
B.Thide - Electromagnetic Field Theory
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7.3 THE RADIATION FIELDS |
103 |
the total electric field:
E(t;x) = Z 1d!E!(x) e i!t |
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= |
4 "0 ZV |
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(tret0 xx0 |
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d3x0 |
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0} |
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d3x |
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[j{zret0 0 |
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Retarded Coulomb field |
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4 "0c ZV |
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x x0 |
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j |
x0)] |
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x0)} |
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[j(tret0 ;x0){z(x |
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(7.23) |
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Intermediate field |
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ZV |
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4 "0c |
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[j˙(tret0 ;x0) |
{z(x x0)] (x x0) 3} |
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Intermediate field |
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4 "0c2 |
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{z |
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} |
Radiation field
Here, the first term represents the retarded Coulomb field and the last term represents the radiation field which carries energy over very large distances. The other two terms represent an intermediate field which contributes only in the near zone and must be taken into account there.
With this we have achieved our goal of finding closed-form analytic expressions for the electric and magnetic fields when the sources of the fields are completely arbitrary, prescribed distributions of charges and currents. The only assumption made is that the advanced potentials have been discarded; recall the discussion following Equation (3.35) on page 45 in Chapter 3.
7.3 The Radiation Fields
In this section we study electromagnetic radiation, i.e., the part of the electric and magnetic fields fields, calculated above, which are capable of carrying energy and momentum over large distances. We shall therefore make the assumption that the observer is located in the far zone, i.e., very far away from the source region(s). The fields which are dominating in this zone are by definition the radiation fields.
From Equation (7.12) on page 100 and Equation (7.23) above, which give
Draft version released 31st October 2002 at 14:46. |
Downloaded from http://www.plasma.uu.se/CED/Book |
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104 |
ELECTROMAGNETIC FIELDS FROM ARBITRARY SOURCE DISTRIBUTIONS |
the total electric and magnetic fields, we obtain |
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˙ |
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Brad(t;x) = Z 1d!Brad!(x) e i!t = |
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ZV |
j(tret0 |
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d3x0 |
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4 c |
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(7.24a)
Erad(t;x) = Z 1d!Erad |
!(x) e i!t |
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4 "0c2 |
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where |
@t t=tret0 |
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j˙(tret0 ;x0) |
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def |
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˙ 0 0 0 0
[j(tret;x ) (x x )] (x x ) d3x0 jx x0j3
(7.24b)
(7.25)
Instead of studying the fields in the time domain, we can often make a spectrum analysis into the frequency domain and study each Fourier component separately. A superposition of all these components and a transformation back to the time domain will then yield the complete solution.
The Fourier representation of the radiation fields Equation (7.24a) and Equation (7.24b) above were included in Equation (7.11) on page 100 and Equation (7.22) on page 102, respectively and are explicitly given by
B! (x) = |
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2 Z 1 dt B |
rad |
(t;x) e |
i!t |
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rad |
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= i 4 0 ZV |
! |
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x0 |
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eikjx x0j d3x0 |
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= i 40 |
ZV j!x |
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eikjx x0j d3x0 |
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(x ) |
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E! (x) = |
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2 Z 1 dt E |
rad |
(t;x) e |
i!t |
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rad |
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= i 4 "0c ZV |
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[j!(x0) (x |
x00 |
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eikjx x0j d3x0 (7.26b) |
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4 "0c ZV |
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jx x0j2 |
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[j!(x0) k] |
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eik x |
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ˆ 0 j j where we used the fact that k = kk = k(x x ) x x` .
=
If the source is located inside a volume V near x0 and has such a limited spatial extent that max jx0 x0j jx x0j, and the integration surface S , centred
Downloaded from http://www.plasma.uu.se/CED/Book |
Draft version released 31st October 2002 at 14:46. |
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