Chapter 25 Problem 85

Problem

The Tevatron accelerator at Fermilab (Illinois) is designed to carry an $11$ mA beam of protons traveling at very nearly the speed of light ($3\times 10^{8}$ m/s) around a ring 6300 m in circumference. How many protons are in the beam?

Solution

Take the cross-sectional area of the ring to be $A$. Then by definition, $I=neAv_{d}$; $n$ being number of protons per volume $n={\frac {N}{Al}}$. We are trying to find $N$, the total number of protons.

$I=neAv_{d}={\frac {N}{Al}}eAv_{d}={\frac {N}{l}}ev_{d}$

$\rightarrow N={\frac {Il}{ev_{d}}}={\frac {(11\times 10^{-3})(6300{\text{ m}})}{(1.60\times 10^{-19}{\text{ C}})(3.00\times 10^{8}{\text{ m/s}})}}=1.4\times 10^{12}{\text{ protons}}$