ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩαβγδεζηθικλμνξοπρςστυφχψω+<=>|~±×÷′″⁄⁒←↑→↓⇒⇔ ∂Δ∇∈∏∑(ε1 e1 + ε2 e2 )
∓∔⁄∗∘∙√∞∫∮∴
≂≃≄≅≆≠≡≪≫≤≥
e−iωt
ε
−
∇ ∙ E = 0∇ × E = −∂B /∂t∇ ∙ B = 0∇ × B = + μ0ε0 ∂E /∂t
Homework Assignment #9 due Halloween
/1/ Problem 16.19.
/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 Maxwell equations.
/4/ Calculate 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
MAXWELL’S EQUATIONS IN VACUUM∇ ∙ E = 0∇ × E = −∂B /∂t∇ ∙ B = 0∇ × B = + μ0ε0 ∂E /∂t
--Spherical Waves--
How would you make a plane wave?
Take a plane of charge, and oscillate it back and forth in a direction tangent to the plane.
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.)
1 ∂2w c2 ∂t2
Zangwill shows how to construct spherical waves in general. Today we’ll consider one special case, which is particularly important.
Let w(r,t) be a solution of the scalar wave equation,
∇2 w − = 0
Now construct an electromagnetic wave; let s be a constant vector, and
BTM(r,t) = − s × ∇
ETM(r,t) = (s ∙ ∇) ∇w − s
Theorem: These fields obey the Maxwell equations. “Transverse Magnetic”: s ∙ B(r,t) = 0
The simplest scalar spherical wave
1 ∂2w c2 ∂t2
1 ∂w c2 ∂t
amplitude
Exercise: show that it satisfies the 3D wave equation.
Properties of the radiation fields:
❏ Singular at r = 0 (IRRELEVANT)❏ Asymptotically ~ 1/r (VERY
RELEVANT)❏ 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 .
The asymptotic fields, for large r.
(asymptotic) ^
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 total power radiated by the source
P = ∫ Srad ∙ er dA
where dA = r2 dΩ = r2 sin θ dθ dφ
Exercise…
P = cos2(kr- ωt)
The asymptotic Poynting vector
Srad = (Erad × Brad) /μ0
Exercise…
Srad = er cos2(kr−ωt)
The intensity is largest at the equator;zero at the poles.
sin2θ p2 ω4
r2 μ0 c5 8π p2 ω4
3 μ0 c5
The limiting fields for small rIn the limit as r approaches 0, the electric field has the form of an electric dipole.
cos(kr-ωt)
cos(ωt)
ΑΒΓΔΕΖΗΘΙΚΛΜΝΞΟΠΡΣΤΥΦΧΨΩαβγδεζηθικλμνξοπρςστυφχψω+<=>|~±×÷′″⁄⁒←↑→↓⇒⇔ ∂Δ∇∈∏∑(ε1 e1 + ε2 e2 )
∓∔⁄∗∘∙√∞∫∮∴
≂≃≄≅≆≠≡≪≫≤≥
e−iωt
ε
−
Homework Assignment #9 due Halloween
/1/ Problem 16.19.
/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 Maxwell equations.
/4/ Calculate the Poynting vector for the complete fields of the oscillating point-like electric dipole.
/5/ and /6/ to be announced Monday