ps07 - ASSIGNMENT 7 Physics 218 Due Oct 29 Fall 2004...

ASSIGNMENT 7
Physics 218 Due Oct 29 Fall 2004
Remember the microwave lab on Oct 25, Oct 27, or Nov 2 and Prelim II on Nov 4!
1. Assume that we have an EM wave with electric and magnetic fields given by:
E
x
( z, t ) = E
0
e
i ( kz − ωt )
and B
y
( z, t ) = B
0
e
i ( kz − ωt )
where E
0
and B
0
are complex and it is understood that the actual fields are the real parts of
these expressions. Let E
0
= E
r
+ iE
i
and B
0
= B
r
+ iB
i
, where E
r
, E
i
, B
r
, and B
i
are real.
(a) Write E
x
( z, t ) = a
E
cos( kz − ωt + φ
E
) and B
y
( z, t ) = a
M
cos( kz − ωt + φ
M
) where a
E
, φ
E
,
a
M
, and φ
M
are real, and determine these parameters in terms of the real and imaginary
parts of E
0
and B
0
given above.
(b) Calculate the electric and magnetic field energy densities u
E
( z, t ) and u
M
( z, t ) in terms of
the parameters determined in Part (a).
(c) Calculate the energy densities u
E
and u
M
averaged over a cycle.
(d) Now relate the answers in Part (c) to the magnitudes of the complex numbers E
0
and B
0
,
or – better yet – to these complex numbers and their complex conjugates E
∗
0
and B
∗
0
.
(e) Now find a real expression for the z component of the Poynting vector, s ( z, t ). Note that φ
E
and φ
M
are not necessarily equal, e.g. , in a conductor.
(f) Now calculate the average s , of s ( z, t ) averaged over 1 cycle.
(g) Now relate your answer in Part (f) to some product involving E
0
or B
0
or their complex
conjugates. Your answer may involve taking the real or imaginary part of some product.
These “rules” for dealing with complex amplitudes are often taught in some form in electrical
engineering courses. They are good things to have in your bag of tricks for doing calculations!
2. In the lecture we derived the expressions for the amplitudes of reflected and transmitted EM
waves that are incident perpendicular to a boundary between two materials 1 and 2 with
impedances Z
1
and Z
2
and – in the case of two non-conducting dielectrics with κ
M 1
= κ
M 2
= 1
– indices of refraction n
1
and n
2
. We found reflection and transmission coefficients,
R ≡
E
R
E
I
=
n
1
− n
2
n
1
+ n
2
T ≡
E
T
E
I
=
2 n
1
n
1
+ n
2
where E
I
, E
R
, and E
T
are the incident, reflected, and transmitted electric field amplitudes,
respectively. This is the figure that was used in the lecture to illustrate the relationships among