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Unformatted text preview: lution of the problem
� We now consider the scattering of a plane wave by an infinitely
long, perfectly conducting cylinder � When the incident wave hits the
cylinder, surface currents (and
charges) are induced. � ki These currents create the ‘scattered’
field. At any point, the total electric
field is E ( r ) � Ei ( r ) � E s ( r ) Ei ( r ) � Ei0 exp � � jki � r �
Scattering from cylindrical objects 35 Exact solution of the problem
� We saw how the incident plane wave can be represented in
terms of cylindrical vector solutions � The scattered field (outside the
cylinder) can also be expanded in
terms of those solutions � But: for the scattered field we should ki use vectors with the right condition at
the infinity � �� Scattering from cylindrical objects 36 Exact solution of the problem
� We should use the Hankel function of the 2nd kind for these
waves which satisfy the radiation condition (behave as
outgoing waves at infinity) � Also, since the system is uniform in zdirection, kz is the same
as that of the incident wave: k z � ki , z . Of course the wave will
be scattered along different angles (different �’s) Es (r ) � M H
m , ki , z � � m ��� H
am M m,ki ,z ( � , � , z ) � 1
H
ˆ
� �� �� m ,ki ,z z �
�
�
k Scattering from cylindrical objects � � m ��� N H
bm N m,ki ,z ( � , � , z ) H
m , ki , z 1
H
� �� M m,ki ,z
k
37 Exact solution of the problem
� The vector functions are explicitly given by H
M m , ki , z � exp � � jm� � jki , z z � �
jm (2)
ˆ ki , � H (2)� � k � � �
m
i,�
� ˆ k � H m � ki , � � � �
�
k
�
�
H
N m, ki ,z (r ) � exp � � jm� � jki , z z � �
�
jki , z ki , � (2)
ki2,� (2)
mki , z (2)
� � ki , � � � � ˆ
ˆ
Hm
H m � ki , � � � � z 2 H m � ki , � � ��
�� ˆ
2
2
k
k�
k
�
�
� Note that we have used Scattering from cylindrical objects k � � k 2 � k z2 � k 2 � ki2z � ki ,�
,
38 Exact solution of the problem
� The next step is to choose the coefficients of the expansion
such that the condition at the surface of the cylinder is satisfied � Zero tangential component of total electric field E z ( a, � , z ) � E� (a , � , z ) � 0
� a 1st Result: bm � �vm J m � ki , � a �
H (2)
m �k a� Ez E� i ,� k J m � ki , � a �
ˆ
� � E � vi �
( � j )m exp � jm�i �
(
ki ,� H m2) � ki ,� a �
0
i Scattering from cylindrical objects 39 Exact solution of the problem
� 2nd result: am � �um �
J m � ki , � a � (2)
� H m �� � k i , � a �
�
�
�
k J m � ki ,� a �
0ˆ
� j Ei � hi
( � j )m exp � jm�i �
(
ki ,� H m2)� � ki ,� a � � � � Note that a TE incident wave excites TE polarized waves
(horizontal polarization), and a TM wave excites TM polarized
waves (vertical polarization) Scattering from cylindrical objects 40 Exact solution of the problem (TE polarization)
� TE scattering (horizontal electric field): EsTE ( r ) � � � m ��� H
am M m , ki , z ( � , � , z ) �
um J m � ki ,� a � � ��� m ��� � � ˆ jk
� E � hi
ki , �
0
i (2)
H m � � ki , � a �
� � m ��� Scattering from cylindrical objects H
M m , ki , z ( � , � , z ) �
J m � ki , � a � (2)
H m � � ki , � a � H
( � j ) m exp �...
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 Spring '13
 akbabi
 Electromagnet

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