Pre-Quantum Electrodynamics
Point Chargeems.ms.lf.pc
Force on a point charge \(q\) moving at velocity \({\bf v}\) in magnetic field \({\bf B}\):
\[ {\bf F}_{mag} = q {\bf v} \times {\bf B} \label{eq:LorentzForce} \]
Units of \({\bf B}\): \(N/(A~m)\) is called a {\bf tesla} (symbol: \(T\)). Total electromagnetic force:
\[ {\bf F}_{mag} = q ({\bf E} + {\bf v} \times {\bf B}) \label{eq:EMForce} \]
\paragraph{Example 5.1:} cyclotron motion. Field \({\bf B}\) pointing into page. Charge \(q > 0\) moves counterclockwise with speed \(v\) on a circle of radius \(R\). Magnetic force points inwards. Equating, obtain the {\bf cyclotron formula} \[ q v B = m \frac{v^2}{R} ~~\rightarrow~~ p = mv = q B R. \label{Gr(5.3)} \] The {\bf cyclotron frequency} is \[ \omega = 2\pi \frac{v}{2\pi R} = \frac{q B}{m} \label{Gr(5.4)} \]
\paragraph{Example 5.2:} cycloid motion. Recommendation: {\it look at it!!}
Important point: {\bf magnetic forces do no work}. Work: \[ dW_{mag} = {\bf F}_{mag} \cdot d{\bf l} = q ({\bf v} \times {\bf B}) \times {\bf v} dt = 0 \label{Gr(5.11)} \]
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Created: 2022-02-14 Mon 20:35