Download fundamental topics in physics

THE UNIVERSITY OF HULL
___________________
Department of Physical Sciences (Physics)
Level 3 Examination
January 2008
FUNDAMENTAL TOPICS IN PHYSICS
(Module No. 04306)
TUESDAY 15 JANUARY 2008
2 hours (13.30 – 15.30)
Answer THREE questions, TWO from Section A and ONE from
Section B.
Do not open or turn over this exam paper, or start to write anything until told
to do so by the Invigilator. Starting to write before permitted to do so may
be seen as an attempt to use Unfair Means.
Module 04306
CONTINUED
Page 1 of 7
SECTION A: ELECTRODYNAMICS
1.
An electromagnetic (EM) plane wave travelling in a vacuum in the z-direction has an E field
of amplitude E0 polarised in the x-direction, i.e., E  E0iˆ cost  kz and an H field given by
H  H 0 cos(t  kz) , where H0 is the vector amplitude of the H field.
H
(i) Using the Maxwell equation   E    0
, show that the H field is polarised in the
t
y-direction with an amplitude of H 0  k / 0 E0  E0 / 0 c  , where c   k is the speed
of light.
[8 marks]
(ii) The Poynting vector for an EM wave is given by N  E  H .
a. Discuss briefly the physical significance of the Poynting vector.
b. Using your result in (i), calculate the magnitude and direction of the Poynting vector
for the EM wave.
c. Show that the magnitude of the time averaged Poynting vector is E02 /( 2 0 c) .
[6 marks]
(iii) The Sun has a power output of 3.8×1026W. If the earth-sun distance is 1.5×1011m, use
your result in (ii) to estimate the average amplitude of the E and H fields produced by the
sun’s radiation at a distance equal to the earth’s orbit.
[6 marks]
[c = 3 ×108m s-1;  0  4 10 7 Hm -1 ; SI unit for E field is Vm-1; SI unit for H field is Am-1]
Module 04306
CONTINUED
Page 2 of 7
2.
An electromagnetic wave is incident at a boundary between two dielectric media of
refractive index n1 and n2 respectively. The amplitude reflection and transmission
coefficients are
 E '
n cos i  n2 cos t
rN   0   1
 E0  N n1 cos i  n2 cos t
and,
(1)
 E '' 
2n1 cos i
t N   0  
 E0  N n1 cos i  n2 cos t
(2)
when the electric field of the incident beam is polarised perpendicular to the plane of
incidence. i and t are the angles of incidence and transmission respectively.
(i) Use equation (1) to show that the reflected beam is in phase with the incident beam
when n1 > n2, and that the two beams are out of phase by  when n1 < n2.
[4 marks]
(ii) Use equation (1) to find an expression for RN , the coefficient of reflection for energy
flow, at normal incidence. Evaluate RN for
(a) a GaAs/air, and
(b) a glass/air
interface, taking the refractive index of GaAs and glass as 3.5 and 1.5 respectively. Hence,
give a reason why the cleaved facets of GaAs are often used to form a laser cavity
whereas the polished ends of a Nd3+:glass rod are unlikely to be used.
[6 marks]
(iii) The refractive index of a good conductor at wavelength 0 can be expressed as
n
0
(1  i)

(3)
where is  the skin depth. Use equations (1), (2) and (3) to show that for normal incidence
at a boundary between a dielectric and good conductor
(a)
the reflected wave has approximately the same amplitude as the incident
wave and is completely out of phase with it, and
(b)
the transmitted beam is extremely weak and out of phase by -/4 with the
incident beam.
State any approximations used.
[6 marks]
(iv) Copper has a skin depth of 6.6  10-5 m at a frequency of 1 MHz. What is the phase
velocity of the transmitted wave in copper? Find the intensity transmission coefficient from
air to copper at normal incidence.
[4 marks]
[Velocity of light in vacuo c = 3  108 ms-1]
Module 04306
CONTINUED
Page 3 of 7
3.
When an external electric field Eex is applied to a medium, dipoles are induced in individual
molecules and the induced molecular dipole moment p is linearly dependent on the local
electric field Eloc given by
E loc  E ne ar  E L  E p  E e x .
(1)
where EL and Ep are the Lorenz and macroscopic polarisation fields respectively.
(i) Use the Lorenz model to explain the origin of the components of Eloc. You should use a
diagram to clarify your explanation.
[8 marks]
(ii) Using Coulomb’s Law and integration with spherical coordinates, show that
EL = P/30, where P is the macroscopic polarisation and 0 the permittivity of free space.
[6 marks]
(iii) Explain why local field effects are not usually considered when describing the
polarisation of gases
[2 marks]
(iv) In a gas, the dielectric constant r is related to the molecular polarisability  of a molecule by
the formula
 r 1 
N
0
where N is the number of molecules per unit volume. Find  for CO2 at STP (P = 105 Nm-2,
T = 273K) if r = 1.00099.
[4 marks]
[Boltzmann’s constant k = 1.381  10-23 JK-1; 0 = 8.854  10-12 Fm-1]
Module 04306
CONTINUED
Page 4 of 7
4.
(i) Sketch the generalised equivalent circuit representation for an element of transmission
line. Give a brief explanation of each component in this equivalent circuit. How is this
simplified for an ideal transmission line?
[4 marks]
(ii) The voltage, V, and current, i, on an ideal transmission line satisfy
V
i
 L
x
t
i
V
 C
x
t
and
where C is the capacitance per unit length and L the inductance per unit length of line.
(a) Use these equations to show that V and i obey wave equations where the phase
velocity u 
1
.
LC
(b) Assuming a solution to the wave equation of the form V  Vo sin[  (t  x / u)] show
that V and i are in phase and that Z, the ratio V/i, is Z 
L
.
C
(c) What is Z called? Give a brief explanation of its physical significance.
[9 marks]
(iii) An overhead power transmission line with C = 4.03  10-12 F.m-1 and
L = 2.76  10-6H.m-1 connects a dynamo and step-up transformer to a sub-station located
150km away. If a lightning strike produces a sudden short-circuit at the dynamo explain
qualitatively what happens on the line. How long will it take the wave produced to reach the
sub-station and what fraction of the voltage wave is reflected at the sub-station if it forms a
load of 900?
[7 marks]
Module 04306
CONTINUED
Page 5 of 7
SECTION B: ACOUSTICS
5.
(i) The wave equation for the transverse displacement y on a string of mass per unit
length L under tension T is
2 y 1 2 y

x 2 c 2 t 2
where the phase velocity is c2 = T/L. Assuming a general solution for y of the form
y(x,t) = Aej(t + kx)+ Bej(t - kx) show that the normal modes (eigenfunctions) for a string of
length L, rigidly fixed at both ends are
yn(x,t) = Ansin(n x/L) expjn t
where n = 1, 2, 3 …. is an integer. Hence derive the expression for n, the characteristic
angular frequency of the string.
Discuss how, in principle, yn(x,t) could be used to analyse the harmonic content of the note
from a plucked guitar string and from a struck piano string and explain why these have
different ‘acoustic’ qualities.
[10 marks]
(ii) A recent paper reports a fixed/fixed string formed by a single carbon nanotube. If
L = 1.110-16kgm-1 and T = 810-10N calculate the wave velocity on the string. If
L = 300nm what is its fundamental frequency and does this lie in the ‘acoustic’ range?
[4 marks]
(iii) Why do gases support only longitudinal acoustic waves? The pressure amplitude po in
a plane acoustic wave varying as p = po sin(t –kx) in air is measured as po = 0.01Pa.
Calculate the corresponding particle velocity amplitude and determine the particle
displacement if  = 200rad.s-1 (for air take  = 1.25 kgm-3and c = 342m/s).
[6 marks]
Module 04306
CONTINUED
Page 6 of 7
6.
(i) A sound wave travels from a material of specific acoustic impedance Z1 into one
of specific acoustic impedance Z2. From the conditions that the acoustic pressure
and the particle velocity are continuous at the boundary show that the reflection
coefficient rp for pressure is:
rp 
pro Z 2  Z1

pio Z1  Z 2
[7 marks]
(ii) A sound wave travels from air (Zair = 428 rayls) into concrete (Zcon= 8 106 rayls).
Determine rp and express this in decibels.
[3 marks]
(iii) (a) What is understood in acoustics by the term reverberation? (b) Discuss briefly
how the Sabine reverberation time T is defined. Calculate T for a cubic enclosure
of dimensions 3m  3m  3 m if its entire inner surface is coated with acoustic plaster
(sound absorption coefficient  = 0.2). You may assume T = 55.2V/ac where V is the
enclosure volume, a 
 i Si where Si is the area of wall with sound absorption

i
coefficient I and c = 340ms-1 is the sound speed for air. (c) Comment on the
acoustic quality of this enclosure.
[6 marks]
(iv) Give a short account of the operating principle and constructional features of
either the moving coil (‘dynamic’) microphone or the electret microphone.
[4 marks]
Module 04306
END
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