Download ∇ Homework Assignment #9 due Halloween

ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
due Halloween
αβγδεζηθικλμνξοπρςστυφχψω
+<=>|~±×÷′″⁄⁒←↑→↓⇒⇔ ∂Δ∇∈∏∑
/1/ Problem 16.19.
(ε1 e1 + ε2 e2 )
Homework Assignment #9
∇∙E=0
∇ × E = −∂B /∂t
∇∙B=0
∇ × B = + μ ε0 ∂E /∂t
/2/ Calculate the z component of the Poynting vector, for the0 fields
given by the paraxial
∓∔⁄∗∘∙√∞∫∮∴
equations (16.115) and (16.125).
≂≃≄≅≆≠≡≪≫≤≥
/3/ Show that the “transverse magnetic” fields given by equation (16.175) obey the four
e−iωt
Maxwell equations.
/4/εCalculate the Poynting vector for the complete fields of the oscillating point-like electric
dipole.
−
/5/ and /6/ to be announced Monday
Chapter 16 : Waves in vacuum
How would you make a plane wave?
MAXWELL’S EQUATIONS IN VACUUM
Take a plane of charge, and oscillate it
back and forth in a direction tangent to
the plane.
∇∙E=0
∇ × E = −∂B /∂t
∇∙B=0
∇ × B = + μ0ε0 ∂E /∂t
--Spherical Waves--
An infinite source can make an infinite
wave.
But a small source will make spherical
waves.
Think about water waves.
This figure shows that a spherical wave,
far from the source, looks like a plane
wave.
(Analogously, the surface of the Earth
looks flat, especially in southern
Michigan.)
Zangwill shows how to construct spherical
waves in general. Today we’ll consider one
special case, which is particularly important.
The simplest scalar spherical wave
Let w(r,t) be a solution of the scalar wave
equation,
2
∇2 w − 12 ∂ w
=0
c
∂t2
Now construct an electromagnetic wave; let s be
a constant vector, and
BTM(r,t) = − s × ∇ 12 ∂w
c
∂t
2
1
∂
w
ETM(r,t) = (s ∙ ∇) ∇w − s 2
c
∂t2
Theorem: These fields obey the Maxwell
equations. “Transverse Magnetic”: s ∙ B(r,t) = 0
amplitude
Exercise: show that it satisfies
the 3D wave equation.
The electric and magnetic fields
The asymptotic fields, for large r.
Properties of the radiation fields:
❏
❏
Singular at r = 0 (IRRELEVANT)
Asymptotically ~ 1/r (VERY
RELEVANT)
❏
❏
❏
❏
❏
❏
(asymptotic)
^
Erad oscillates in the θ direction.
Brad oscillates in the φ direction.
They propagate as harmonic
waves in the r direction.
They travel at the speed of light.
|Brad| = |Erad| /c.
The Erad and Brad field
oscillations are in phase.
These are in fact the asymptotic
fields of an oscillating dipole at the
origin, p(t) = p cos(ωt) ez .
Asymptotically the fields approach a
plane wave.
The electric field vectors are in the θ
direction, i.e., tangent to the lines of
longitude;
The magnetic field vectors are in the
φ direction, i.e., tangent to the lines of
latitude.
The asymptotic Poynting vector
The total power radiated by the source
Srad = (Erad × Brad) /μ0
P = ∫ Srad ∙ er dA
where dA = r2 dΩ = r2 sin θ dθ dφ
Exercise…
Srad = er
sin2θ p2 ω4
r2
μ 0 c5
2
cos (kr−ωt)
The intensity is largest at the equator;
zero at the poles.
Exercise…
P=
8π
3
p 2 ω4
μ 0 c5
cos2(kr- ωt)
The limiting fields for small r
In the limit as r approaches 0, the electric
field has the form of an electric dipole.
cos(kr-ωt)
cos(ωt)
The limiting magnetic field
ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩ
due Halloween
αβγδεζηθικλμνξοπρςστυφχψω
+<=>|~±×÷′″⁄⁒←↑→↓⇒⇔ ∂Δ∇∈∏∑
/1/ Problem 16.19.
(ε1 e1 + ε2 e2 )
Homework Assignment #9
/2/ Calculate the z component of the Poynting vector, for the fields given by the paraxial
∓∔⁄∗∘∙√∞∫∮∴
equations (16.115) and (16.125).
≂≃≄≅≆≠≡≪≫≤≥
/3/ Show that the “transverse magnetic” fields given by equation (16.175) obey the four
e−iωt
Maxwell equations.
/4/εCalculate the Poynting vector for the complete fields of the oscillating point-like
electric dipole.
−
/5/ and /6/ to be announced Monday