Halliday, Resnick & Walker — Relativity: Problem 21

21 In Module 28-4, we showed that a particle of charge \( q \) and mass \( m \) will move in a circle of radius \( r = mv/|q|B \) when its velocity \( \vec{v} \) is perpendicular to a uniform magnetic field \( \vec{B} \). We also found that the period \( T \) of the motion is independent of speed \( v \). These two results are approximately correct if \( v \ll c \). For relativistic speeds, we must use the correct equation for the radius: \[ r = \frac{p}{|q|B} = \frac{\gamma mv}{|q|B}. \] (a) Using this equation and the definition of period (\( T = 2\pi r/v \)), find the correct expression for the period. (b) Is \( T \) independent of \( v \)? If a 20.0 MeV electron moves in a circular path in a uniform magnetic field of magnitude 2.20 T, what are (c) the radius according to Chapter 28, (d) the correct radius, (e) the period according to Chapter 28, and (f) the correct period?