Classical Electrodynamics — PHY5347

HOMEWORK 13

(January 14, 2013)

Due on Thursday, January 24, 2013

PROBLEM 37

Cosmic Rays—mostly very energetic protons—produce a copious amount of pions asthey collide with molecules in the Earth’s atmosphere. With a lifetime of only 26nanoseconds, these pions decay rapidly into muons. The muons—with a lifetime of2.2 microseconds—travel through Earth’s atmosphere and eventually get detected byparticle detectors placed at the surface of the Earth (or even underground). Assumethat a large number of muons traveling at 99.99% of the speed of light are produced100 kilometers above the Earth’s surface.

(a) Assuming simple Netwonian mechanics, predict the fraction of those muons thatwill reach the surface of the Earth?

(b) What is the fraction of muons that will actually reach the surface of the Earth?Note: Answer this question from the perspective of an observer at rest on thesurface of the Earth.

(c) What is the fraction of muons that will actually reach the surface of the Earth?Note: Answer this question from the perspective of an observer moving to-gether with the muons.

PROBLEM 38

Consider the collision of an energetic positron with an electron at rest in the labora-tory frame. The collision is so violent that the electron-positron pair gets convertedinto a pair of muons, i.e., e+ + e− → µ+ + µ−.

(a) Compute the minimum kinetic energy of the positron in the laboratory framefor the above reaction to proceed.

(b) For that minimum kinetic energy, what is the momentum of the µ+ as measuredin the laboratory frame.

(c) For that minimum kinetic energy, what is the momentum of the µ− as measuredin the laboratory frame.

Useful information: The rest mass of the electron and muon are given by:

mec2 =0.511 MeV and mµc

2 =105.658 MeV .

PROBLEM 39

As seen by an observer in the laboratory frame, two events have the following space-time coordinates:

xµ1 = (ct1 =0, x=0) and xµ2 = (ct2 =10, x=0) .

(You can ignore the y and z coordinates throughout this problem).

(a) Could you find an inertial reference frame in which the time order of the eventswould be reversed? That is t′1>t

′2.

Now consider the following two events as “seen” by an observer in the laboratoryframe:

xµ1 = (ct1 =0, x=0) and xµ2 = (ct2 =0, x=10) .

(b) Could you find an inertial reference frame in which t′1>t′2.

(c) Could you find an inertial reference frame in which t′1<t′2.

(d) Use a short sentence to summarize what have you learned from this problem.