Next: About this document ...
Five pages  Page one

University of the Witwatersrand
Physics IIE (Engineering) : PHYS284 : 2003
Examination : June 2003

Instructions: Answer all questions.
Time: 2 hours = 120 minutes
Total Marks: 120 marks
 1.
 a)
 Given that the Lorentz Transformation is :
derive the formula for the relativistic time dilation
(6)
 b)
 The Global Positioning System (GPS) consists of a network of 24 satellites
in roughly 12hour orbits, each carrying atomic clocks on board. The orbital
radius of the satellites is about 26,600 km.
The satellites have orbital speeds of about 3.9 km/s in a frame centred
on the Earth. The onboard atomic clocks have a period of about 1 nanosecond (ns)
and they have a rate accuracy of 1 ns/day. The GPS receiver determines its
current position and heading by comparing the time signals it receives via
a radio transmission from a
number of the GPS satellites (usually 4 to 12) and triangulating on the
known positions of each satellite.
 i)
 Consider the distance that light can travel in 1 ns and determine the accuracy
with which the timing information from the
radio transmisions must be known in order to correspond to the typical position
accuracy of 5m.
(3)
The timing signals of the clocks are affected by both General and Special
Relativity. (The effects of General Relativity are about nine times larger than the
those of Special Relativity, but they are not considered here.)
 ii)
 Considering the Special Relativity effects, will the satellite based clock
tick slower or faster than the ground based clock as seen from the ground based clock ?
Elaborate on your answer briefly.
(3)
 iii)
 Now show that the time difference between the satellite based clock, and
the ground based clock, amounts to about 7 s per day.
(4)
2/Page two...
Five pages  Page two
PHYS284
June 2002 Physics IIE (Engineering)
 iv)
 What positional error does this time error correspond to ?
(3)
Engineers preset the satellite clocks to new tick rates which account for the effects
of both General and Special Relativity. This is done in such a way that once
the satellites are in their known orbits with their known speeds, they then
remain in better synchronisation with the earth bound clocks.
A regular signal corrects any remaining error in their mutual synchronisation. A further
relativistic calculation is required by the GPS receiver, once it has acquired four or
more satellites, to make additional clock corrections.
 v)
 Consider what this calculation may take into account.
(3)
 vi)
 Discuss the sources of error in a GPS system.
(3)
Total for Question 1 [25]
 2.
 A quantum particle approaches a rectangular barrier whose height is greater
than the energy of the particle as in the figure below.
Consider the Schrödinger Wave Equation
with the potential
The solutions will have an oscillatory or an exponential form
where
depending on the particular region of the figure
being considered.
 a)
 Specify which of the forms belong to which regions and give the structure of the
terms.
(6)
3/Page three...
Five pages  Page three
PHYS284
June 2003 Physics IIE (Engineering)
 b)
 Applying matching conditions at the region interfaces as well as boundary
conditions, we find the transmission probability through the barrier is
Deduce the dependence of the transmission on particle energy, barrier height and barrier width.
What is the name given to this purely quantum process ?
(4)
 c)
 Discuss the flow of electrons between two partly oxidised contacts,
as they approach each other, as in a simple switch, in terms of this theory.
(4)
 (d)
 The Scanning Tunneling Microscope directly exploits these insights.
 i)
 Explain the principle of operation by referring to
a particle energy drawing showing clearly the three regions corresponding to the
sample, the tip and the airgap in an idealised barrier penetration problem.
(4)
 ii)
 The transmission probability of electrons through the barrier in a tunneling microscope
is characterised by nm.
Suppose the electronic detection system is sensitive to electrical
current changes of 0.1%. What surface height sensitivity does this correspond to ?
(5)
Total for Question 2 [23]
 3.
 (a)
 Consider the radial wave equation for the Hydrogen atom,
as well as the radial wave function for the 1s state
Show that the Bohr radius and ground state energy for the hydrogen atom are given by
(8)
 (b)
 Verify that these two quantities are indeed well predicted, by calculating their values
in units of Å and eV respectively.
(5)
4/Page four...
Five pages  Page four
PHYS284
June 2003 Physics IIE (Engineering)
 (c)
 Verify that the average value of for a 1s electron is by calculating
the expectation value of given by the equation
with
and
with given as above. Also note that
(7)
Total for Question 3 [20]
 4.
 (a)
 Starting with the wavefunctions for two identical noninteracting
quantum particles,
show that the probability for two fermions to occupy the same quantum
state is zero, and
that the probability for two bosons to occupy the same quantum state is
twice that for two classical particles to occupy the same state.
(Note : Use
and
.)
(6)
 (b)
 Use this result to state and explain the relative pressure
exerted by similar gases of classical molecules, bosons, or
fermions for the same temperature.
(4)
 (c)
 Sketch the FermiDirac distribution
at low but finite temperature
and explain why only electrons near the Fermilevel
are expected to participate in transport properties.
(4)
 (d)
 Show that if the average occupancy of a state of energy
is at any temperature, then the average occupancy of a state of energy
is .
(6)
Total for Question 4 [20]
 5.
 The Fermi theory predicts Ohm's law for metallic conductors with the resistivity given as
Find the mean free path of electrons between collisions in copper at 20C
if its resistivity
at that temperature is
m
(
eV,
cm).
(5)
Total for Question 5 [5]
5/Page five...
Five pages  Page five
PHYS284
June 2003 Physics IIE (Engineering)
 6.
 (a)
 Classify metals, semiconductors and insulators on the basis of their band structure.
(6)
 b)
 Consider Si, a semiconductor with a bandgap of 1.11 eV.
Sketch the band structure diagram and indicate
the position of the Fermilevel.
(3)
 c)
 The Si is now doped with phosphorous atoms.
Using the following data,

= 
11.7 for Si 

= 
0.19 
calculate the ionisation energy of the phosphorous atom in silicon.
(5)
 d)
 Make a new band structure diagram which includes the dopant level
and the new position of the Fermilevel.
Label fully.
(5)
 e)
 The figure below is a logarithmic plot of the charge carrier concentration
versus inverse temperature.
There are three regions in the plot. Discuss the physics which accounts for the
behaviour of the carrier concentration in each region. Indicate the energy represented
by the slope of the straight lines in the regions I and II.
(8)
Total for Question 6 [27]
Total Marks [120]
No more pages
Next: About this document ...
Connell
20040311