# Problem Set 3

## Due Week February 5, 2001

1. Bending of Light by Gravity:: Calculate semi-classically the deflection of light from a distant star which grazes the surface of the sun on its way to a telescope on the Earth. Use the impact parameter approximation. The effective photon gravitational mass is h f / c^2. The radius of the Sun is 7 x 10^8 m. [Einstein's General Theory of Relativity gives an answer exactly twice as large.]

2. In a betatron, electrons in the plane z = 0 are maintained in an orbit of constant radius r = a by an axially symmetric magnetic field which is in the z direction in the z = 0 plane. The field at r = a is B(a,t) and the enclosed flux phi = 2 pi a^2 B(a,t). [This equation expresses the "betatron condition".] If the entire field increases in time while maintaining its shape and this condition, the electrons are accelerated by the electric field of induction while maintaining a fixed radius. Neglecting radiation due to the centripetal acceleration of the charge, prove all these statements assuming that the electrons may have any speed less than the speed of light.

3. Particle Kinematics Consider a beam of protons (mass = 0.938272 GeV/c^2) each with momentum 8.0 Gev/c. (GeV = 10^9 eV)

(a) Find the proton kinetic and total energy in GeV. (Formula & evaluate)

(b) Determine ratio of the velocity of the protons to the speed of light by formula and evaluate to three significant figures.

(c) Find the magnetic field necessary to bend them with a radius of 10 m.

(d) Find the strength of electric field needed to do the same. (Find the tangent circle or consider the protons in orbit around a large spherical charge.)

4. Energy and Momentum Transformation (a) Find by formula and numerical evaluation the energy and momentum of that same beam of protons (p = 8.0 GeV/c) from the frame of reference of a proton in another beam with the same momentum heading directly towards the first beam. (i.e. colliding beams)

(b) If two protons collide head-on, what is the maximum rest mass particle that they can create. (Again show by formula and evaluate.)

(c) Why would one go to the trouble of making colliding beams rather than just accelerating the first beam up to sufficient energy to make a particle or group of particles of the same maximum mass?

5. Equivalence of Mass and Energy The binding energy of a hydrogen atom (a proton and an electron) is Eb = 13.6 electron-Volts (eV).

(a) By what fraction does the total mass of the electron plus proton change when combined into a hydrogen atom?

(b) How accurate is conservation of mass in chemical reactions?

(c) If the typical scale of nuclear reactions is of order 1 MeV (million electron volts), how accurate is conservation of mass in nuclear reactions?

(d) If the typical scale of high-energy physics reactions is of order 100 MeV (million electron volts), how accurate is conservation of mass in high energy reactions?

6. A little more Relativistic Kinematics The pi-zero meson (rest mass times c^2 of 134.9764 MeV) decays into two photons with a very short lifetime (8.4 x 10^-17 seconds).

(a) What is the energy of each photon in the pi-zero rest frame? And what are their relative directions?

If in the laboratory the pi-zero is moving so it has a total energy twice its rest mass, and decays into two photons, for ease in calculation, that are emitted in the pi-zero rest frame at right angles to the direction of motion.

(b) What is their energy in the laboratory frame?

(c) What is the angle each makes to the pi-zero direction? E.g what would be the angle between them.

(d) How would this change if they are emitted at an angle theta to the line of flight in the rest frame? What would be the energy and angle to the original motion or opening angle between the photons as a function theta? How does your answer for the angle agree with the relativistic aberration of light formula?