Field of a system of charges at large distances
Field of a system of charges at
large distances
LL2 section 66
Exact retarded potentials (62.9) and (62.10)
Approximate form at
large distances
r.n
is neglected in the
denominators
But not in the retarded times
t
r
= t - R/c = t – R
0
/c +
r.n
/c
because
and
j
might change a lot during the time
r.n
/c
even if that time is short compared to R
0
/c.
Note that both
and
A
go as
1/R
0
“Wave-zone”. Field looks like a plane
wave in a small region.
Large distances
Need R
0
>> size of the distribution
Need R
0
>>
.
Plane waves:
Evaluated at the
retarded time
t’ = t - R/c
Since A goes as 1/R
0
, the field in the wave zone also goes as 1/R
0
We need only
A
, not
, to find
the fields in the wave zone.
(47.3)
, where t’-
r
0
(t’).
n
/c = t – R
0
/c
determines t’
If the source is a single point charge, use the Lienard-Wiechert potentials
(63.5), exact
At large distances R R
0
(constant)
Then equation (63.1) for
the retarded time t’
Energy flux density
Energy Flux = amount of energy per unit
time passing through area df = R
0
2
do
Units of dI = speed x energy density x area = Power
Since H
2
~ 1/R
0
2
, dI is the same for all distances if t - R
0
/c is the same for them
Solid angle
LL calls this “Intensity”
dI, which is usually
reserved for <S>
t
Spectral resolution of radiated waves
Substitute the
monochromatic parts
from the Fourier
expansion
Time-dependent fields
What is the total energy radiated from a non-periodic event, e.g. a collision?
dE
n
= energy radiated into d
o
with frequencies in the range
to
+ d
From (49.8) for any time dependent field f
From (66.6), the radiated intensity is
The part of the total
radiated energy that is
within the interval d
/2
th
coefficient in Fourier expansion of current density
Substitute into (66.7)
Line integral along
trajectory of particle
The field of a system of charges at large distances is explored, focusing on the approximate form, wave zone, plane waves, Lienard-Wiechert potentials, energy flux density, spectral resolution of radiated waves, and total energy radiated from non-periodic events like collisions. The content delves into exact and approximate calculations, providing insights into electromagnetic field behavior in different scenarios.
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Presentation Transcript
Field of a system of charges at large distances LL2 section 66
Approximate form at large distances Exact retarded potentials (62.9) and (62.10) r.n is neglected in the denominators But not in the retarded times tr = t - R/c = t R0/c + r.n/c because and j might change a lot during the time r.n/c even if that time is short compared to R0/c. Note that both and A go as 1/R0
Large distances Wave-zone . Field looks like a plane wave in a small region. Need R0 >> . Need R0 >> size of the distribution
Plane waves: We need only A, not , to find the fields in the wave zone. (47.3) Evaluated at the retarded time t = t - R/c Since A goes as 1/R0, the field in the wave zone also goes as 1/R0
If the source is a single point charge, use the Lienard-Wiechert potentials (63.5), exact At large distances R R0 (constant) Then equation (63.1) for the retarded time t , where t -r0(t ).n/c = t R0/c determines t
Energy flux density Energy Flux = amount of energy per unit time passing through area df = R02do LL calls this Intensity dI, which is usually reserved for <S>t Solid angle Since H2 ~ 1/R02, dI is the same for all distances if t - R0/c is the same for them Units of dI = speed x energy density x area = Power
Substitute the monochromatic parts from the Fourier expansion Time-dependent fields
What is the total energy radiated from a non-periodic event, e.g. a collision? dEn = energy radiated into do with frequencies in the range to + d
From (49.8) for any time dependent field f From (66.6), the radiated intensity is The part of the total radiated energy that is within the interval d /2
th coefficient in Fourier expansion of current density Substitute into (66.7)
Line integral along trajectory of particle
