Download 2003 Exam

Information which may be useful:

B
t

HJ f  D
t
  D f
  E 
  B 0
 ( F )d   F da
 (  F) da   F dl
v
s
v
Spherical
l
p field  o o S o E  B
S  
T 

(U
 Umech )
t field

(p
p
)
 t field mech
(r):
Coordinates
F
1  2
1

1
F  2
r Fr 
Sin( ) F 

r Sin( ) 
r Sin( ) 
r r
Cylindrical
 
(rz):
Coordinates
1 Fz F  Fr Fz  1  (rF ) 1 Fr 
ˆ 
ˆr 
zˆ
  F  


+



r 
z 
z
r
r r
r  
D  o E P
H
1
o
B M
linear media : D  E, B   H
 o  8.854  10-12 C/ Nm 2 o  4  10-7 N / A 2

1
1 

2

2 
2
2
  
  
 
k˜  
1    1  i  1    1 




 
 
 
2 


 



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(Each question is worth 20 marks)
Question 1.
a) An electric field is applied to a dielectric sphere of radius a, which induces a
polarisation of P(r) = r rˆ (for 0≤r≤a), where rˆ is a unit vector in the radial direction
and r is the radial coordinate. Using spherical coordinates calculate the bound
charges, b [C/m2] and b [C/m3].
[8 marks]
b) Given that the relative permittivity of the material is, r = 2.5, calculate the electric
field E(r) and electric flux density D(r) inside the material. [4 marks]
c) An infinitely long circular cylinder of diameter, a, carries a magnetization parallel to
its axis of M = r2 zˆ (for 0≤r≤a), where zˆ is the unit vector parallel to the cylinder
axis and r is the radial coordinate from the centre of the cylinder. Calculate the
bound current densities Jb [A/m2] and Kb [A/m].
[8 marks]
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Question 2.
Consider a rectangular waveguide constructed from an infinitely conductive material
with dimensions a and b as shown below.
a) 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) whether the
phase and group velocities are greater than or less than c. [5 marks]
b) Given that a = 2.80 cm and b = 2.20 cm and you only want to excite one TE mode,
what range of frequencies could you use? [5 marks]
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˜ (r,t)  E˜ eik I rt ,
c) An electromagnetic monochromatic plane wave given by E
I
0I
incident on a flat surface gives rise to reflected and transmitted waves of the form,
˜ (r,t)  E
˜ eik R rt and E
˜ (r,t)  E˜ ei kT rt  respectively. Assuming
E
R
0R
T
0T
that the polarization of the wave is s-polarized (perpendicular to the plane of
˜  E˜ yˆ , E
˜  E˜ yˆ and E
˜  E˜ yˆ . Show that
incidence) such that E
0I
0I
0R
0R
0T
0T
1  
E˜ 0R  r E˜ 0I and E˜ 0T  t E˜ 0I , where r 
1  

cos( T )
and  
cos( I )
and t 
2
, where
1 
1 2
. [10 Marks]
1 2
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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 relation 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:

fj
Nq 2 
˜
r   1
 2


m o j  j   2  i j  
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]
d) Assuming the loss term above is zero, calculate the phase and group velocity within
 ˜
the dielectric, given that k˜ 
r .
c
[5 marks]
e) What is anomalous dispersion? [3 marks]
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5
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    A . Why is this
so? Define the scalar potential for the static and time dependent cases, how do they
differ? [6 marks]
Consider two infinite conductor plates with charge density 0 c/m2 as shown.
b) In the rest frame S0(x0,y0,z0), what is the magnetic flux density (B0) and electric field
intensity (E0) between the plates. [5 marks]
c) The moving frame S(x,y,z) has a velocity of v0 xˆ with respect to S0(x0,y0,z0) along
the x axis as shown. Determine the magnetic flux density (B) and electric field
intensity (E) between the plates in the moving frame, in terms of 0, the Lorentz
contraction factor 0, v0, 0 and 0. (Remember you are dealing with vectors)
[9 marks]
(END OF PAPER)
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