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Unformatted text preview: e following general theorem � Assume that �(r) is a solution of the ‘scalar’ wave equation
solved in any coordinate system � Also consider a known vector field a(r), such that �� a (r ) � 0
Scattering from cylindrical objects 8 Vector wave equation
� Now, let us introduce the new vector fields 1
M (r ) � �� �� (r )a (r )�
k
� 1
N (r ) � �� M (r )
k It can then be shown that the 1st vector field is a solution of the
vector wave equation if � � � �� � a � 2 � � � � � � a � � 0
�
�
� For the derivation use � � � � �� a � � � � � �� � a �
� ��� � a � a� 2� � � a �� � �� � � �� �� � a
� ��� � a � a� 2� � � � a ��� � � 2 � �� � � � a
Scattering from cylindrical objects 9 Vector wave equation
� The 2nd vector field is then automatically a solution as well � These two vector fields are linearly independent from each other � They both have zero divergence � Every solution of the vector wave equation in a homogeneous
medium can be written as a combination of these vectors,
generated by different solutions of the scalar wave equation � One cannot generate more functions by taking more curls. Note
that 1
� � N ( r ) � � � � � M ( r ) � kM ( r )
k
Scattering from cylindrical objects 10 Vector wave equation
� Solution 1: 1
E (r ) � M ( r ) � � � �� ( r )a( r )�
k
1
1
j
H (r ) � �
� � E (r ) � �
� � M (r ) � N (r)
j��0
j��0
�
� Solution 2: 1
E (r ) � N (r ) � � � M (r )
k
1
j
H (r ) � �
� � N (r ) � M (r )
j��0
�
Scattering from cylindrical objects 11 Vector wave equation
� Example: choose the scalar function to be a solution of the wave
equation (Helmholtz equation) in the Cartesian system � This would be a simple scalar plane wave � 2 � k 2 �� � 0 � � � exp � � jk � r �
�
� ˆ
Take, as an example, a constant a(r), i.e., a (r ) � x ˆˆ
M (r ) � � jk � x exp � � jk � r � � � ˆ ˆˆ
N (r ) � �k � k � x exp � � jk � r �
� M ˆ
k N x These correspond to two independent polarizations Scattering from cylindrical objects 12 Vector waves in cylindrical coordinates
� For cylindrical objects, a proper choice is to use the constant
vector field ˆ
a (r ) � z
� This field satisfies all the requirements and results in 1
1
ˆ � � �� � z
ˆ
M � �� �� z
k
k 1
1
ˆ
N � 2 �� � �� � z � � 2
k
k
1
�2
k �
�2
�
ˆ
� �� � z � �z �� �
�
� �2 ˆ �
�
k � z � �� �
�
�z
�
� Scattering from cylindrical objects 13 Vector waves in cylindrical coordinates
� We next use cylindrical coordinates �� m,k z � ˆ �� m,kz
�� �ˆ �� m,kz
��� ˆ
�z �� m,kz
�z 1
ˆ �� m, kz � ˆ �� m,kz
ˆ
M m,kz � ��...
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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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