1. The surface of a piece of sodium metal is illuminated in a vacuu...

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1. The surface of a piece of sodium metal is illuminated in a vacuum with monochromatic light of various wavelengths and the stopping potentials required to halt the most energetic photoelectrons are observed as follows: Wavelength (nm) Stopping Potential (V) 253.6 2.60 283.0 2.11 303.9 1.81 330.2 1.47 366.3 1.10 435.8 0.57 Use these data to make a plot in such a way as to produce a straight line as predicted by the photoelectric equation. Obtain a numerical value for Planck's constant from your plot. (This is a more direct way of measuring h than the atomic spectrum of hydrogen we use in the lab.) 10³ 2. The figure to the right shows the results of an experiment in which alpha particles were scattered through an angle of 60° by a thin lead foil. Also plotted is the theoretical Rutherford scattering formula we derived in 10² class. Use this information to provide as Rutherford formula precise an estimate as you can for the upper bound on the radius of the lead nucleus. (NOTE: This is NOT a head-on collision!) 10 1 10 15 20 25 30 35 40 45 a particle energy (MeV) 3. In this problem, we will investigate another piece of evidence in support of the Planck quantization hypothesis as an explanation for the photoelectric effect: the time lapse between light striking a metal surface and the emission of the first photoelectrons. A beam of ultraviolet light of power 1.6 x 10-12 Watts is suddenly turned on and shines on a metal surface, ejecting photoelectrons. The beam has a cross-sectional area of 1.0 cm², and a wavelength corresponding to a photon energy of 10. eV. The metal work function is 5.0 eV. (a) Estimate of the time (in years) needed classically for the first photoelectron to be emitted based on the time for the work function energy to accumulate over the area of one atom (r= 1.0 Â). Assume the energy in the beam of light is uniformly distributed over its cross-section. (b) In the quantum picture of the process, it is possible for photoelectron emission to begin immediately -- as soon as the first photon strikes the emitting surface. To obtain a time to compare with your estimate in (a), calculate the average time (in seconds) between arrival of successive 10. eV photons. (This would also be the average time delay between switching on the beam and the emission of the first photoelectron.) Mass m 5. in circular orbit k www. Spring A particle of mass m attached to a spring of spring constant k moves in a circular path of radius r around a fixed point on a frictionless surface. It is assumed that the spring is massless and has zero unstretched length such that the potential energy of the spring is given by U = kr²/2. Apply the Bohr hypothesis of quantized angular momentum to show that the allowed energies for the mass are given by En = nhf, (n = 1, 2, 3, ) where f = w/2tc and w is the classical frequency of a mass oscillating on a spring (=(k/m) 12). 6. According to classical electromagnetic theory, an electron that experiences acceleration a will radiate energy at the instantaneous rate = dt 6TE0C The theory also predicts that an oscillating electron will emit radiation whose frequency is the same as the oscillation frequency of the electron. (a) What is the acceleration of the electron in the ground state of the Bohr hydrogen atom? (b) What is the rate of energy loss of this electron? (c) As an order-of-magnitude estimate, assume that this energy-loss rate remains constant, and calculate the order of magnitude of the time for this electron to crash into the proton. 7. The Bohr "correspondence principle" is often used as a check on quantum mechanical calculations. The principle states that when n (the integer in the Bohr energy level equation for single-electron atoms, or, more generally, as we will see later, the principal quantum number) is small, quantum physics yields results that are very different from those of classical physics; however, when n is very large, the differences are insignificant, and the two "correspond". Investigate this principle for the Bohr model of the hydrogen atom as follows: According to classical electromagnetic theory for the hydrogen atom, the frequency of the emitted em radiation is equal to the frequency of the orbiting electron. (a) Calculate the orbital frequencies forb(1) and forb(2) of a classical electron in the n = 1 and n = 2 orbits of a Bohr atom. (b) Now calculate the frequency, fph(2 1) of the actual photon emitted in the 2 1 transition. Show this frequency is not equal to either forb(1) or forb(2), their average, or their difference. (c) More generally, show that the orbital frequency of a Bohr electron in hydrogen is (d) Show that for very large n, the radiated frequency calculated from the Bohr energies for a transition from ni = n + 1 to nf = n reduces to the orbital frequency in (c). 8. Refine the analysis of the Bohr model as follows: The the n=1 electron in a hydrogen atom at rest in free must recoil state, in emitting the -X direction. a photon in the +x direction. space Since undergoes the photon a transition has momentum, from the n=2 the state atom to (a) Calculate the energy of the photon (in eV). (b) Using conservation of energy and momentum, calculate the recoil velocity of the atom. (c) Calculate the kinetic energy of the recoiling atom (in eV). Compare it to the photon energy.

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