Exam (elaborations) TEST BANK FOR Applied Quantum Mechanics By A. F. J. Levi (Solution manual)

PROBLEM 1 A metal ball is buried in an ice cube that is in a bucket of water. (a) If the ice cube with the metal ball is initially under water, what happens to the water level when the ice melts? (b) If the ice cube with the metal ball is initially floating in the water, what happens to the water level when the ice melts? (c) Explain how the Earth’s average sea level could have increased by at least 100 m compared to about 20,000 years ago. (d) Estimate the thickness and weight per unit area of the ice that melted in (c). You may wish to use the fact that the density of ice is 920 kg m-3, today the land surface area of the Earth is about 148,300,000 km2 and water area is about 361,800,000 km2. PROBLEM 2 Sketch and find the volume of the largest and smallest convex plug manufactured from a sphere of radius r = 1 cm to fit exactly into a circular hole of radius r = 1 cm, an isosceles triangle with base 2 cm and a height h = 1 cm, and a half circle radius r = 1 cm and base 2 cm. PROBLEM 3 An initially stationary particle mass m1 is on a frictionless table surface and another particle mass m2 is positioned vertically below the edge of the table. The distance from the particle mass m1 to the edge of the table is l. The two particles are connected by a taught, light, inextensible string of length L l. (a) How much time elapses before the particle mass m1 is launched off the edge of the table? (b) What is the subsequent motion of the particles? (c) How is your answer for (a) and (b) modified if the string has spring constant k? PROBLEM 4 The velocity of water waves in shallow water may be approximated as where g is the acceleration due to gravity and h is the depth of the water. Sketch the lowest frequency standing water wave in a 5 m long garden pond that is 0.9 m deep and estimate its frequency. PROBLEM 5 (a) What is the dispersion relation of a wave whose group velocity is half the phase velocity? (b) What is the dispersion relation of a wave whose group velocity is twice the phase velocity? (c) What is the dispersion relation when the group velocity is four times the phase velocity? PROBLEM 6 A stationary ground-based radar uses a continuous electromagnetic wave at 10 GHz frequency to measure the speed of a passing airplane moving at a constant altitude and in a straight line at 1000 km hr-1. What is the maximum beat frequency between the out going and reflected radar beams? Sketch how the beat frequency varies as a function of time. What happens to the beat frequency if the airplane moves in an arc? v = gh Applied quantum mechanics 3 PROBLEM 7 How would Maxwell’s equations be modified if magnetic charge g (magnetic monopoles) were discovered? Derive an expression for conservation of magnetic current and write down a generalized Lorentz force law that includes magnetic charge. Write Maxwell’s equations with magnetic charge in terms of a field . PROBLEM 8 The capacitance of a small metal sphere in air is . A thin dielectric film with relative permittivity uniformly coats the sphere and the capacitance increases to . What is the thickness of the dielectric film and what is the single electron charging energy of the dielectric coated metal sphere? PROBLEM 9 (a) A diatomic molecule has atoms with mass m1 and m2. An isotopic form of the molecule has atoms with mass m'1 and m'2. Find the ratio of vibration oscillation frequency w / w' of the two molecules. (b) What is the ratio of vibrational frequencies for carbon monoxide isotope 12 ( ) and carbon monoxide isotope 13 ( )? PROBLEM 10 (a) Find the frequency of oscillation of the particle of mass m illustrated in the Fig. The particle is only free to move along a line and is attached to a light spring whose other end is fixed at point A located a distance l perpendicular to the line. A force F0 is required to extend the spring to length l. (b) Part (a) describes a new type of child’s swing. If the child weighs 20 kg, the length l = 2.5 m, and the force F0 = 450 N, what is the period of oscillation?