mp-chapt-4-sol - Chapter 4. Problem Solutions 1. The great...

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Inha University Department of Physics Chapter 4. Problem Solutions 1. The great majority of alpha particles pass through gases and thin metal foils with no deflections. To what conclusion about atomic structure does this observation lead? 3. Determine the distance of closest approach of 1.00-MeV protons incident on gold nuclei. Sol The fact that most particles pass through undetected means that there is not much to deflect these particles; most of the volume of an atom is empty space, and gases and metals are overall electrically neutral. Sol For a "closest approach", the incident proton must be directed "head-on" to the nucleus, with no angular momentum with respect to the nucleus (an "Impact parameter" of zero; see the Appendix to Chapter 4). In this case, at the point of closest approach the proton will have no kinetic energy, and so the potential energy at closest approach will be the initial kinetic energy, taking the potential energy to be zero in the limit of very large separation. Equating these energies, m. 10 14 1 J 10 60 1 C) 10 60 1 79 C m N 10 99 8 4 1 or 4 13 13 2 19 2 2 9 initial 2 2 initial - - - × = × × × = = = . . . )( ( ) / . ( , min min K Ze r r Ze K o o pe pe
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Inha University Department of Physics 5. What is the shortest wavelength present in the Brackett series of spectral lines? 7. In the Bohr model, the electron is in constant motion. How can such an electron have a negative amount of energy? Sol The wavelengths in the Brackett series are given in Equation (4.9); the shortest wavelength (highest energy) corresponds to the largest value of n . For n →∞ , Sol While the kinetic energy of any particle is positive, the potential energy of any pair of particles that are mutually attracted is negative. For the system to be bound, the total energy, the sum of the positive kinetic energy and the total negative potential energy, must be negative. For a classical particle subject to an inverse-square attractive force (such as two oppositely charged particles or two uniform spheres subject to gravitational attraction in a circular orbit, the potential energy is twice the negative of the kinetic energy. m 1.46 m 10 46 1 m 10 097 1 16 16 6 1 - 7 m l = × = × = - . . R
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Inha University Department of Physics 9. The fine structure constant is defined as a = e 2 /2 e o hc . This quantity got its name because it first appeared in a theory by the German physicist Arnold Sommerfeld that tried to explain the fine structure in spectral lines (multiple lines close together instead of single lines) by assuming that elliptical as well as circular orbits are possible in the Bohr model. Sommerfeld's approach was on the wrong track, but a has nevertheless turned out to be a useful quantity in atomic physics. (a) Show that a = v 1 / c , where v , is the velocity of the electron in the ground state of the Bohr atom. (b) Show that the value of a is very close to 1/137 and is a pure number with no dimensions. Because the magnetic behavior of a moving charge depends on its velocity, the
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mp-chapt-4-sol - Chapter 4. Problem Solutions 1. The great...

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