PHY 3300, Fall 2014, Prof. Cacket Cackettt Homew Homework ork Set 3: Partic Particle le Nature of Matter Matter Homework due: Wednesday, 24 September, in class
1. For each of the following physicists, briefly describe their principle discoveries and the year of each discovery. a) J. J. Thomson, b) Ernest Rutherford, c) James Franck and Gustav Hertz, d) Neils Bohr. [4 points] points] 2. The Thomson experiment. An electron beam traverses traverses a region of uniform electric electric field inside capacitor plates with the length l length l along along the beam axis and separation separation d d between between the plates. The electron velocity vx is constant along along the direction direction of the beam, before and after entering entering the capacitor. The electron electron velocity along the direction of the electric field is zero before entering the capacitor and vy after exiting. There is no magnetic field. a) Find the deflection angle θ angle θ of the electron beam after exiting the capacitor in terms of v of v x and v and v y . b) Find vy in terms of vx , d, the voltage V across V across the capacitor plates, and fundamental constants. (Don’t just copy the equation out of the text - please show how to derive it). Now suppose that a uniform magnetic field of strength B is applied in a direction perpendicular to both the electric field and the beam direction. direction. The direction direction and magnitude magnitude of B of B are such that there is no net force on an electron. c) Sketch the directions of E of E , B and the beam direction. Take into account the negative charge of the electron. d) Find the magnitude of B in B in terms of the given properties of the setup, vx , and fundamental constants. [6 points] points] 3. a) What is the “plum pudding” pudding” model of the atom and how did Rutherford’ Rutherford’ss experiment experiment disprove disprove it? b) State the Rutherford scattering formula and define all the variable and constants. c) Sketch a graph of the scattering rate versus angle, with other variables fixed. [5 points] points] 4. A parallel beam of α α particles with fixed kinetic energy is normally incident on a piece of gold foil. a) If 90 α 90 α particles per minute are detected at 25 , how many will be counted at 45, 70 and 110 ? ◦
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b) If the kinetic energy of the incident α particles is doubled, what rate of α of α particles will be observed at 25 ? ◦
c) If the α particles were incident on a copper foil of the same thickness, what rate of scattered α particles would be observed at 25 ? ◦
[Note that ρ that ρ(Cu) (Cu) = 8.9 g/cm3 and ρ and ρ(Au) (Au) = 19.3 g/cm3 , m A (Cu) = 63.54 amu, m amu, m A (Au) = 196.97 amu, Z (Cu) (Cu) = 29, Z 29, Z (Au) (Au) = 79] [8 points] points] 5. It is observed observed that that α particles with kinetic energies of 15.0 MeV and higher, incident on a foil target of Ga (Z = = 31), do not obey the angular dependence of Rutherford’s scattering law. Estimate the nuclear radius of this element from this observation. [2 points] points] 6. a) What is the difference difference between between an emission emission spectrum and an absorption absorption spectrum? spectrum? b) What is the difference between a continuous and a discrete spectrum? c) Give examples examples of continuo continuous us and discrete discrete spectra d) What is a gas-discharge lamp and what type of spectrum does it produce? 1
e) What type of spectrum is produced by a star? f) What element was discovered in a star before is was discovered on Earth? [6 points] 7. Bohr derived a formula for the energy levels of the hydrogen atom, E n =
−
k 2 e4 me 1 2(h/2π)2 n2
where n = 1, 2, 3, . . . a) By direct substitution, find the numerical value of the constant factor in eV units. Find the energy in eV of a photon emitted in the following transitions: b) n = 3 → n = 2 (first transition in the Balmer series). c) n = 2 → n = 1 (first transition in the Lyman series). [3 points] 8. A hydrogen atom initially at rest in the n = 3 state decays to the ground state with the emission of a photon. a) Calculate the energy in eV of the emitted photon. b) Calculate the momentum of the emitted photon. c) By conservation of momentum, the atom, as a whole, must be moving after the photon is emitted. Find the kinetic energy in eV of the atom. Note that the atom moves non-relativistically. [Helpful numbers: the mass of a hydrogen atom is 1.008 amu, and 1.0 amu = 931.5 MeV/ c2 ] [3 points] Honors Option Question: (only for those taking the Honors option)
This question is about the Millikan Oil Drop experiment. Although only briefly mentioned in class, use the extensive details in the textbook to answer the following question. 1. Actual data from one of Millikan’s early experiments are as follows: a = 0.000276 cm ρ = 0.9561 g/cm3 Average time of fall = 11.894 s Rise or fall distance = 10.21 nm Plate separation = 16.00 mm Average potential difference between plates = 5085 V Sequential rise times in seconds: 80.708, 22.336, 22.390, 22.368, 140.566, 79.600, 34.748, 34.762, 29.286, 29.236 Find the average value of e by requiring that the difference in charge for drops with different rise times be equal to an integral number of elementary charges.
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