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Unformatted text preview: H
H s ( r ) � � � vm (2)
M m,ki , z ( � , � , z )
� m ��� H m � ki ,� a �
Scattering from cylindrical objects 47 Exact solution of the problem (TM polarization)
H TM
s ,� ˆ
( r ) � � E � vi �
0
i � � m m ��� H TM
s ,� exp � � jki , z z �
� ki , � � ( � j ) m J m � ki , � a �
(2)
H m � ki , � a � (2)
H m � ki ,� � � exp � � jm �� � �i � �
�
� j
ˆi � exp � � jki , z z �
(r ) � � � E � v
�
0
i � � m ��� ( � j ) m J m � ki , � a �
(2)
H m � ki , � a � Scattering from cylindrical objects (2)
H m � � ki ,� � � exp � � jm �� � �i � �
�
� 48 Scattering by a perfectly conducting infinite cylinder
� Remember that this is the full solution everywhere. We are
actually interested in the scattering in the far field limit. � We again use the asymptotic relationship
m
j m exp � � jk � � �
2
j� � j exp � � jk � � �
�
(2)
H m ( k � � � 1) �
exp �
� � jC0
�
�
k� �
k� �
�4� � We find that in the far field limit the fields are parallel to: E (ki , � � � 1) � ˆ
TE
s Scattering from cylindrical objects E TM
s ki , �
� ki , z
ˆ�
(ki , � � � 1) � � �
k
�k
ˆ
Same as θ �
ˆ
z�
�
49 Scattering by a perfectly conducting infinite cylinder
� More specifically: E TE
s � ˆ
� ˆ Ei0 � hi � jC0 exp( � jk � � � jk z z )
k� �
� � m ��� E TM
s ki , �
� ki , z
ˆ�
� ��
k
�k (2)
H m � � ki , � a � exp � jm ��i � � � �
�
� � � jC0 � exp( � jki ,� � � jki ,z z )
�0
ˆ
ˆ
z � � Ei � vi �
ki ,� �
�
� � m ��� Scattering from cylindrical objects �
J m � ki , � a � J m � ki ,� a �
H (2)
m �k a �
i ,� exp � jm ��i � � � �
�
�
50 Scattering by a perfectly conducting infinite cylinder
� The interesting point here is the behavior of far field as
function of the angle � , which is governed by the functions �
J m � ki ,� a � � � m ��� H � � ki , � a � � J m � ki ,� a � � m ��� � (2)
m H (2)
m �k a �
i ,� exp � jm ��i � � � �
�
� TE exp � jm ��i � � � �
�
� TM Remember that
(2)
(2)
H � m � (�1) m H m J � m � (�1) m J m Scattering from cylindrical objects 51 TE Scattering by a thin conducting cylinder
� To get some insight let us consider the limit of a thin cylinder k i , � a � ka � 1
� For small arguments (sufficient to consider positive or zero m’s) z
�
J0 � z � � �
2
(2)
H0 � � z � � � j2
�z Scattering from cylindrical objects m
z m�1
�
J m�0 � z � � � J m�1 ( z ) � J m � z � � m
z
2 (m � 1)!
m (2)
j 2m m !
(2)
(2)
H m � 0� � z � � � H m �1 ( z ) � H m � z � � �
z
� z m �1 52 TE Scattering by a thin conducting cylinder
� To the lowest order we have for the TE case
� � m ��� �
J m � ki , � a � H � � ki , � a �
(2)...
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This note was uploaded on 02/26/2013 for the course EE 25227 taught by Professor Akbabi during the Spring '13 term at Sharif University of Technology.
 Spring '13
 akbabi
 Electromagnet

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