7.8 Waves in and out of media

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General Physics part 1 / PhysPraktikum / Woan G. The Cambridge Handbook of Physics Formulas (CUP, 2000)(ISBN 0521573491)(230s)_PRef_.pdf

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152	Electromagnetism

ωt), where k = 2π/λ.

bAll ﬁeld components propagate with a further phase factor equal to

The brightness temperature of a source of speciﬁc intensity Iν is Tb = λ

Iν /(2kB).

Waves in lossless media

∂2E

electric ﬁeld

Electric ﬁeld

2E =

(7.193)

permeability (= µ0µr)

∂t2

permittivity (= 0 r)

Magnetic ﬁeld

∂2B

magnetic ﬂux density

B = µ ∂t2

(7.194)

time

η = √

(7.195)

Refractive index

rµr

wave phase speed

Wave speed

v =

(7.196)

refractive index

√

speed of light

µ0

impedance of free

Impedance of free space

Z0 =

376.7 Ω

(7.197)

space

Wave impedance

Z =

= Z0

µr

(7.198)

wave impedance

magnetic ﬁeld strength

Radiation pressurea

Radiation			N						G	momentum density
momentum		G =	N					(7.199)	N	Poynting vector

			c2
density									c	speed of light
									pn	normal pressure
Isotropic			1						u	incident radiation
Isotropic		pn =	1					(7.200)
radiation		pn =		3 u(1 + R)				(7.200)		energy density
										energy density
									R	(power) reﬂectance
										coe cient

Specular		pn = u(1 + R)cos2 θi						(7.201)	pt	tangential pressure
reﬂection		pt = u(1 − R)sinθi cosθi						(7.202)	θi	angle of incidence
									Iν	speciﬁc intensity
From an		pn =		1 + R			Iν (θ,φ)cos2 θ dΩ dν		ν	frequency
				1 + R					ν	frequency
extended
extended						c			Ω	solid angle
source	b					c		(7.203)	Ω	solid angle
source								(7.203)	θ	angle between dΩ
										and normal to plane

From a point				L(1 + R)					L	source luminosity
source,c		pn =		L(1 + R)				(7.204)		(i.e., radiant power)

					4πr2c
luminosity L					4πr2c				r	distance from source
luminosity L									r	distance from source

aOn an opaque surface.

bIn spherical polar coordinates. See page 120 for the meaning of speciﬁc intensity. cNormal to the plane.

θi

θ dΩ

(normal)

φ y

7.8 Waves in and out of media	153

Antennas

Spherical polar geometry:

x φ

Field from a short

Er =

[˙p]

[p]

cosθ

(7.205)

distance from

2π 0

r2c

+ r3

angle between r and

(

[¨p]

[˙p]

[p]

dipole

) electric

Eθ =

sinθ

(7.206)

[p]

retarded dipole

dipole in free

4π 0

r c

spacea

moment

Bφ =

µ0

[¨p]

[˙p]

sinθ

(7.207)

[p] = p(t− r/c)

4π rc

speed of light

Radiation

R =

ω2l2

2πZ0

(7.208)

dipole length ( λ)

resistance of a

6π 0c3

angular frequency

short electric

wavelength

dipole in free

impedance of free

789

ohm

(7.209)

space

ΩA

beam solid angle

ΩA = 4π Pn(θ,φ) dΩ

normalised antenna

Beam solid angle

(7.210)

power pattern

Pn(0,0) = 1

dΩ

di erential solid

angle

Forward power

G(0) =

4π

(7.211)

antenna gain

gain

ΩA

Antenna e ective

Ae =

λ2

(7.212)

e ective area

area

ΩA

Power gain of a

G(θ) =

sin2 θ

(7.213)

short dipole

Beam e ciency

e ciency =

ΩM

(7.214)

ΩM

main lobe solid

ΩA

angle

Antenna

4π Tb(θ,φ)Pn(θ,φ) dΩ

antenna

TA =

(7.215)

temperature

temperatureb

ΩA

sky brightness

temperature

154									Electromagnetism
Reﬂection, refraction, and transmissiona
parallel incidence				perpendicular incidence				E	electric ﬁeld
								E	electric ﬁeld
E i			E r		ηi			B	magnetic ﬂux density
				E i	ηi		E r	ηi	refractive index on
				E i			E r	ηi	refractive index on
	θi	θr			θi	θr			incident side
	θi	θr			θi	θr
Bi			Br					ηt	refractive index on
				Bi			Br		transmitted side
								θi	angle of incidence
			E t		ηt			θr	angle of reﬂection
			E t					θt	angle of refraction
		θt				θt		θt	angle of refraction
		θt				θt	E t
			Bt			Bt	E t
Law of reﬂection				θi = θr			(7.216)

Snell’s lawb	ηi sinθi = ηt sinθt	(7.217)
			θB Brewster’s angle of
Brewster’s law	tanθB = ηt/ηi	(7.218)	incidence for
Brewster’s law	tanθB = ηt/ηi	(7.218)	plane-polarised
			plane-polarised
			reﬂection (r = 0)

Fresnel equations of reﬂection and refraction

r	=	sin2θi − sin2θt				(7.219)
		sin2θi + sin2θt
t =			4cosθi sinθt			(7.220)
		sin2θi + sin2θt
R = r2						(7.221)
		ηt cosθt			2
T	=			t		(7.222)

			ηi cosθi
Coe cients for normal incidencec
		2
R =		(ηi − ηt)2				(7.227)
		(ηi + ηt)
T =			4ηiηt			(7.228)
		(ηi + ηt)2
R + T = 1						(7.229)

−

sin(θi − θt)

sin(θi + θt)

t =

2cosθi sinθt

sin(θi + θt)

= r2

ηt cosθt

ηi cosθi

r =

ηi − ηt

ηi + ηt

t =

2ηi

ηi + ηt

t− r = 1

(7.223)

(7.224)

(7.225)

(7.226)

(7.230)

(7.231)

(7.232)

	electric ﬁeld parallel to the plane of		electric ﬁeld perpendicular to the
	incidence		plane of incidence
R	(power) reﬂectance coe cient	r	amplitude reﬂection coe cient
T	(power) transmittance coe cient	t	amplitude transmission coe cient

aFor the plane boundary between lossless dielectric media. All coe cients refer to the electric ﬁeld component and whether it is parallel or perpendicular to the plane of incidence. Perpendicular components are out of the paper.

bThe incident wave su ers total internal reﬂection if ηi sinθi > 1.

ηt

cI.e., θi = 0. Use the diagram labelled “perpendicular incidence” for correct phases.

7.8 Waves in and out of media	155

Propagation in conducting mediaa

						σ	electrical conductivity
Electrical		nee2				ne	electron number density
conductivity	σ = neeµ =	nee2			(7.233)
				τc		τc	electron relaxation time
				τc		τc	electron relaxation time
(B = 0)			me			µ	electron mobility
(B = 0)						µ	electron mobility
						B	magnetic ﬂux density
						me	electron mass
Refractive index	η = (1 + i)					me	electron mass
Refractive index			σ		1/2	−e	electronic charge
of an ohmic			σ		(7.234)	−e	electronic charge
conductorb			4πν 0		(7.234)	η	refractive index
						0	permittivity of free space
Skin depth in an	δ = (µ0σπν)−1/2					ν	frequency
Skin depth in an					(7.235)	δ	skin depth
ohmic conductor					(7.235)	δ	skin depth
ohmic conductor						µ0	permeability of free space
						µ0	permeability of free space

aAssuming a relative permeability, µr, of 1.

bTaking the wave to have an e−iωt time dependence, and the low-frequency limit (σ 2πν 0).

Electron scattering processesa

Rayleigh

σR

Rayleigh cross section

ω4α2

radiation angular frequency

scattering

σR =

(7.236)

cross sectionb

particle polarisability

6π 0c

permittivity of free space

8π

Thomson

σT =

(7.237)

σT

Thomson cross section

4π 0mec2

electron (rest) mass

scattering

8π

classical electron radius

cross sectionc

re2 6.652 × 10−29 m2

speed of light

(7.238)

Ptot

electron energy loss rate

Inverse

urad

radiation energy density

Compton

Ptot = 3 σTcuradγ

(7.239)

Lorentz factor = [1 − (v/c)2]−1/2

scatteringd

electron speed

Compton

λ − λ =

scatteringe

(1 − cosθ)

(7.240)

λ,λ

incident & scattered wavelengths

mec

mec2

ν,ν

incident & scattered frequencies

hν =

(7.241)

photon scattering angle

1 − cosθ + (1/ε)

electron Compton wavelength

mec

cotφ = (1 + ε)tan

(7.242)

= hν/(mec2)

σKN Klein–Nishina cross section

πr2

&1 −

ε+ 1)

ln(2ε+ 1) +

σKN =

−

(7.243)

Klein–Nishina

ε2

2(2ε+ 1)2

cross section

σT

(ε 1)

(7.244)

(for a free

πr2

electron)

ln2ε+

(ε 1)

(7.245)

aFor Rutherford scattering see page 72.

bScattering by bound electrons.

dScattering from free electrons, ε 1.

e Electron energy loss rate due to photon scattering in the Thomson limit (γhν mec ).

From an electron at rest.

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