Department of Physics, Portland State University

Ph 311 (Introduction to Modern Physics): Final Exam

7 December 1999, 8:00 - 9:50 am Dr. R. Egerton

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electron charge = 1.60 ´ 10-19 C electron rest mass = 9.11 ´ 10-31 kg

electron rest energy = 0.511 MeV speed of light in vacuum = 3.00 ´ 108 m/s

Rydberg energy R = 13.6 eV Planck's constant h = 6.626 x 10-34 J s

Formula sheet permitted. Show clearly how numerical answers were calculated.
  1. (a) Muons circulating in a storage ring were found to have a half-life of 3.3 m s, compared to a half-life of 2.2 m for a stationary muon. Calculate the speed of the circulating muons.

    2. (b) If the radius of the muon orbit is 93.5m, how many orbits do the muons make before half of them have decayed ?

  2. (a) What physical principle did Einstein introduce (in 1905) to explain Lennard's experiments on the photoelectric effect?

    3. (b) Light of wavelength 500nm is incident on a metal whose work function is 1.98 eV. Calculate the maximum kinetic energy (in eV) of the photoelectrons produced.

  3. (a) Sketch a labelled diagram showing the wavelength dependence of the photon intensity produced by an x-ray tube.

    4. (b) X-rays of wavelength 0.063 nm are incident on a crystal and are scattered through an angle of 46 deg. Calculate the smallest spacing of the atomic planes which could give rise to this diffraction.

    (c) A positron with a kinetic energy of 0.578 MeV collides with a stationary electron. Calculate the frequency of the photons produced.

  4. (a) State concisely what basic physical principle distinguishes the Bohr model of an atom from the Rutherford model.

(b) Calculate the energy of the photons produced when a singly-ionized helium ion (He+, Z = 2) returns to the ground state from its first excited state.

(c) Why is the above calculation inappropriate for a neutral helium atom ?

  1. (d) What is the least amount of energy which must be supplied to a He+ ion in order to produce a doubly-ionized (He2+) ion ?
  • 5. For each of the following physical principles, name an instrument which makes use of the principle and (in one sentence) describe its purpose.
  • (a) Stefan's law
  • (b) The photoelectric effect
  • (c) Elastic scattering of x-ray photons
  • (d) Centripetal acceleration of an electron
  • (e) Pair annihilation

    Ph 311 Final Exam (7 December 1999) ANSWERS (R. Egerton)

    1.(a) dilated time = (gamma) (proper time), so (gamma) = 3.3 / 2.2 = 1.5

    = (1-v^2/c^2)^-1/2, giving (1-v^2/c^2) = 4/9

    v/c = [(5/9)^(1/2)]c = (0.745)(3.00E8 m/s) = 2.23E8 m/s

    (b) In the laboratory frame: distance gone = (speed)(time) = (2.23E8)(3.3E-6) = 737.4 m

    = (number of orbits) (2 pi R) , so number of orbits = 737.4/[(2) (3.142) (93.5)] = 1.255

    2. (a) UV or visible light can be regarded as a stream of photons, each of which has a quantized energy = hf = hc/lambda

    (b) hf = hc/lambda = (6.626E-34)(3.00E8)/(500E-9) = 3.97E-19 J = 2.48 eV

    kinetic energy = hf - phi = 2.48 - 1.98 = 0.50 eV

    3.(a) The diagram should show a continuous (bremsstrahlung) background, which increases with decreasing wavelength but then falls to zero at short-wavelength cutoff, on which a superimposed sharp characteristic peaks.

    (b) n (lambda) = 2 (d) sin(thetaB), where thetaB = (scattering angle)/2 = 23 deg

    The smallest interplanar spacing d corresponds to n = 1,

    giving d = (lambda)/[2 sin(23deg)] = (0.063E-9)/[(2)(0.391)] = 0.081 nm

    (c) Mass-energy conservation: hf = K + (m0 c^2) + 0 + (m0 c^2)

    = (0.578 + 0.511 + 0.511) MeV = 1.60 MeV = 2.56E-13 Joule = 2hf

    frequency = (1/2)(2.56E-13 / 6.62E-34) = 1.93E20 Hz

    4.(a) The Bohr model introduces quantization of angular momentum: mvr = n (h / 2 pi),

    as a condition for allowed orbits (stationary states of the atom).

    (b) hf = (Z^2) R [ 1 / 1^2 - 1 / 2^2 ] = 4 (13.6 eV) (0.75) = 40.8 eV

    (c) The calculation would be inaccurate because it neglects the electrostatic force between the two electrons which are present in a neutral helium atom.

    (d) The energy of the system must be increased from -(Z^2)R to zero,

    and is therefore 0 - (-4)(13.6eV) = 54.4 eV

    5. (a) total-radiation pyrometer, measures temperature of a remote object.

    (b) photomultiplier tube, measures the flux of incident photons or charged particles

    (c) x-ray diffraction camera, measures atomic spacings in crystalline solids

    (d) synchrotron radiation source, produces x-rays for diffraction experiments (for example)

    (e) Positron Emission Tomography. PET produces an image of organs in the body, showing the spatial distribution of an injected radioisotope which can be related to the functioning of that organ (e.g. different mental states in the brain).