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6.013/ESD.013J Electromagnetics and Applications, Fall 2005
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Massachusetts Institute of Technology
Department of Electrical Engineering and Computer Science
6.013 Electromagnetics and Applications
Problem Set #6
Fall Term 2005
Issued: 10/12/05
Due: 10/26/05
______________________________________________________________________________
Reading Assignment:
Quiz 1: Thursday, Oct. 20, 2005 in lecture, 10-11AM. Covers material up to and
including P.S. #5.
Problem 6.1
An electric field is present within a plasma of dielectric permittivity with conduction constituent
relation
J f
2

p E

t
,
where
q 2n
2p 
m
with q, n and m being the charge, number density (number per unit volume) and mass of each
charge carrier.
(a) Poynting’s theorem is
w

S  EM E 
Jf
t
For the plasma medium, E
Jf , can be written as
w
E
Jf  k .
t
What is wk ?
(b) What is the velocity v of the charge carriers in terms of the current density J f and
parameters q, n and m defined above?
(c) Write w k of part (a) in terms of v, q, n, and m. What kind of energy density is wk ?
(d) Assuming that all fields vary sinusoidally with time as:
E
r, t Re Eˆ
r e jt 


write Maxwell’s equations in complex amplitude form with the plasma constitutive law.
(e) Reduce the complex Poynting theorem from the usual form
1 ˆˆ*
1 ˆ
ˆ* 

Jf
2 E( r) H ( r) 2 jwEM 2 E


1
to
1 ˆ
ˆ* 

) 0
2 E( r) H ( r) 2 j(wEM wk 


What are wEM and wk ?
(f) Show that
1
1
2
2
wEM wk  H  
( )E
4
4
What is 
( ) and compare to the results from Problem 5.3b?
Problem 6.2
A TEM wave ( E x , H y ) propagates in a medium whose dielectric permittivity and magnetic
permeability are functions of z, ( z ) and ( z ) .
(a) Write down Maxwell’s equations and obtain a single partial differential equation in H y .
(b) Consider the idealized case where ( z ) a e z and ( z ) a ez . Show that the equation
of (a) for H y reduces to a linear partial differential equation with constant coefficients of
the form
2
H y
z2
2

Hy
H y

 2 0

z

t
What are and ?
(c) Infinite magnetic permeability regions with zero magnetic field extend for z<0 and z>d. A
jt
current sheet Re 
ix K0 e 
is placed at z = 0. Take the magnetic field of the form
ˆy e( jtz ) ]
H Re[ i y H
and find values of that satisfy the governing equation in (b) for 0<z<d.
(d) What are the boundary conditions on H ?
(e) Superpose the solutions found in (c) and find H that satisfies the boundary conditions of
(d).
(f) What is the electric field for 0<z<d?
2
Problem 6.3
j (t k y y )
ˆ
A sheet of surface charge with charge density f Re[
] is placed in free space
0e
(0 , 0 ) at z 0 .
j ( t k y y )
ˆ
f Re[
]
0e
0 , 0
y
x
0 , 0
z
The complex magnetic field in each region is of the form
j ( k y y kz z )

i
Hˆ
1e
ˆ
H  j( k yk z) x
y
z
ix
Hˆ

2e
z 0
z 0
(a) What is k z ?
ˆ
(b) What is the complex electric field for z 0 and z 0 in terms of Hˆ
1 , H 2 , k y , k z and ?
(c) Using the boundary conditions at z 0 , what are Hˆ
and Hˆ
?
1
2
(d) For what range of the frequency will the waves for z 0 and z 0 be evanescent?
(e) What surface current flows on the charge sheet at z 0 ?
3
Problem 6.4
A TM wave is incident onto a medium with a dielectric permittivity 2 from a medium with
dielectric permittivity 1 at the Brewster’s angle of no reflection, B . Both media have the same
magnetic permeability 1 2 . The reflection coefficient for a TM wave is
Eˆ
cos i 2 cos t
r
R  1
ˆ
1 cos i 2 cos t
Ei
(a) What is the transmitted angle t when i B ? How are B and t related?
(b) What is the Brewster angle of no reflection?
2 , 

1 ,
t
i B
Ei

Hi
(c) What is the critical angle of transmission C when 1 2 ? For the critical angle to
exist, what must be the relationship between 
1 and 
2 ?
4
(d) A Brewster prism will pass TM polarized light without any loss from reflections.

i B
0 , 0

, 0
n

0
For the light path through the prism shown above what is the apex angle ? Evaluate for
glass with n 1.45.
(e) In the Brewster prism of part (d), determine the output power in terms of the incident
power for TE polarized light with n 1.45 . The reflection coefficient for a TE wave is
 cosi 1 cost
Eˆ
r
R  2
Eˆ
2 cosi 1 cost
i
5