USAPhO 2013 Semi-Final Problem A3

Past Contests USAPhO 2013 Semi-Final
1

A beam of muons is maintained in a circular orbit by a uniform magnetic field. Neglect energy loss due to electromagnetic radiation.

The mass of the muon is 1.88 \times 10^{-28} \mathrm{~kg}, its charge is -1.602 \times 10^{-19} \mathrm{C}, and its half-life is 1.523 \mu \mathrm{s}.

a. The speed of the muons is much less than the speed of light. It is found that half of the muons decay during each full orbit. What is the magnitude of the magnetic field?

b. The experiment is repeated with the same magnetic field, but the speed of the muons is increased; it is no longer much less than the speed of light. Does the fraction of muons which decay during each full orbit increase, decrease, or stay the same?

The following facts about special relativity may be useful:

  • The Lorentz factor for a particle moving at speed v is
\gamma=\frac{1}{\sqrt{1-v^{2} / c^{2}}}
  • The Lorentz factor gives the magnitude of time dilation; that is, a clock moving at speed v in a given reference frame runs slow by a factor \gamma in that frame.
  • The momentum of a particle is given by
\vec{p}=\gamma m \vec{v}

where m does not depend on v.

  • The Lorentz force law in the form
\frac{d \vec{p}}{d t}=q(\vec{E}+\vec{v} \times \vec{B})

continues to hold.