Unformatted text preview: ¿½ ï¿½ ï¿½ ln z
ï¿½ j 2m ( m ï¿½ 1)!
(2)
H mï¿½0 ï¿½ z ï¿½ ï¿½ ï¿½
ï¿½
zm ki , ï¿½ a : J m ï¿½ ki , ï¿½ a ï¿½ jï¿½
ï¿½ H (2) ï¿½ k a ï¿½ exp ï¿½ jm ï¿½ï¿½i ï¿½ ï¿½ ï¿½ï¿½ ï¿½ 2 ln ï¿½ k a ï¿½
ï¿½
ï¿½
m ï¿½ï¿½ï¿½
m
i ,ï¿½
i ,ï¿½
ï¿½ Scattering from cylindrical objects 61 TM Scattering by a thin conducting cylinder
ï¿½ Resulting far scattered field for a thin conducting cylinder ki , ï¿½
ï¿½ ki , z
Ë†ï¿½
E ( ï¿½ ,ï¿½ , z) ï¿½ E ï¿½ ï¿½
k
ï¿½k
C0
ï¿½
TM
0
Ë†
Es ,v ï¿½ ï¿½ Ei ï¿½ vi ï¿½
ki ,ï¿½ ï¿½ 2 ln ï¿½ ki ,ï¿½ a ï¿½
TM
s ï¿½ TM
s ,v ï¿½
Ë† ï¿½ exp( ï¿½ jki ,ï¿½ ï¿½ ï¿½ jki , z z )
z
ï¿½ In the lowest order approximation there is no angle
dependence for the TM scattered wave! ï¿½ The amplitude in all directions is the same Scattering from cylindrical objects 62 TM Scattering by a thin conducting cylinder
ï¿½ Let us compare the scattering strength in the two cases Ë† ï¿½ k a ï¿½2
k ï¿½ C0 E ï¿½ hi i ,ï¿½
TE
Es ,ï¿½ (ï¿½ ) ï¿½
1 ï¿½ 2cos ï¿½ï¿½ ï¿½ ï¿½i ï¿½
ki , ï¿½ 4
ki , ï¿½ ï¿½
0
i E ï¿½ TM
s ,v C0 Ë†
ï¿½ E ï¿½ vi
0
i ï¿½ ki ,ï¿½ ï¿½ 2 ln ï¿½ ki ,ï¿½ a ï¿½ For equal horizontal and vertical components of the incident
field we have EsTE (ï¿½ )
,ï¿½
EsTM
,v
Scattering from cylindrical objects 2
3k
ï¿½
ï¿½ ki,ï¿½ a ï¿½ ln ï¿½ ki ,ï¿½ a ï¿½
2 ki , ï¿½ 63 TM Scattering by a thin conducting cylinder
ï¿½ This ratio is quite small for thin cylinders ï¿½ But this result is to be expected: a
vertically polarized incident electric field
has a component along the â€˜wireâ€™ and
easily induces electric currents along the EiTE wire. These currents, in turn, generate the
scattered field.
ï¿½ EiTM A horizontally polarized incident wave has
no longitudinal components and cannot
excite such currents Scattering from cylindrical objects 64 TE Scattering by a thick conducting cylinder
ï¿½ Now, let us investigate the other limit, that of a thick conducting
cylinder which satisfies ki ,ï¿½ a ï¿½ a k 2 ï¿½ ki2,z ï¿½ 1
ï¿½ Remember: scattered electric field
ï¿½ E (r ) ï¿½ ï¿½ ï¿½
TE
s ï¿½ E TE
s m ï¿½ï¿½ï¿½ ï¿½
um J m ï¿½ ki ,ï¿½ a ï¿½
(2)
H m ï¿½ ï¿½ ki , ï¿½ a ï¿½ H
M m ,ki , z ( ï¿½ , ï¿½ , z ) Far field limit ï¿½ Ë†
ï¿½ Ë† E ï¿½ hi
0
i ï¿½ jC0 exp( ï¿½ jki ,ï¿½ ï¿½ ï¿½ jk z z ) Scattering from cylindrical objects ki , ï¿½ ï¿½ ï¿½ ï¿½ m ï¿½ï¿½ï¿½ ï¿½
J m ï¿½ ki , ï¿½ a ï¿½ H ï¿½ ï¿½ ki , ï¿½ a ï¿½
(2)
m exp ï¿½ jm ï¿½ï¿½i ï¿½ ï¿½ ï¿½ ï¿½
ï¿½
ï¿½
65 TE Scattering by a thick conducting cylinder
ï¿½ Let us approximate the denominator
(2)
(2)
z ï¿½ 1 ï¿½ H m ï¿½ ï¿½ z ï¿½ ï¿½ ï¿½ jH m ( z ) ï¿½ ï¿½ j m ï¿½1 2
jï¿½ ï¿½
ï¿½
exp ï¿½ ï¿½ jz ï¿½
ï¿½
ï¿½z
4ï¿½
ï¿½ ï¿½
J m ï¿½ ki , ï¿½ a ï¿½ ï¿½ ki , ï¿½ a
jï¿½ ï¿½
ï¿½
ï¿½ H (2)ï¿½ k a exp ï¿½ jm ï¿½ï¿½i ï¿½ ï¿½ ï¿½ ï¿½ ï¿½ j 2 exp ï¿½ jki ,ï¿½ a ï¿½ 4 ï¿½
ï¿½
ï¿½
ï¿½
ï¿½
m ï¿½ï¿½ï¿½
m ï¿½ i ,ï¿½ ï¿½
ï¿½ ï¿½ ï¿½ m ï¿½ï¿½ï¿½ ï¿½ ï¿½
( ï¿½ j )m J m ï¿½ ki ,ï¿½ a ï¿½ exp ï¿½ jm ï¿½ï¿½i ï¿½ ï¿½ ï¿½ ï¿½
ï¿½
ï¿½ Next, use the relation ï¿½ j cos ï¿½ exp ï¿½ ï¿½ jz cos ï¿½ ï¿½ ï¿½ ï¿½ ï¿½ m ï¿½ï¿½ï¿½ Scattering from cylindrical objects ï¿½
(ï¿½ j ) m J m ( z ) exp ï¿½ ï¿½ jmï¿½ ï¿½
66 TE Scattering by a thick conducting cylinder
ï¿½ The TE far scattered field becomes ï¿½ Ë†
E ( ï¿½ , ï¿½ , z ) ï¿½ ï¿½ Ë† Ei0 ï¿½ hi
TE
s ï¿½ a
cos ï¿½ï¿½ ï¿½ ï¿½i ï¿½
ï¿½ ï¿½ exp ï¿½ ï¿½ jki ,ï¿½ ( ï¿½ ï¿½ a ) ï¿½ jki ,ï¿½ a cos ï¿½ï¿½ ï¿½ ï¿½i ï¿½ ï¿½ jki , z z ï¿½
ï¿½
ï¿½
ï¿½ Remember that this result is also based on the farfield behavior
of the vector solutions (used here), which in turn, was based on
the asymptotic behavior of the Hankel function when kï¿½ ï¿½ ï¿½ 1 Scattering from cylindrical objects 67 Numerical results for a thick cylinder
ï¿½ So let us again look at some numerical results for the azimuthal
component of the electric field y
ï¿½ We again plot this function when
Ei ï¿½i ï¿½ 0 ï¿½ ki ï¿½ ï¿½ ki , x ,0, ki , z ï¿½ , ki , ï¿½ ï¿½ ki , x Ë†
Ë†
hi ï¿½ Ë† ï¿½ y
i EsTE
,ï¿½ ï¿½
ki , x x Ë†
Ei0 ï¿½ hi ï¿½ Ei0, y Ë†
Ë†
Ei (r ) ï¿½ Ei0, y y exp ï¿½ ï¿½ jki ï¿½ r ï¿½ ï¿½ Ei0, y y exp ï¿½ ï¿½ jki , x x ï¿½ jki , z z ï¿½ Scattering from cylindrical objects 68 Numerical results for a thick cylinder
ï¿½ The amplitude profile is shown below Ei ki , x a...
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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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