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Five pages  Page one

University of the Witwatersrand
Physics IIE (Engineering) : PHYS284 : 2004
Examination : June 2004

Instructions: Answer all questions.
Time: 2 hours = 120 minutes
Total Marks: 120 marks
 1.
 a)
 Starting from
Show that the relativistic expression for the kinetic energy is
(8)
 b)
 State the physical meaning of the two terms on the right hand side
of the previous expression.
(2)
 c)
 Show that the nonrelativistic expression can be recovered at low velocities.
(5)
 d)
 The Global Positioning System (GPS) consists of satellites with
orbital speeds of about 3.9 km/s in a frame centred
on the Earth. The orbital radius of the satellites is about 26,600 km.
 i)
 Do the satelite bound clocks tick faster or slower than the earth bound clocks,
considering the effects of Special Relativity ?
(1)
 ii)
 If 12 hours have passed on the satellite, what would the elapsed time
have been on the earth, considering the effects of Special Relativity ?
Express your answer as a time difference in microseconds.
(Use an approximate method if your calculator has insufficient accuracy.)
(3)
 iii)
 What position error would this represent, if the corrections were not made ?
(3)
 iv)
 What procedure is implemented to avoid this error ?
(3)
Total for Question 1 [25]
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PHYS284
June 2004 Physics IIE (Engineering)
 2.
 a)
 Consider the Schrödinger Wave Equation
with the potential
as in the figure.
For a particle incident from the left,
show that the solution in the barrier region is not the normal
oscillatory wave function, but has a decreasing exponential form.
(6)
 b)
 The fact that the wave function penetrates the barrier leads to
the quantum phenomenon of tunneling. The probability for this is approximately
 i)
 Indicate the meaning of the symbols , and .
(4)
 ii)
 In the corresponding classical process, the particle has to hop over the barrier
with probability
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PHYS284
June 2004 Physics IIE (Engineering)
 The tunneling process represents a physics limit for the miniaturisation of
features on a chip. This could be either tunneling between neighbouring wires or
across the gate of a transistor. Imagine that the quantum process should not be more
likely than the classical process, so that the limiting case is when they are equal.
This will allow you to estimate a value for the minimum
feature size of a conventional chip.
Estimate realistic values for and and explain your choice.
(4)
 iii)
 Perform the calculation to find .
(6)
Total for Question 2 [20]
 3.
 (a)
 The probability of finding the electron in the hydrogen atom
ground state, at some distance between and is given by
where is the Bohr radius (0.05292 nm). Make a rough plot of in terms of
and calculate the distance from the nucleus where the
electron is most likely to be found.
(15)
 (b)
 The energy levels of certain kinds of twoelectron atoms/ions of
atomic number may be approximated by
where the ground state of hydrogen is
eV.
 i)
 Sketch the twoelectron atom/ion under conditions where the above expression
could be expected to hold.
(5)
 ii)
 Using the full expression for quantised energy levels in hydrogenlike
(that is, oneelectron) atoms/ions of atomic number ,
describe how each term in the above expression arises.
(6)
 iii)
 Why does the accuracy increase when increases ?
(4)
Total for Question 3 [25]
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PHYS284
June 2004 Physics IIE (Engineering)
 4.
 (a)
 A cubic meter of atomic hydrogen at 0C and at atmospheric
pressure contains about
atoms.
Find the ratio of the number of these atoms in their first excited state
() to the number in the ground state () at 10,000C.
(6)
 (b)
 Show that in a system of fermions at K, all states with
are occupied, while all those with
are unoccupied.
(4)
 (c)
 The number of fermions in a fermi gas (for example, electrons in a metal) that have
energies from to
is
Consider electrons in the low temperature limit where all the lowlying
states are occupied. Show that the energy of the last filled state, known
as the fermi energy, is given by
(7)
 (d)
 The density of metallic zinc is 7.13 g/cm and the atomic mass
of the zinc atom is 65.4 u.
The Fermi energy in zinc metal is 11.0 eV.
 (i)
 Work out the effective mass of a delocalised
electron in zinc metal. Express your answer in terms of the free electron mass.
(Zinc has 2 valence electrons.)
(6)
 (ii)
 Why is there a difference ?
(2)
Total for Question 4 [25]
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PHYS284
June 2004 Physics IIE (Engineering)
 5.
 (a)
 In a onedimensional lattice, the energy of an electron in the valence band
as a function of its wave number, , may be approximated by,
where is a positive constant, and is the lattice spacing.
What is the effective mass an electron at the top of the band () ?
(Express your answer in terms of , and ).
(6)
 (b)
 Derive the distribution of holes in an intrinsic semiconductor
from the statement that
.
(6)
 (c)
 A semiconductor is characterised by the energy band structure shown
in the following figure.
 (i)
 Specify which side of the diagram has acceptor dopants, and which side
has electrons as charge carriers.
(2)
 (ii)
 What is the concentration of charge carriers in the
section of the semiconductor ?
(6)
 (iii)
 What is the resistivity in this section ?
(5)
(
eV,
cm, eV
cmVs,
C.)
Total for Question 6 [25]
Total Marks [120]
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Simon H Connell
20050528