Science > Physics > Electrodymanics questions 5: Scattering from a cylinder
| Topic: |
Science > Physics |
| User: |
"Erland Gadde" |
| Date: |
14 Nov 2004 12:04:01 PM |
| Object: |
Electrodymanics questions 5: Scattering from a cylinder |
This problem is driving me nuts. Any help would be greatly
appreciated.
It is discussed in Leonard Eyges: "The Classical Electromagnetic
field", sections 15.7 and 15.8 (1980 Dover edition).
Below, all fields and currents are time harmonic and we don't write
out the time varying factor e^(-iwt). Also, it is tacitly understood
that the real part should always be taken of any complex entity. All
sums below are taken from -infinity to +infinity. The wave number is k
= w/c. Cylindrical coordinates (r,v,z) are used, and gaussian units.
The problem: An incoming electromagnetic wave with electric field
e^(ikr*cos v) polarized the z-direction is scattered against a
perfectly conducting, infinitely long, circular cylinder with radius
a, centered at the z-axis.
Calculate the scattered field and the resulting surface current on the
cylinder.
In section 15.7, Eyges uses Bessel and Hankel functions J_m and H_m,
rewrites the incoming electric field as
e^(ikr*cos v) = sum_m i^m*J_m(kr)*e^(imv), and calculates the
scattered electric field as -sum_m
i^m*(J_m(ka)/H_m(ka))*H_m(kr)*e^imv, in the z-direction (no electric
field components in any other dierction).
In section 15.8, the same field is calculated with integral equations,
and then the surface current J_S (don't confuse with Bessel
functions!) is calculated. Also J_S has only a z-component:
J_S(v)=(c/(2*pi^2*k*a))*sum_l i^l*e^(i*l*v)/H_l(ka)
But then, Eyges remarks that this result can be derived from the
previous result in 15.7 and the equation J_S=(c/4*pi)*n x B.
(Here, J_S is a the surface current _vector_, n is the outgoing normal
(from the cylinder surface) and B is the magnetic field vector(leaving
out the time factor e^(-iwt)).
But I get a completely different result for J_S(v) than the above,
using the latter method! Letting E be the total electric field vector,
obtained by adding the incoming and scattered fields above, the
formula curl E = ik*B, in cylindrical coordinates, gives the
v-component of B as -1/(i*k)*dE_z/dr =
sum_m i^(m+1)*(J'_m(kr)-(J_m(ka)/H_m(ka))*H'_m(kr))*e^imv. B has no
z-component, but an r-component that is unimportant. Hence J_S has
only a z-component: (c/(4*pi))*sum_m
i^(m+1)*(J'_m(kr)-(J_m(ka)/H_m(ka)*H'_m(kr)))*e^imv, and this
completely different from the expression for J_S(v) above!!!! (or is
it? Do Bessel and Hankel functions have some strange property that
make the two expressions equal??)
This seems as an outright contradiction, and it's driving me nuts. It
can't just be that Eyges has made a mistake, becaouse I don'r find any
error in his calculations. Have I made some fundamental error in my
short calculation of J_s above???
As I said, any help would be appreciated.
Regards,
Erland Gadde
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| User: "Yongtao Yang" |
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| Title: Re: Electrodymanics questions 5: Scattering from a cylinder |
14 Nov 2004 12:21:36 PM |
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"Erland" == Erland Gadde <erland@bredband.net> writes:
Erland> In section 15.7, Eyges uses Bessel and Hankel functions J_m and H_m,
Erland> rewrites the incoming electric field as
Erland> e^(ikr*cos v) = sum_m i^m*J_m(kr)*e^(imv), and calculates the
Erland> scattered electric field as -sum_m
Erland> i^m*(J_m(ka)/H_m(ka))*H_m(kr)*e^imv, in the z-direction (no electric
Erland> field components in any other dierction).
Erland> In section 15.8, the same field is calculated with integral equations,
Erland> and then the surface current J_S (don't confuse with Bessel
Erland> functions!) is calculated. Also J_S has only a z-component:
Erland> J_S(v)=(c/(2*pi^2*k*a))*sum_l i^l*e^(i*l*v)/H_l(ka)
Erland> But then, Eyges remarks that this result can be derived from the
Erland> previous result in 15.7 and the equation J_S=(c/4*pi)*n x B.
Erland> (Here, J_S is a the surface current _vector_, n is the outgoing normal
Erland> (from the cylinder surface) and B is the magnetic field vector(leaving
Erland> out the time factor e^(-iwt)).
Erland> But I get a completely different result for J_S(v) than the above,
Erland> using the latter method! Letting E be the total electric field vector,
Erland> obtained by adding the incoming and scattered fields above, the
Erland> formula curl E = ik*B, in cylindrical coordinates, gives the
Erland> v-component of B as -1/(i*k)*dE_z/dr =
Erland> sum_m i^(m+1)*(J'_m(kr)-(J_m(ka)/H_m(ka))*H'_m(kr))*e^imv. B has no
Erland> z-component, but an r-component that is unimportant. Hence J_S has
Erland> only a z-component: (c/(4*pi))*sum_m
Erland> i^(m+1)*(J'_m(kr)-(J_m(ka)/H_m(ka)*H'_m(kr)))*e^imv, and this
Erland> completely different from the expression for J_S(v) above!!!! (or is
Erland> it? Do Bessel and Hankel functions have some strange property that
Erland> make the two expressions equal??)
Yes, here you have r=a for the second expression of J_s and the
Wronskian of Bessel functions says(Assuming you are using the second
kind Hankel function):
J_m(ka)*H'_m(ka) - H_m(ka)*J'_m(ka) = -i*2/(pi*k*a)
So both of the two solution agree with each other.
HTH
--
Yongtao Yang
email:
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| User: "Erland Gadde" |
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| Title: Re: Electrodymanics questions 5: Scattering from a cylinder |
17 Nov 2004 08:21:50 AM |
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Yongtao Yang <yangyongtao@yahoo.com> wrote in message news:<t74fz3c2swf.fsf@cempc10.elmagn.chalmers.se>...
"Erland" == Erland Gadde <erland@bredband.net> writes:
Yes, here you have r=a for the second expression of J_s and the
Wronskian of Bessel functions says(Assuming you are using the second
kind Hankel function):
J_m(ka)*H'_m(ka) - H_m(ka)*J'_m(ka) = -i*2/(pi*k*a)
So both of the two solution agree with each other.
Thanks! I didn't know about this wronskian property for Bessel
functions, but I found some material on the net. Actually, it is the
first kind of Hankel function that is used, but that is right,
according to http://en.wikipedia.org/wiki/Bessel_function
Regards,
Erland Gadde
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