A Circulating Charged Particle — Question 7

Question 7

Relativistic momentum enlarges an orbit beyond the classical radius estimate

For perpendicular motion use the supplied relativistic relations $p=\gamma mv$, $r=p/(|q|B)$ and $\gamma=(1-v^2/c^2)^{-1/2}$. A particle travels at v=.800c in a uniform field; radiation losses are neglected.

1. Compare its radius and period with the classical formulas evaluated at the same speed.

2. If an observer infers |q|/m from the classical expression v/(Br), find the inferred-to-true ratio.

Question 7 - Solution

Relativistic momentum enlarges an orbit beyond the classical radius estimate

(a) Method and calculation

$$\gamma=(1-.800^2)^{-1/2}=5/3,\quad r=\gamma r_{\mathrm{classical}}.$$$$T=2\pi r/v=\gamma\frac{2\pi m}{|q|B}=\tfrac53T_{\mathrm{classical}}.$$

(b) Interpretation and further calculation

$$\left(\frac{|q|}{m}\right)_{\mathrm{inferred}}=\frac v{Br}=\frac1\gamma\left(\frac{|q|}{m}\right)_{\mathrm{true}}.$$$$\eta_{\mathrm{inferred}}/\eta_{\mathrm{true}}=3/5=.600.$$

Physical check

The magnetic force still does no work; the correction comes from the relation between momentum and velocity.