Download Physics 347: Problem Set 2 DUE Tue Sep 9 Each problem is worth

Physics 347
Module 11. Relativistic Particles
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11.1 High-Speed Particles: The relativistic speed limit
(a) According to classical physics, can an electron travel at the speed of light? greater than the speed of light?
(b) What voltage difference would accelerate an electron from rest to the speed of light according to classical
physics?
(c) According to relativistic physics, can an electron travel at the speed of light? greater than the speed of light?
(d) How fast would an electron be traveling if it were accelerated across the voltage difference found in (b)?
(e) At this speed, find the electron’s relativistic mass and kinetic energy. Express the mass as a factor times the
rest mass and the kinetic energy in units of MeV.
11.2 Pair Annihilation & Creation: It takes two to tango.
A proton and an antiproton are very close to one another and essentially at rest when they annihilate one
another. Two gamma rays are created.
(a) Use the Law of Conservation of Momentum to show that the frequencies of the two gamma rays are the
same.
(b) Use the result of (a) and the Law of Conservation of Energy to find the wavelengths of the gamma rays.
(c) Could one of these gamma rays collide with a nucleus and produce an electron-positron pair? Explain.
11.3 Accelerating Particles: Look at that jerk..
A particle of rest mass mo is accelerated linearly by a constant force F. The acceleration of the particle is given
by Eq. (11.24):
Fmo2 c 3
a(t ) 
(mo2 c 2  F 2 t 2 ) 3 / 2
(a) Evaluate the jerk (the time rate of change of the acceleration) of the particle and show that the size of the jerk
is maximized at a time of moc / 2F.
(b) Evaluate the velocity of the particle at this time and show that it is equal to c / 5 . (Note that this time does
not depend on the type of particle or the size of the accelerating force.)
11.4 High-Speed Particles: Allow me to iterate... high-speed particles.
You are going to use an iteration method to analyze the motion of the electron that we looked at in Module 11.
It is accelerated from rest across a voltage difference of 750 kV between two plates that are separated by 10
meters. You can use a programmable calculator, spreadsheet application like Excel, or write your own program.
Your goals are the following:
(a) Find the final speed of the e- after it has traveled the 10 meters.
(b) Find how long it takes the e- to reach this speed as measured by us in the lab.
(c) Plot position s versus lab time t and the speed ratio of u/c versus lab time from t=0 until the time that
the electron travels 10 meters.
(d) Find how long it takes the e- to reach this speed as measured by the e-.
I suggest the following steps:
1. Find the size of the force in terms of the accelerating voltage and plate separation.
2. Express the acceleration in terms of this force and the speed u of the e-.
3. Use Euler’s Method to calculate the lab frame speed and distance. I suggest using a time interval, t,
of 0.1 ns. This seems to work nicely. The number of iterations required with 0.1 ns will be around
500 or so. Euler’s Method utilizes the kinematic equations for constant acceleration:
u (t  t )  u (t )  a(t )t
s (t  t )  s (t )  u (t )t
Note 1: You may ask what happened to the (1/2)a(t)2 in the second equation. Euler’s Method ignores
this term because t should be very small so that the (t)2 term is negligible. If you want to include it in
your iteration, fine. It doesn’t give significantly different results.
Note 2: Read Suggestion 5 before you run this iteration!
4. Run the iteration and look for the time when the position of the e- is 10 meters. Look at the
corresponding speed. Make the plots.
5. What about (d)? Hmmm…. Look, the time interval of 0.1 ns is a nonproper time interval. The emeasures a corresponding t' that is a proper time interval. Thus, t' = t /  = 0.1 ns / . However,
since the speed u is changing over the time interval, then the gamma factor is also changing. What we
can do, though, is take the average speed over the 0.1 ns interval and use this to calculate t'. Notice
that t' will not be a constant time interval! So here is another iteration. You can incorporate this into
the previous one. We know t' starts at zero. To calculate the first t', find the average speed between
t=0 and t=0.1 ns, calculate the gamma factor with this average speed, and use t' = 0.1 ns / . Add this
to the time at the beginning of the interval (t'=0) and you have the time t' at the end of the first interval.
Repeat this procedure until the electron reaches 10 meters and note the time t'.
You can compare your answers to the results and plots in Module 11 where we performed integrations and
differentiations using the rules of calculus.