Reading Assignment : Chapter 20, Retardation and Radiation

Reading Assignment : Chapter 20, Retardation and Radiation
Homework Assignment #12 due Friday November 21
ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦ
∇∙E = ρ/ε0
;
∇∙B = 0
/1/ Why did Sir Edward Appleton get the Nobel Prize? (Be precise.)
∇×E = −∂B/∂t ;
/2/ΧΨΩ
Problem 18.18.
/3/αβγδεζηθικλμνξοπρςστυφχψω
(A) Derive Equation 18.133, starting from the Kramers-Kronig
(B) ∂E/∂t
Now do
∇×B = − μ0 jrelations.
+ (1/c2)
Problem 18.20.
+<=>|~±×÷′″⁄⁒←↑→↓⇒⇔
μ0 ε0
/4/ Principal Value.
∂Δ∇∈∏∑
−∂ρ/∂t
What
is ∫-11 dx/x? What is P.V.∫-11 dx/x? What is ( ∫-1-ε+∇∙j
∫2ε1 ) =
dx/x
in the limit ε → 0?
e + ε2 eLight?
)
/5/(ε
Faster
1 1 than
2
Let Q(x,t) be a field component propagating as a wave1in the
x > 0; and assume Q = 0 in this
∂2domain
ψ
domain for t < 0. Write Q(x,t) = ∫ Q^(ω) exp{ i (n(ω) x - c2t) ω/c2} dω/2π , where n(ω) is the complex
∓∔⁄∗∘∙√∞∫∮∴
c ∂t
index of refraction. (A) Evaluate the signal Q(x,t) for x > ct. (B) Many systems have frequency bands
≂≃≄≅≆≠≡≪≫≤≥
where
the phase velocity is greater than c; i.e. Re n(ω) < 1. How does that affect the result in (A)?
∇ 2 ψ − = − f(r,t)
/6/＾＾
Absorption band.
Suppose
ÊÊÊ a material absorbs electromagnetic radiation only in the band ω1 < ω < ω2. For simplicity
assume ε’’ ( = Im ε^) is a constant η in this frequency range, and is ε’’ = 0 outside the range. Use the
ŝ
Kramers-Kronig relations to calculate ε’ ( = Re ε^). Show a graph of this function and compare the
∞−−−−−∞
result
to the Lorentz single-oscillator model.
Two additional problems : Problem 20.1 and Problem 20.2.
Radiation from a source with finite
dimensions
Retardation
Theory of Radiation.
Start with the inhomogeneous wave
equation
Then B = ∇ × A and ∇ × E = −∂B/∂t.
●
. j(r,t) has finite spatial extent; L
●
. far
from the source, i.e., for r >> L, we
may approximate A, B, E etc by the
“asymptotic fields”;
●
.B
●
. Notation:
2
~ 1/r and E ~ 1/r; thus S ~ 1/r are
required by conservation of energy
Brad and Erad are the
asymptotic fields.
A(r,t) = μ0 ∫ G(+)( r,t | r’t’ ) j(r’,t’) d3r’ dt’
∫
δ(t−t’−|r−r’|/c)
4π|r−r’|
j(r’,t−|r−r’|/c) d3r’
4π|r−r’|
This is exact, given the current density.
A(r,t) = μ0
Now consider the radiation zone: |r−r’| >> L
1 / | r−r’ | ~ 1 /r ;
but use a better approximation in the delta function
| r−r’ | ~ r − er ∙ r’ ;
The asymptotic potential is
Arad(r,t) = (μ0 /4πr ) ∫ j(r’,t−τ) d3r’
where τ = τ(r,r’) = ( r − er ∙ r’ ) /c .
Arad(r,t) = (μ0 /4πr ) ∫ j(r’,t−τ) d3r’ ;
τ = τ(r,r’) = ( r − er ∙ r’ ) /c .
Radiation
T = period of oscillation of the source
Now calculate the radiation fields,
Brad and Erad.
Final results:
.
●
Erad, Brad and er form an orthogonal triad.
●
.
|Erad| = c |Brad|
●
.
just like a plane wave on the distant sphere.
Fields in the radiation zone
Example: Larmor’s formula
Consider a nonrelativistic particle moving
with velocity v(t) and acceleration a(t).
Radiation pattern
from an
accelerating particle
Calculate the power radiated by the
particle.
The relativistic formula:
Integrated power =
Application: synchrotron radiation.
Multipole radiation
Define α(r,t) by (∂/∂t) ∫ j(r’,t−τ) d r’ ,
where τ = ( r − er∙r’ )/c.
Approximate
E1
j(r’,t−τ) ≅ j(r’,t−r/c) +
M1 & E2
(er∙r’ /c) [(∂/∂t) j(r’,t−r/c)]
3
Electric dipole (E1) radiation
The continuity equation implies
∫
j(r’,t’) d3r’ = d p(t’)
dt
..
αE1(r,t) = pret
(dP/dΩ)E1
(e. dipole moment)
..
= μ0/(16π2c) | er × pret |2
Radiation pattern
produced by a
vertically oriented
dipole at the
origin.
Magnetic dipole (M1) and electric
quadrupole (E2) radiation
m(t’) = ½ ∫ r’ x j(r’,t’) d3r’
..
αM1(r,t) = mret x er /c
..
(dP/dΩ)M1 = μ0/(16π2c3) | er × mret |2
..
αE2(r,t) = Qret ∙ er /c
2 2
..
(dP/dΩ)E2 = μ0/(16π c ) | er × (Qret ∙ er ) |2
Electric dipole dominates unless it is 0.
Electric quadrupole radiation:
Left: an axial quadrupole; two counter dipoles on the z axis.
Right: a lateral quadrupole; two counter dipoles on the xy plane.