Semester 1 Examination Physics 560.3XX
The University of Western Australia
FIRST SEMESTER EXAMINATION JUNE 2001 560.312/3/4/5/6 560.324/5/6 560.336
THIRD YEAR PHYSICS
Electromagnetic Radiation This paper contains: 5 pages 4 questions
Time allowed: TWO HOURS
INSTRUCTIONS TO CANDIDATES
Candidates should attempt all questions, writing their answers in the booklets provided. The paper contains 4 questions, each worth 20 marks.
This examination paper must not be taken from the Examination Venue. All pages must be named in the space provided. Candidates may use the extra answer book for working. This will be collected and must also be named.
Information which may be useful:
Page 1 Semester 1 Examination Physics 560.3XX
Page 2 p field =eomo S=eo E ҙ B
Semester 1 Examination Physics 560.3XX
Page 3 ¶ СЧS=- (U field+Umech) ¶t t ¶ СЧ(-T ) =- (p field+p mech) ¶t Semester 1 Examination Physics 560.3XX
Page 4 ( СЧF) dt = FЧda (Сҙ F)Чda= FЧdl volumeт volumeт surfaceт lineт Semester 1 Examination Physics 560.3XX
Spherical Coordinates (r): 1 ¶ 2 1 ¶ 1 ¶Ff СЧF = (r Fr ) + (Sin(q) Fq ) + r 2 ¶r r Sin(q) ¶q r Sin(q ) ¶f Cylindrical Coordinates (rz): ¶F ¶(rF ) ж1 ¶Fz f ц ж¶ Fr ¶Fz ˆ ж1 f 1 ¶Fr ц С ҙ F = з - ч rˆ + - цf + з - ч zˆ иr ¶f ¶z ш и ¶z ¶r ш иr ¶r r ¶f ш
Page 5 Semester 1 Examination Physics 560.3XX
(Each question is worth 20 marks)
Question 1. a) An electric field is applied to a dielectric sphere of radius R, which induces a
polarisation of P(r) = r2rˆ , where rˆ is a unit vector in the radial direction. Using
2 3 spherical coordinates calculate the bound charges, b [C/m ] and b [C/m ].
[8 marks]
b) Given that the relative permittivity of the material is, r = 10, calculate the electric
field E(r) and electric flux density D(r) inside the material. [4 marks]
c) An infinitely long circular cylinder carries a uniform magnetization parallel to its
axis of M = Mzzˆ , where Mz is a constant and zˆ is the unit vector parallel to the
2 cylinder axis. Calculate the bound current densities Jb [A/m ] and Kb [A/m].
[8 marks]
Page 6 Semester 1 Examination Physics 560.3XX
Question 2. (5 marks each)
Consider a rectangular waveguide constructed from an infinitely conductive material with dimensions a and b as shown below. The axis of the waveguide is along the z direction. Note: in a conductor the complex wave number is:
ж 1 1 ц зй 2 щ2 й 2 щ2 ч ˜ em ж s ц ж s ц k = w зк 1 + з ч +1ъ + i к 1 + з ч -1ъ ч 2 зк иe w ш ъ к иe w ш ъ ч зл ы л ы ч и ш
y z
x
b
a a) Given that a=2.28 cm and b = 1.01 cm, what TE modes will propagate in this
waveguide if the driving frequency is 15 GHz? [5 marks]
b) Suppose you only wanted to excite one TE mode, what range of frequencies could
you use? [5 marks]
c) Draw a ray diagram in the y-z plane, which illustrates how the wave fronts
propagate in the waveguide. Show graphically (or with another method) if the
phase and group velocities are greater than or less than c. [5 marks]
d) What would happen to the propagating modes if the metal is a real conductor (i.e.
small Ohmic losses are present)? Calculate the skin depth at 15 GHz if the
waveguide is made from copper of conductivity, =107. [5 marks]
Page 7 Semester 1 Examination Physics 560.3XX
Question 3.
a) What is dispersion? [2 marks]
b) Show that the phase and group velocity of light in vacuum equals the speed of light
c, how does this differ in a dispersive material? [3 marks]
c) Due to vibrations of electrons in the lattice of a dielectric, the complex permittivity
may be written as;
2 Nq ж fj ц e˜ (w)= 1+ з r m e е 2 2 ч o j иw j - w - ig jw ш
Explain in detail what you know about the physics described by this equation,
with regards to a plane wave travelling in the dielectric medium (a derivation is
not required, but please include in the explanation the meaning of each variable in
the above formula). [7 marks]
w d) Assuming the loss term above is zero, given that k˜ = e˜ , calculate the phase c r
and group velocity within the dielectric. [5 marks]
e) What is anomalous dispersion? [3 marks]
Page 8 Semester 1 Examination Physics 560.3XX
Question 4.
a) In the time dependent and static cases, the Magnetic Flux Density B, may be
represented by the curl of a vector potential field given by, B = С ҙ A . Why is this
so? Define the scalar potential for the static and time dependent cases, how do they
differ? [6 marks]
˜ i(kI Чr-wt) b) An electromagnetic monochromatic plane wave given by E I (r,t) = E˜ 0I e
gives rise to reflected and transmitted waves of the form,
˜ i(kR Чr-wt) ˜ i(kT Чr-wt) E R(r,t) = E˜0R e and ET (r,t) = E˜ 0T e respectively. Assuming
that the polarisation of the wave is p-polarised (parallel to the plane of incidence),
a - b 2 show that E˜ = r E˜ and E˜ = t E˜ where r = and t = , where 0R 0I 0T 0I a + b a + b
cos(qT ) m1 e2 a = and b = . [10 Marks] cos(qI ) e1 m2
qT
c) Does r2 + t2 = 1? Explain the reason why or why not. [4 Marks]
Page 9 Semester 1 Examination Physics 560.3XX
(END OF PAPER)
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