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.

The electric and magnetic fields

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)

The limiting magnetic field

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