\perp^2}{\\rho} \\quad \\Rightarrow \\quad \\rho = \\frac{m v_\\perp}{|q| B}\]

Worked Example

For a 1-keV proton in \(B = 3\) T: - \(v_\\perp \\approx \\sqrt{2\\times 1.6\\times 10^{-16}\\,\\text{J} / 1.67\\times 10^{-27}\\,\\text{kg}} \\approx 4.4\\times 10^5\\,\\text{m/s}\) - \(\\rho \\approx (1.67\\times 10^{-27} \\times 4.4\\times 10^5) / (1.6\\times 10^{-19} \\times 3) \\approx 1.5\\times 10^{-3}\\,\\text{m}\)

Common Mistakes

Using parallel velocity instead of perpendicular velocity.
Forgetting the \(\\sqrt{2}\) factor when converting from temperature to thermal speed.

Quiz Questions

If \(B\) doubles, what happens to \(\\rho\)?
Why do electrons gyrate faster than ions?