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Problems
1. Atomic polarizability model. When an atom is placed in an electric field, the electrons are
pulled in one direction and the nucleus in the other direction. This creates a dipole moment in the atom. We may estimate the dipole moment by assuming that the electron cloud is a uniform sphere of charge (radius a ) and calculating the restoring force when the nucleus is moved off center. Use this model to show that the polarizability of a hydrogen atom is
3
α = 4 πε 0 a
#2. Plum pudding model. Prior to the discovery of the atomic nucleus by Rutherford, it was suggested by J. J. Thomson (who discovered the electron) that the electrons were imbedded in a background of positive charge, like the raisins in a plum pudding. Consider a hydrogen atom comprising an electron (charge qe ) imbedded in a uniform sphere of charge with radius R. By
symmetry, the force on the electron vanishes when the electron is at the center of the sphere, but when it moves away from the center of the sphere it experiences a restoring force proportional to the distance from the center. This causes the electron to oscillate. Show that the frequency of the oscillation is
2 3 (^40)
q e R
Find the wavelength λ of the corresponding radiation if R = 0.05 nm.
#3. Alpha scattering. Rutherford discovered the nucleus by watching alpha particles (He nuclei) scatter from gold atoms. What is the minimum energy (in eV) required for an alpha particle (charge 2e) to reach the surface of a gold nucleus (charge 79e, radius 7 fm)?
#4. Van de Graaff generator. Consider the Van de Graaff generator used in our demonstration experiments. Assume that the terminal is a smooth sphere of radius 10 cm. (a) If the maximum surface field is 3 MV/m, limited by breakdown of the air (corona discharge), what is the maximum voltage? (b) What is the charge on the terminal at maximum voltage? (c) If the electrons are brought to the terminal as a continuous current from infinity, how much work must be done to raise the terminal from zero to the final voltage? (d) If the current carried by the charging belt is 100 nA, how powerful must the motor be to charge the terminal to full voltage?
#5. Electrostatic mass of an electron. We have calculated the classical radius of an electron based on the energy in the electrostatic field surrounding an electron viewed as a spherical shell of charge with the so-called classical radius. What is the additional electrostatic energy if the charge on the electron is uniformly distributed throughout its volume inside the classical radius?
#6. Electrostatic suction. The plates of a parallel-plate capacitor are aligned vertically with the bottom edges submerged in oil with dielectric constant 4.5 and density 0.8 g/cc. The plates are separated by 1 cm and 10 kV is maintained on the plates. How far is the oil drawn up into the region between the plates (ignore surface tension and meniscus effects)? Where does the energy come from to raise the liquid against the pull of gravity?
#7. Magnetic resonance. A spining top with angular momentum L and magnetic dipole moment μ is placed in a magnetic field B. Assume that the magnetic moment is in the same direction as
the angular momentum, with a magnitude μ = g L for some constant g. If the top is a nucleus, we
call g the gyromagnetic ratio. Show that the frequency of precession of the top about the magnetic field is
ω= − g B
#8. Force between magnetic dipoles. Find the force between two magnetic dipoles with dipole moments μ 1 and μ 2. Assume that μ 2 is located at a distance z on the axis of μ 1 and that μ 2 is
parallel to μ 1. Hint: find the field at μ 2 due to μ 1 , then find the energy of μ 2 in that field.
Differentiate to find the force.
#9. Rogowski coil. A Rogowski coil is a toroidally wound coil used to measure the AC current through a wire in your house or even a high-energy electron beam without actually touching the wire or the electron beam. Consider a toroid with a cross sectional area A and n turns per unit length along a major circumference L. The major circumference need not form a circle provided that the winding is uniform (that is, n is a constant). The wire or electron beam whose current is to be measured passes through the Rogowski coil. Show that the EMF induced in the coil by the time-variation of the current in the wire is
0 EMF nA^ dI dt
= μ
10. Solenoid forces. The magnetic fields in superconducting solenoids (like those used in MRI
machines in the hospital) exert enormous forces on the coils of the magnet. Consider a superconducting solenoid of length 2 m and radius 0.3 m, with a magnetic field of 7 T (as in the most advanced machines).
(a) What is the total energy in the magnetic field? Assume the the field is uniform in the solenoid and vanishes outside. How many pounds of TNT is this equivalent to (TNT has an explosive energy of 4.6 MJ/kg)?
(b) If the length increases by dL, what happens to the strength of the magnetic field (Hint: for a supercoonducting coil, the EMF is zero)? What happens to the energy in the magnetic field? The
Show that the elapsed time on the rocket (the proper time) for the trip is
arccosh 2 1 3 years c gL g c
τ = ⎛⎜^ + ⎞⎟=
where L is the distance to the star in the stationary (earth and star) frame.
Hint: In the rocket frame of reference the astronaut experiences a velocity change dv '= gd τ in the time d τ. In the earth frame, the velocity after the time d τ is vx = v + dv. Substitute into the velocity addition law, expand the result to first order in the differential quantities (the zero-order
terms cancel) and integrate to find β τ( ). Integrate dx = β cdt = βγ cd τ to find x ( τ ).
#15. Electric field transformation. Consider a line charge with density λ in the rest frame.
(a) What is the electric field at radius r from the line charge?
(b) Now consider a frame moving at the velocity v parallel to the line charge. In this frame, the line charge becomes a line charge and a current. What are the line charge density and current in the moving frame?
(c) What are the electric and magnetic fields in the moving frame? Compare your answers with the transformation formulas for electric and magnetic fields.
#16. Political science/relativistic law question. A truce has been arranged between the Federation and the Romulans. According to the treaty, the first captain to fire in an engagement is executed! The evidence is as follows: Captain Picard of the Federation starship Enterprise detects emissions from a Romulan ship as it begins to uncloak, and realizes that the Romulan is about to fire at him. The strength of the emissions shows that the Romulan was about 1000 light- seconds away when they were radiated, so the Romulan must already have fired his photon torpedo. Picard therefore fires a photon torpedo (which travels at the speed of light) at the Romulan. 100 s later, the Enterprise is hit by the photon torpedo fired by the Romulan, and some time after that, in the starship coordinate system, the Romulan is hit by the photon torpedo fired by the Enterprise. Both ships travel at constant velocity throughout the engagement. With both ships disabled, the engagement ends, and a court of inquiry is convened. The two captains agree on the evidence presented above, but each captain argues, on the basis of his electronic logs, that the other fired first. Is it possible that both are telling the truth, at least as they see it in their own coordinate system? If so, what was the minimum relative speed of the Romulan and the Enterprise, and in which direction (toward each other or away) were they traveling? If not, which captain should be executed? Draw a world diagram in the coordinate system of the Enterprise to explain this. This is a tricky question, and you need every bit of the evidence.
#17. Ether drag. Before Einstein's theory of relativity, it was thought that light propagated through a medium called the "luminiferous ether," and that the earth's motion through the ether should be detectable. It wasn't, so people proposed the "ether drag." According to this theory, the
earth (or any medium) would drag the ether along so the motion through the ether would be impossible to observe. To investigate this, Fizeau measured the speed of light in moving water using an interferometer. Assume that light propagates through water at the velocity (relative to the water) c/n , where n is the index of refraction. If the water moves at the velocity v parallel to the light, what is the velocity of the light in the lab frame? If the frequency of the light is f , what is the phase shift of light propagating a distance L through the moving water relative to light traveling through the air, for which n =1?
By the way, Fizeau got good agreement with his experiments provided that the ether drag coefficient depended on the index of refraction. Unfortunately, this meant that the ether drag velocity depended on the wavelength of the light. There would have to be a separate ether for each color!
#18. Magnetic field transformation. For a wire carrying a current I , there is a magnetic field B around the wire. In a reference frame K' moving at velocity v parallel to the wire, the magnetic field transforms into a magnetic field and an electric field as described in the notes. What is the electric field in the moving frame? Gauss' law says that there must be a net charge along the wire to produce this electric field. What is this charge density? Where does it come from?
#19. Nuclear decay. An excited nucleus decays to its ground state and emits a gamma-ray photon, recoiling in the process. If the rest mass of the excited nucleus is m * and the rest mass of the ground-sate nucleus is m , what is the wavelength of the gamma-ray photon? You must account for the kinetic energy of the nucleus after it emits the photon.
#20. Cyclotron motion. Newton's equation of motion is still correct if we write it in the form F = d p / dt , where^ p^ is the momentum and^ F^ is the force, and the Lorentz force on a charged
particle is still correct in relativistic mechanics. For a particle going around in a circle, the rate of change of the momentum is the angular frequency times the momentum. Find the angular frequency ω for cyclotron motion of a charged particle in a magnetic field B and the radius R of
the circle as a function of the relativistic total energy E = γ mc^2.
#21. Particle colliders. In high-energy particle-physics colliders, such as the Large Hadron Collider at CERN, which is now coming into operation, two beams of particles are circulated in opposite directions and collide head-on. It would seem simpler to accelerate one beam to high energy and collide it with stationary particles, but this is never done in high-energy experiments, for good reason. Consider two particles, one coming from the left with momentum p and one coming from the right with momentum -p. Since the total momentum is zero in this frame, this is called the center-of-momentum frame. In relativistic physics, this is more useful than the center-
of mass frame. The total energy of the two particles in this frame is E = 2 γ mc^2. Next, look at
these same particles in a reference frame moving to the right at the speed of the particle coming from the left. In this frame, the particle coming from the left has only its rest energy, but the other particle has much more energy.
#26. Bohr's correspondence principle states that in the limit of large quantum numbers, quantum mechanical behavior must look like classical behavior. Consider an electron in a 1- dimensional box of length L with an infinitely high potential barrier. Find the energy difference between two adjacent levels and show that in the limit when the quantum number is very large the frequency of a photon emitted by a transition between these levels is the same as the frequency at which a classical electron would bounce back and forth (one round trip) in the box.
#27. Uncertainty principle. If we confine a particle to a box of length L , the uncertainty in the position is some fraction of this length. What is the minimum uncertainty in the momentum? How does this compare to the particle momentum in the ground state?
#28. Neutron star. In the free-electron theory of a metal, the electrons fill all available energy levels of the confining potential up to the Fermi energy. This is called a degenerate electron gas. If the box containing the electrons is allowed to expand, the Fermi energy decreases along with the electron density, and the work done (corresponding to the energy lost) is attributed to a “degeneracy pressure.” Neutron stars are a degenerate gas of neutrons, and the degeneracy pressure keeps the neutron star from collapsing into a black hole under the gravitational force. However, the pressure is so great that the Fermi energy corresponds to a relativistic velocity. Using the relativistic energy relation
E^2 = p^2 c^2 + m c^2
Show that the Fermi energy (including the rest energy) is
3 2/ 2 1 3 8
C F
n mc λ π
E
where λ C = h / mc is the Compton wavelength of a neutron. For a neutron star, the density of
neutrons per Compton wavelength cubed is on the order of unity.
