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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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118	Thermodynamics

5.7 Radiation processes

Radiometrya

Radiant energyb

Qe = Le cosθ dA dΩ dt J

(5.145)

∂Qe

(5.146)

Radiant ﬂux

Φe =

∂t

(“radiant power”)

= Le cosθ dA dΩ

(5.147)

Radiant energy

We =

∂Qe

Jm−3

(5.148)

densityc

∂V

Me =

∂Φe

Wm−2

(5.149)

∂A

Radiant exitance

= Le cosθdΩ

(5.150)

Ee =

∂Φe

Wm−2

(5.151)

∂A

Irradiance

= Le cosθdΩ

(5.152)

Ie =

∂Φe

Wsr−1

(5.153)

∂Ω

Radiant intensity

Le cosθ dA

(5.154)

Le =

∂2Φe

Wm−2 sr−1

(5.155)

Radiance

cosθ

dAdΩ

∂Ie

(5.156)

cosθ

∂A

Qe radiant energy

Le radiance (generally a function of position and direction)

θangle between dir. of dΩ and normal to dA

Ωsolid angle

Aarea

ttime

Φe radiant ﬂux

We radiant energy density

dV di erential volume of propagation medium

Me radiant exitance

(normal)	θ	dΩ
	θ

dA x

φ y

Ee	irradiance
Ie	radiant intensity

aRadiometry is concerned with the treatment of light as energy.

bSometimes called “total energy.” Note that we assume opaque radiant surfaces, so that 0 ≤ θ ≤ π/2. cThe instantaneous amount of radiant energy contained in a unit volume of propagation medium. dPower per unit area leaving a surface. For a perfectly di using surface, Me = πLe.

ePower per unit area incident on a surface.

5.7 Radiation processes	119

Photometrya

Luminous energy

Qv = Lv cosθ dA dΩ dt lms

(5.157)

(“total light”)

Φv =

∂Qv

lumen (lm)

(5.158)

∂t

Luminous ﬂux

= Lv cosθ dA dΩ

(5.159)

Luminous

Wv =

∂Qv

lmsm−3

(5.160)

densityb

∂V

Mv =

∂Φv

(lmm−2)

(5.161)

Luminous

∂A

exitancec

= Lv cosθdΩ

(5.162)

Ev =

∂Φv

lmm−2

(5.163)

Illuminance

∂A

(“illumination”)d

= Lv cosθdΩ

(5.164)

∂Φv

(5.165)

Luminous

Iv =

∂Ω

intensitye

Lv cosθ dA

(5.166)

Luminance

Lv =

∂2Φv

cdm−2

(5.167)

cosθ

dAdΩ

(“photometric

∂Iv

brightness”)

(5.168)

= cosθ ∂A

Luminous e cacy

K =

Φv

lmW−1

(5.169)

Φe

Luminous

V (λ) =

K(λ)

(5.170)

e ciency

Kmax

Qv luminous energy

Lv luminance (generally a function of position and direction)

θangle between dir. of dΩ and normal to dA

Ωsolid angle

Aarea

ttime

Φv luminous ﬂux

Wv luminous density

Vvolume

Mv luminous exitance

z			5
(normal)	θ	dΩ	5
	θ

dA x

φ y

Ev illuminance

Iv luminous intensity

Kluminous e cacy

Le	radiance
Φe	radiant ﬂux
Ie	radiant intensity

Vluminous e ciency

λwavelength

Kmax spectral maximum of

K(λ)

aPhotometry is concerned with the treatment of light as seen by the human eye.

bThe instantaneous amount of luminous energy contained in a unit volume of propagating medium. cLuminous emitted ﬂux per unit area.

dLuminous incident ﬂux per unit area. The derived SI unit is the lux (lx). 1lx = 1lmm−2. eThe SI unit of luminous intensity is the candela (cd). 1cd = 1lmsr−1.

120	Thermodynamics

Radiative transfera

Flux density

Fν = Iν (θ,φ)cosθ dΩ Wm−2 Hz−1

(through a

plane)

(5.171)

Mean intensityb

Jν =

Iν (θ,φ) dΩ

Wm−2 Hz−1

(5.172)

4π

Spectral energy

Iν (θ,φ) dΩ

Jm−3 Hz−1

densityc

uν =

(5.173)

Speciﬁc

Wkg−1 Hz−1 sr−1

emission

jν =

(5.174)

coe cient

Gas linear

absorption

αν = nσν

m−1

(5.175)

coe cient

lν

(αν 1)

Opacityd

κν =

αν

kg−1 m2

(5.176)

Optical depth

τν = κν ρ ds

(5.177)

1 dIν

= −κν Iν + jν

(5.178)

Transfer

equatione

dIν

= −αν Iν + ν

(5.179)

Kirchho ’s lawf

Sν ≡

jν

(5.180)

κν

αν

Emission from

Iν = Sν (1 − e−τν )

a homogeneous

(5.181)

medium

	z		dΩ

	(normal)	θ

	x		y

	φ
Fν	ﬂux density
Iν	speciﬁc intensity
	(Wm−2 Hz−1 sr−1)
Jν	mean intensity
uν	spectral energy density

Ωsolid angle

θangle between normal and direction of Ω

jν	speciﬁc emission
	coe cient
ν	emission coe cient
	(Wm−3 Hz−1 sr−1)

ρdensity

αν linear absorption coe cient

nparticle number density

σν	particle cross section
lν	mean free path
κν	opacity
τν	optical depth, or
	optical thickness
ds	line element

Sν	source function

aThe deﬁnitions of these quantities vary in the literature. Those presented here are common in meteorology and astrophysics. Note particularly that the ambiguous term speciﬁc is taken to mean “per unit frequency interval” in the case of speciﬁc intensity and “per unit mass per unit frequency interval” in the case of speciﬁc emission coe cient. bIn radio astronomy, ﬂux density is usually taken as S = 4πJν .

cAssuming a refractive index of 1. dOr “mass absorption coe cient.” eOr “Schwarzschild’s equation.”

fUnder conditions of local thermal equilibrium (LTE), the source function, Sν , equals the Planck function, Bν (T ) [see Equation (5.182)].

5.7 Radiation processes	121

Blackbody radiation

1050

1010

)

1010 K

)

1040

−1

νm(T ) = c/λm(T )

109 K

λm(T )

−1

105

−1

108 K

109 K

108 K

−2

107 K

−2

mW/

10−5

106 K

/mW

1020

107 K

106 K

105 K

1010

105 K

10−10

104 K

10−10

104 K

2.7K

brightness

10−20

brightness

10−15

103 K

100K

2.7K

10−20

10−12 10−10

10−8

10−6 10−4 10−2

106 108 1010

1012 1014

1016 1018

1020

1022

10−14

102

frequency (ν/Hz)

wavelength (λ/m)

Bν (T ) =

2hν3

exp

hν

−1

(5.182)

Bν

surface brightness per

unit frequency

Planck

−

(Wm

−

2 Hz

1 sr

−

dν

Bλ

−

functiona

Bλ(T ) = Bν (T )

dλ

(5.183)

surface brightness per

unit wavelength

hc2

exp

(Wm−2 m−1 sr−1)

= 2λ5

λkT − 1 −

(5.184)

Planck constant

Spectral energy

uν (T ) =

4π Bν (T )

Jm−3 Hz−1

(5.185)

speed of light

Boltzmann constant

density

uλ(T ) =

4π Bλ(T )

Jm−3 m−1

(5.186)

temperature

uν,λ

spectral energy density

Rayleigh–Jeans

Bν (T ) =

2kT

(5.187)

law (hν kT )

λ2

Wien’s law

2hν

−hν

Bν (T ) =

exp

(5.188)

(hν kT )

Wien’s

5.1 10−3 mK

for Bν

λm

wavelength of

displacement

λmT = "2.9 × 10−3 mK

for Bλ

(5.189)

maximum brightness

law

∞

Stefan–

M = π

Bν (T ) dν

(5.190)

exitance

Boltzmann

Stefan–Boltzmann

law

2π5k4

Wm−

constant (

= 15c2h3 T

= σT

(5.191)

5.67 × 10−

Wm−

K−

)

Energy density

u(T ) =

4 σT 4

Jm−3

(5.192)

energy density

Greybody

M = σT

= (1 − A)σT

(5.193)

mean emissivity

albedo

aWith respect to the projected area of the surface. Surface brightness is also known simply as “brightness.” “Speciﬁc intensity” is used for reception.

bSometimes “Stefan’s law.” Exitance is the total radiated energy from unit area of the body per unit time.

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	z		dΩ

	(normal)	θ

	x		y

	φ
Fν	ﬂux density
Iν	speciﬁc intensity
	(Wm−2 Hz−1 sr−1)
Jν	mean intensity
uν	spectral energy density

	z		dΩ

	(normal)	θ

	x		y

	φ
Fν	ﬂux density
Iν	speciﬁc intensity
	(Wm−2 Hz−1 sr−1)
Jν	mean intensity
uν	spectral energy density

	z		dΩ

	(normal)	θ

	x		y

	φ
Fν	ﬂux density
Iν	speciﬁc intensity
	(Wm−2 Hz−1 sr−1)
Jν	mean intensity
uν	spectral energy density