Particles, Particles, Particles — Question 1

Question 1

Curvature and speed together identify a charged particle mass

A singly charged track has radius r=2.00 m in a perpendicular field B=1.00 T. Use $p[\mathrm{GeV}/c]=.300B[\mathrm T]r[\mathrm m]$. An independent measurement gives $\beta=v/c=.800$.

1. Find momentum and total energy.

2. Infer rest energy and explain why curvature alone is insufficient to identify mass.

Question 1 - Solution

Curvature and speed together identify a charged particle mass

(a) Method and calculation

$$\boxed{pc=.300(1.00)(2.00)=.600\,\mathrm{GeV}}.$$$$\beta=pc/E\Rightarrow\boxed{E=.600/.800=.750\,\mathrm{GeV}}.$$

(b) Interpretation and further calculation

$$\boxed{mc^2=\sqrt{E^2-p^2c^2}=\sqrt{.750^2-.600^2}=.450\,\mathrm{GeV}}.$$$$p=|q|Br\quad\text{fixes rigidity};\qquad m=\frac p{\gamma v}\quad\text{also requires }v.$$

Physical check

The field measures momentum per charge, not kinetic energy. The result assumes known charge magnitude and negligible energy loss across the fitted track.