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L A N C A S T E R U N I V E R S I T Y
2008 EXAMINATIONS
Part II
PHYSICS - Paper 3.B
- Candidates should answer all those sections identified with the modules for which they are registered.
- An indication of mark weighting is given by the numbers in square brackets following each part.
- Use a separate answer book for each section.
PHYSICAL CONSTANTS
Planck’s constant h = 6. 63 × 10 −^34 J s ℏ = 1. 05 × 10 −^34 J s Boltzmann’s constant kB = 1. 38 × 10 −^23 J K−^1 Mass of electron me = 9. 11 × 10 −^31 kg Mass of proton mp = 1. 67 × 10 −^27 kg Electronic charge e = 1. 60 × 10 −^19 C Speed of light c = 3. 00 × 108 m s−^1 Avogadro’s number NA = 6. 02 × 1023 mol−^1 Permittivity of the vacuum ǫ 0 = 8. 85 × 10 −^12 F m−^1 Permeability of the vacuum μ 0 = 4 π × 10 −^7 H m−^1 Gravitational constant G = 6. 67 × 10 −^11 N m^2 kg−^2 Bohr magneton μB = 9. 27 × 10 −^24 J T−^1 (or A m^2 ) Bohr radius a 0 = 5. 29 × 10 −^11 m Gas constant R = 8 .31 J K−^1 mol−^1 Acceleration due to gravity g = 9.81 m s−^2 1 standard atmosphere = 1. 01 × 105 N m−^2
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Section A: Module 321: Atomic & Nuclear Physics (The time allocated for this section is 60 minutes. Candidates should answer question A1 and either question A2 or question A3.)
Compulsory question:
A1. (a) (i) Define the angular momentum operator L~ for a particle in three dimen- sional space. (ii) Write down the operator ~L^2 in Cartesean coordinates. (iii) What are the possible eigenvalues of Lz for a 3p electron? [8] (b) The electron configuration of phosphorus is 1s^2 2s^2 2p^6 3s^2 3p^3. Consider the shell model of this atom. (i) How many shells are occupied by electrons in phosphorus? (ii) What is the effective nuclear charge experienced by the innermost shell? (iii) Given that the ionisation energy of Hydrogen is 13.6 eV, estimate the ion- ization energy (energy needed to remove an electron) from the innermost shell of phosphorus. [12] (c) (i) Explain the terms α-decay, α-particle and β-decay. (ii) Define the term half-life. The half-life of Polonium-210 is 138 days. A sample of radioactive material contains 0.1 mg of Polonium-210. How long ago was the sample prepared if initially it contained 3 mg of Polonium-210? [10]
Answer one of the following two questions:
A2. The Hamiltonian of the spin-orbit interaction for the Hydrogen atom is:
HSO =
2 m^2 c^2
r
dV (r) dr
S~ · ~L
(a) Describe qualitatively how the spin-orbit interaction arises for electrons in the Hydrogen atom. [10] (b) Sketch energy diagrams for the splitting of the n = 3 level of the Hydrogen atom due to the spin-orbit interaction. Indicate the degeneracy of each level. [10] (c) Assuming that the mean distance from the electron to the nucleus in the n = 3 level of Hydrogen is three times the Bohr radius, and that the angular momen- tum is ℏ, estimate the magnitude in eV, of the spin-orbit splitting of this level. [10]
Answer one of the following two questions:
B2. The Fermi-Dirac distribution function can be written as
f (ε) =
e(ε−μ)/kB^ T^ + 1
(a) Define the terms ε and μ and describe the physical meaning of f (ε). Sketch this distribution at zero temperature. State the values for f (ε) at T = 0 for (i) ε < μ, (ii) ε > μ and (iii) ε = μ. Indicate how the distribution changes as the temperature is raised. What is the characteristic energy range for partially filled states? [10] (b) Given that the density of states in k-space is g(k) = V k^2 /π^2 in a gas of N spin 1 2 fermions with mass^ m^ confined to a volume^ V^ , calculate the density of states as a function of energy, g(ε). Hence derive the formula for the Fermi energy εF at zero temperature. [12] (c) Estimate the electronic heat capacity per unit volume for a typical metal, with an electron density of N/V = 10^29 m−^3 , at room temperature. [8]
B3. (a) Write down the Boltzmann distribution function for N localized particles with non-degenerate energy levels ǫj. Define all the terms used. [6] (b) Consider N spin-1 non-interacting magnetic atoms in thermal equilibrium at temperature T in a magnetic field B. The energy spectrum of such a system consists of three levels with energies ǫ 1 = −μB B, ǫ 2 = 0 and ǫ 3 = μB B, where μB is the Bohr magneton. (i) Write down the distribution function and partition function of this system. Show that for such a system the internal energy can be written as
U =
−NμB sinh
(μB kT
1 2 + cosh^
( (^) μB kT
Sketch the dependence of the internal energy as a function of temper- ature and state the characteristic temperature for the variation. Using this sketch, describe qualitatively the temperature dependence of the heat capacity. [14] (ii) Calculate the Helmholtz free energy F and the entropy S for this system. Sketch the entropy as a function of temperature. State the value of the entropy in the high and low temperature limits. [10]
Section C: Module 323: Physics of Fluids (The time allocated for this section is 60 minutes. Candidates should answer question C1 and either question C2 or question C3.)
Compulsory question:
C1. (a) The Reynolds number Re of a viscous fluid flow is
Re =
U 0 L 0 ρ 0 η 0
(i) Define the symbols ρ 0 and η 0 and explain the physical meaning of quantities U 0 and L 0. (ii) Describe the qualitative difference between fluid flows with large Reynolds numbers and small Reynolds numbers. (iii) Explain the concept of hydrodynamic similarity and its importance for air- craft design. [12] (b) Explain how vorticity is generated by a moving aircraft wing and describe the importance of vorticity for flight. State the Kutta-Joukowski formula relating lift and vorticity. [10] (c) The angular frequency ω of a periodic surface wave with wavenumber k on deep water is ω =
gk where g is the gravitational acceleration. Obtain an expression for the wave’s phase velocity in terms of its wavelength and explain how a localized non-periodic disturbance far out at sea can lead to approximately periodic surface waves at the shore. [8]
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C3. (a) (i) Describe the differences in the physical behaviour of an ideal fluid flowing near a solid impenetrable sphere at rest and a viscous fluid flowing near the same sphere at rest. In each case, state the appropriate boundary conditions. (ii) Describe the contact force experienced by the sphere in a steady potential flow of an ideal fluid. Identify the properties of a viscous flow at high Reynolds number responsible for drag on the sphere and draw a diagram of the flow. [12] (b) A steady incompressible fluid with a finite viscosity η 0 flows through a pipe of radius a. The velocity field v = vz (r)z, where the unit vector z points along the pipe, depends only on the radial distance r from the pipe’s axis of symmetry. (i) Sketch the profile of the velocity field v across the pipe’s cross-section. (ii) When one uses cylindrical polar coordinates (r, ϕ, z) the Navier-Stokes equations can be reduced to
η 0
r
d dr
r
dvz dr
= A 0
where A 0 is a constant. Verify that this equation is satisfied by
vz =
A 0
4 η 0
r^2 + B 0 ln(r) + C 0
where B 0 , C 0 are constants of integration. Determine B 0 and C 0 using the appropriate boundary condition. [10] (c) Consider a long cylindrical rod initially at rest immersed in a Newtonian viscous fluid. The rod starts to spin around its axis of symmetry, gradually reaching a constant (slow) angular velocity Ω. (i) Give the appropriate boundary condition on the fluid velocity field v at the cylinder and sketch the streamlines of the steady state flow. (ii) Explain how the steady state flow is achieved by considering the contact force between adjacent infinitesimal elements of fluid. [8]
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