FINAL EXAM — PHYS 601 — Fall 2014
Thursday, December 18, 2014, 8:00–10:00 am
Professor Victor Yakovenko
Office: 2115 Physics
Do not forget to write your name!
Total score is 46 points.
1. 8 points. A homogeneous cube of mass m and sides ` is initially at rest in unstable
equilibrium with one edge in contact with a horizontal plane. The cube is given a
small angular displacement and allowed to fall due to a vertical gravitational force of
acceleration g. Consider two cases:
(i) The edge in contact with the plane cannot slide.
(ii) The edge can slide along the plane without friction.
For each of these cases, do the following:
(a) [4 points] Write down the Lagrangian L(φ, φ̇) of the system, where φ is the tilt
angle, and the conserved energy function h(φ, φ̇) of the system.
(b) [4 points] Using conservation of energy, calculate the angular velocity ω = φ̇ of
the cube when one face contacts the horizontal plane.
Info: The moment of inertia of a solid cube about its center of mass is I = m`2 /6.
2. Based on Physics Qualifier Problem I.1 given in January 2010.
15 points
A particle of mass m moves in an isotropic three-dimensional harmonic oscillator potential with natural frequency ω0 . In addition, the particle has electric charge q and
moves in crossed applied uniform electric and magnetic fields E = E0 x̂ and B = B0 ẑ,
where E0 and B0 are constants.
(a) [3 points] Write the non-relativistic Lagrangian for this system.
(b) [3 points] Find the stationary position of the particle.
(c) [3 points] Obtain the equations of motion for oscillations about this equilibrium.
(d) [3 points] From the equations of motion, find the normal modes of oscillations,
including their frequencies and normal coordinates.
(e) [3 points] Describe the normal modes of oscillations in plain language and qualitatively explain their frequency differences from ω0 .
3. Based on Physics Qualifier Problem I.4 given in January 2014.
23 points
Consider scattering of a photon of frequency ν (with energy E = hν and wavelength
λ = c/ν) off an electron of mass m assumed to be free and at rest.
(a) [7 points] Find the wavelength λ0 of the photon after the scattering in terms of
λ, m, and the scattering angle θ of the photon in the reference frame where the
electron is initially at rest (the Compton formula).
2
Final Exam, PHYS 601, Fall 2014, Prof. Yakovenko
If the electron is initially not at rest and moves with some velocity v, it can transfer
some energy to the photon under certain conditions, which you will investigate. For
simplicity, assume that the initial photon and the electron move along the same line.
E0
E
v u
before
K
PuP
PP
θ
PP
q
P
after
(b) [7 points] Calculate the final energy E 0 of the photon in terms of the initial
photon energy E, the initial energy γmc2 and momentum βγmc of q
the electron,
and the scattering angle of the photon θ, where β = v/c and γ = 1/ 1 − (v/c)2 .
(c) [3 points] Show that the photon cannot gain energy (E 0 < E) if initially the
electron and the photon move in the same direction.
Show that the photon energy is unchanged for forward scattering (θ = 0).
(d) [3 points] Therefore, the most promising configuration for energy gain is headon collision (β < 0) with backward scattering (θ = −π). Work out ∆E/E =
(E 0 − E)/E in terms of E, β and γ, and show that ∆E becomes positive when
the electron momentum exceeds the photon momentum E/c.
(e) [3 points] When the electron energy is large (γ 1) and the initial photon
energy is small (E mc2 /γ), show that ∆E/E ≈ 4γ 2 – a very large increase.