Pre-Quantum Electrodynamics
Point Chargesems.ms.lf.pc
- PM 2.7, 10.2
- Gr 5.1-2
The magnetic force on a point charge \(q\) moving at velocity \({\bf v}\) in magnetic field \({\bf B}\) is given by
\[ {\bf F}_{mag} = q {\bf v} \times {\bf B} \label{eq:LorentzForce} \]
Units of \({\bf B}\): \(N/(A~m)\) is called a tesla (symbol: \(T\)). The total electromagnetic force on a point charge \(q\) moving at velocity \({\bf v}\) is given by the Lorentz force:
\[ {\bf F} = q ({\bf E} + {\bf v} \times {\bf B}) \tag{LorFo}\label{LorFo} \]
Cyclotron motion
Consider a magnetic field \({\bf B}\) pointing into the page. A charge \(q > 0\) moves counterclockwise in the plane of the page with speed \(v\) on a circle of radius \(R\). The magnetic force points inwards. Equating the centrifugal and centripetal accelerations, we obtain the cyclotron formula
\[ q v B = m \frac{v^2}{R} ~~\rightarrow~~ p = mv = q B R \] in which the cyclotron frequency is
\[ \omega = 2\pi \frac{v}{2\pi R} = \frac{q B}{m} \]
Due to the perpendicularity between the velocity and the magnetic force, magnetic forces do no work, as we can see by computing a differential work element
\[ dW_{mag} = {\bf F}_{mag} \cdot d{\bf l} = q ({\bf v} \times {\bf B}) \times {\bf v} dt = 0 \]
Except where otherwise noted, all content is licensed under a
Creative Commons Attribution 4.0 International License.
Created: 2024-02-27 Tue 10:31