Black Body Radiation and Radiometic Paametes: All mateials absob and emit adiation to some extent. A blackbody is an idealization of how mateials emit and absob adiation. It can be used as a efeence fo eal souce popeties. An ideal blackbody absobs all incident adiation and does not eflect. This is tue at all wavelengths and angles of incidence. Themodynamic pincipals dictates that the BB must also adiate at all s and angles. The basic popeties of a BB can be summaized as: 1. Pefect absobe/emitte at all s and angles of emission/incidence. Cavity BB. The total adiant enegy emitted is only a function of the BB tempeatue. 3. Emits the maximum possible adiant enegy fom a body at a given tempeatue.
4. The BB adiation field does not depend on the shape of the cavity. The adiation field must be homogeneous and isotopic. T If the adiation going fom a BB of one shape to anothe (both at the same T) wee diffeent it would cause a cooling o heating of one o the othe cavity. This would violate the 1 st Law of Themodynamics. T T A B Radiometic Paametes: 1. Solid Angle d whee is the suface aea of a segment of a sphee suounding a point. d A
is the distance fom the point on the souce to the sphee. The solid angle looks like a cone with a spheical cap. z d sind y sin x An element of aea of a sphee sin dd Theefoe d sindd The full solid angle suounding a point souce is: d sind d cos 4 O integating to othe angles < : The unit of solid angle is steadian. 1 cos
. Radiant Flux and Enegy Density: Radiant enegy Q (J); Enegy flux is the ate of adiant enegy tansfeed fom one point o suface. Enegy flux (Φ) is measued in watts. Can be spectal o a total ove all wavelengths. (Note that Flux is the same as optical powe.) The enegy density (u) is the enegy pe unit volume. dq / dt u dq/ dv Note that Φ is the total flux integated ove all wavelengths. It consists of the integated spectal flux o powe with: Note that d but d d d since this epesents an equal incement of powe. The scaling facto between units is found by using the elation: c / c d d c 3. Iadiance: The powe pe unit aea illuminating a collection o detection suface.
E d det 4. Spectal Intensity: The powe emitted pe unit solid angle fom a souce. (Wavelength dependent) d( ) I d whee dω is an incement of solid angle. Note that in physics intensity efes to the magnitude of the Poynting vecto of an EM field. This intepetation esembles iadiance as defined hee. Note that if we conside a eceiving suface element, the solid angle subtended by this suface elative to a point souce is dcos / this case the iadiance is elated to the intensity by:. In
d E d I cos / I cos E 5. Spectal Emitance: The powe pe unit souce aea emitted fom a souce. (Wavelength dependent.) M d( ) sc 6. Spectal Radiance: The powe emitted pe unit of pojected souce aea pe solid angle. (Wavelength dependent.) L I d cos d cos sc sc
L d Souce Aea n Spectal intensity, emittance, and adiance ae tems efeing to an optical souce. They can also be integated ove all wavelengths to obtain the total intensity, emittance, and adiance fom the souce. Radiance is the most geneal of the souce adiometic tems. The adiant enegy dq(λ) emitted fom an aea ove time dt, wavelength inteval d, and solid angle d in the diection, is elated to the monochomatic adiance by: cos dq L dddt. Note that this epesents the enegy emitted fom the souce element at an angle elative to the suface nomal ˆn. cos is the pojected aea of the souce. The optical flux o powe emitted fom an element of souce aea can be estimated using the appoximate elation: Lcos A.
whee Q dt A Lambetian souce emits light with the following chaacteistic: I( ) L cos sc I( ) I cos o o, with Io L o. sc I o I o cos A BB acts as a Lambetian emitte. Fo a BB the emittance and adiance ae elated accoding to: M L cosd / L sincosd 1 L cosd cos L.
Note that the definition of the solid angle:. d sin dd is used with Relations between Paametes: Fom a point souce: IW ( / s) 4 At a distance fom the point souce the iadiance on a plane pependicula to is: I E. 4 The iadiance was obtained by integating ove the aea suounding a point that emits powe. It can be seen that the powe deceases as 1/ which is a well known adiation popety. Even if a souce has a finite diamete it is still possible to use the invesesquae distance elation to appoximate the powe emitted fom a souce Souce Obseve D S povided that the atio of the souce diamete-to distance is small enough.
Sc distance/diamete Subtended Angle Invese Sq. Law Eo atio 1.6 ~3 o ~1% 5 ~11.3 o ~1% 1 ~5.7 o ~.5% 16 ~3.6 o ~.1% Note that the limit of angula esolution fo the human eye is ~.1 o. Theefoe fo visual applications a atio of distance/souce diamete of 16 is adequate fo appoximating an extended souce as a point souce. The angle subtended by the sun is.5 o theefoe it is definitely a point souce when viewed fom the Eath. Consevation of Radiance: The adiance theoem is an impotant law of adiomety and states that adiance is conseved with popagation though a lossless optical system. The adiance measued at the souce and eceive is compaed. Souce Receive o 1 o 1 Conside a souce aea o and eceive aea 1 sepaated by a distance. The coesponding solid angles ae: d o = the solid angle subtended by 1 at o
d o 1cos1 d 1 = the solid angle subtended by o at 1 d cos 1 If L is the adiance of the adiation field measued at in the diection of 1, the flux tansfeed fom to 1 is: cos d L d Similaly the flux tansfeed fom 1 to is: cos d L1 1 1 d 1. The adiance L 1 measued at 1 is theefoe: L 1 d cos d 1 1 1 And since the flux oiginates fom o : d L cos d L1 cos d cos d L L 1 1 1 1 1 1 1 This implies that the adiance of the souce is the same egadless of whee it is measued and is conseved. Anothe inteesting esult can be obtained by e-witing the expession:
1cos d Lcosd Lcos L d cos 1 1 1 The esult shows that the tansmitted flux can be obtained by taking the pojected aea and solid angle poduct at the souce o eceive. This allows using a pespective eithe fom the souce o eceive to pefom an analysis. Lambetian Disk Souce: It is desied to compute the iadiance fom a disk Lambetian souce of adius R and unifom adiance L at an aea element 1 that is paallel to the suface of the disk and is located axially at a distance z fom the cente of the disk. An annula aea element on the suface on the souce is given by: z sind cos 3 The element of solid angle subtended by 1 fom any point on is given by: d 1 cos z /cos
The flux tansfeed fom to 1 is given accoding to the definition of adiance as: The iadiance at 1 becomes: d L1 sincosd 1/ d E Lsincosd 1 R Lsin 1/ L R z with R z 1 1/ tan Note that when z << R the iadiance appoaches a value of L. This is the same value found peviously fo the adiant exitance of a Lambetian souce. When z >> R the iadiance appoaches E R L/ z in this case the iadiance obseves an invese squae law. At vey lage z the souce looks like a point souce with the intensity of I R L and the iadiance fo a point souce is:
R RL E L R z z I z which illustates the invese squae law dependence. Example: Spheical Lambetian Souce: Conside the iadiance on an aea located a distance fom the cente of a spheical Lambetian souce with unifom adiance L. To detemine this we will use the symmety of the situation athe than integating ove the suface of the souce. R The adiant exitance of a Lambetian souce is given by: M L The total flux emitted by the souce is 4 RL At a distance fom the cente of the souce it adiates unifomly ove an aea 4. Theefoe the iadiance at a distance is given by:
E R L. The iadiance is seen to follow an invese squae law at a distance. The coesponding intensity becomes: I 4 R L. An obseve at looking back at the souce will see a unifom disk with half angle: Planck s Radiation Law: R. 1 1/ sin The spectal adiant exitance fom a BB is given by: M c exp h / kt1m Hz 3 h 1 W M exp hc / kt 1m m hc 1 W 5 Note that due to the elation between and but M M
M d M d c c d d Stefan-Boltzmann Law: The total emittance fom a BB is only a function of the BB tempeatue. It can be found by integating M ove all wavelengths. M T 4 ( W / m ) The Stefan-Boltzmann constant: s 5 4 k 8 W 5.6731 3 4 15ch m K 3 Boltzmann Constant: k 1.381 J / K Emissivity: A BB is a pefect emitte with unity emissivity (ε =1). All pactical bodies have < 1. The cuve below shows the diffeence between a BB and a non-ideal souce. The emissivity fo a non-ideal souce can be obtained though the following expeiment. Conside an object that is illuminated with incident adiation Fom Enegy Consevation: whee RT 1 o
R T 1 3 ; ; (BB) S The above figue shows the spectum of a BB and a non-ideal souce.
Object Incident 3 Tansmitted 1 Reflected Absobed The emissivity can be shown to be equal to the absoption though the use of the Kichoff Radiation Law. Conside: An object at tempeatue T suounded by an enclosue also at tempeatue T. Enclosue Temp (T) Object Temp (T) At themal equilibium:. Emitted by Object Absobed by Object If the total flux incident on the object is then the Absobed Flux = Emitted Flux
and. Theefoe the emissivity coefficient (ε) can be obtained fom the absoption coefficient (α) and using RT 1 1R T, and fo an opaque object 1 R. The emissivity will in geneal be wavelength dependent which would equie evaluating the eflectance at each wavelength with a spectomete. Howeve ove a limited spectal ange it can be consideed a constant. Once the emissivity is found the spectal emitance of a non-ideal (BB) souce can be found by multiplying M by to account fo non-ideal emittes.
Wien s Displacement LAW: The wavelength at which the Planck function is a maximum M times the tempeatue of the coesponding BB is a constant. MT 89mK The figue above shows the elative powe emitted fom a themal souce at T = o K, 5 o K, and 5 o K as a function of wavelength. The peak wavelength can be pedicted using the Wein displacement law.
Example 1: Fist ode adiometic popeties of the sun. Peak sola adiation appeas at.48m. Theefoe o 89m K o T 65 K..48m The sun appeas as a blackbody souce with a tempeatue nea 6 o K. The peak adiant exitance can be computed: M.48m 4 5 8 3.741 1.441.48 exp 1.4865 8 1 W / m m The adiance of the sun at.48 m using M 1 8 Lpeak W / m s m L : The spectal flux fom the sun collected on the Eath:.488m Lpeak cosod A (Lambetian Assumption) sun d is the angle subtended by the Eath elative to the sun, and is an incement of aea on the sun s suface. Since the sun is a sphee it appeas as a full disk to an obseve and the obseve appeas nomal to it. Theefoe o coso 1 Note: Expanding the poduct of the diffeentials, d d col sun sun col sun col sun col Using a collection aea A 1m and = 931 6 miles, D sun =.865 million miles, then the angle subtended by the sun is col
.865/ 5 sun 6.81 s. 93 The flux fom the sun hitting 1 squae mete on Eath:.48m sun sun col A1m peak 1 8 6.5 1 5 1 s m L 16 W / m A The sola output ove all wavelengths: M T 4 7.35 1 W 7 at T = 6 o K m M L fo a Lambetian emitte. 7 7.351 6.8 1 5 1 159W s m A1m fo a 1 m collecto. Example : Flux collected by a detecto on a suface.
I o o Collected Ei cos(3 ) ACollecto Ei cos(3 )(1 cm ) o I(3 ) I, E, E i 4 4 cm o o 3 Iocos3 Collected 1W I L A cm o o souce E i 1 1 cm st st o 1 W / st cos(3 ) cm 1cos(3) 1cm cm o.188w mw W