EMG-12

Cyclotron frequency

ω = q B / m independent of speed (non-relativistic).

Reading speed

Charged particlesLawrence cyclotron

Governing equation

\omega_c=\dfrac{qB}{m},\quad f_c=\dfrac{qB}{2\pi m}

where

q: Charge (e units) (e)
B: Magnetic field (T)
m: Mass (u) (u)
f_c: Cyclotron frequency (MHz)
r: Radius at 1 keV/u (order) (mm)

Lecture brief

Historical brief

Coulomb, Gauss, Ampère, Faraday and Maxwell (1861–65) unified charge, current and light. The lab computes fields, induction, Poynting flux and the electromagnetic wave in SI. This sheet (EMG-12 — Cyclotron frequency) is the form associated with Lawrence cyclotron. Working symbols: $q$ , $B$ , $m$ $\rightarrow$ $f_c$ , $r$ . The Lorentz centripetal requirement q v B = m v² / r cancels v and leaves ω = v/r = q B / m, the cyclotron frequency.

Purpose

Purpose: compute

f_c

r

from

q

B

m

in Electromagnetism via

\omega_c=\dfrac{qB}{m},\quad f_c=\dfrac{qB}{2\pi m}

ω = q B / m independent of speed (non-relativistic). Use it when a real electromagnetism question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

q = 1.000\,\mathrm{e}

B = 1.500\,\mathrm{T}

m = 1.000\,\mathrm{u}

, the governing relation

\omega_c=\dfrac{qB}{m},\quad f_c=\dfrac{qB}{2\pi m}

yields

f_c = 23.034\,\mathrm{MHz}

r = 0.000\,\mathrm{mm}

. Uniform B, v ⟂ B, non-relativistic electron or ion. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

Charge (e units) q1 eMagnetic field B1.5 TMass (u) m1 u

Outputs

Cyclotron frequency f_c23.034 MHz
Radius at 1 keV/u (order) r0.000 mm

Reading speed

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Narration of this film

Uniform B, v ⟂ B, non-relativistic electron or ion.

The Lorentz centripetal requirement q v B = m v² / r cancels v and leaves ω = v/r = q B / m, the cyclotron frequency.

Reading speed

Watch on YouTube