A New Look at Momentum — Question 8

Question 8

A reflecting particle beam exerts pressure through momentum flux

A dilute beam of identical particles has laboratory number density n and speed $v=0.60c$ normal to a stationary wall. Each particle reverses its velocity elastically. The density refers to the incoming beam alone. Use $\mathbf p=\gamma m\mathbf v$, $\gamma=(1-v^2/c^2)^{-1/2}$, and $\mathbf F=d\mathbf p/dt$. Here m is invariant mass.

1. Find the wall pressure from the arrival rate and impulse per particle.

2. Compare with the pressure predicted by classical momentum at the same laboratory density and speed.

Question 8 - Solution

A reflecting particle beam exerts pressure through momentum flux

(a) Method and calculation

$$\frac{\dot N}{A}=nv,\qquad |\Delta p_{\mathrm{particle}}|=2\gamma mv.$$$$\boxed{P=2n\gamma mv^2}.$$

(b) Interpretation and further calculation

$$\beta=.60:\quad\boxed{P=.900\,nmc^2}.$$$$P_{\mathrm{cl}}=2nmv^2=.720\,nmc^2,\quad\boxed{P/P_{\mathrm{cl}}=1.25}.$$

Physical check

The particle impulse points away from the wall; the equal and opposite impulse on the wall creates the positive pressure.