ATMOS 5140 Lecture 3 –Chapter 2 - University of Utah

Lecture3– Chapter2

• MoreonMaxwell’sEquations• QuantumPropertiesof Radiation• FluxandIntensity• SolidAngle

ATMOS5140

Maxwell’sEquations

• Changingelectricfieldinducesamagneticfield• Changingmagneticfieldinducesanelectricfield• Doesnotneedmaterialmedium(vacuumworks)• QuantitativeInterplaydescribedbyMaxwell’sEquations

Direction of propagation

Electric field vector

Magnetic field vector

Maxwell’sEquations

Gauss’sLawofElectricField

Gauss’sLawofMagneticField

Faraday’sLaw

Ampere’sLawandMaxwell’sadditionofdisplacementcurrentdensity

D=ElectricdisplacementE=ElectricfieldB=MagneticinductionH=MagneticfieldJ=ElectricCurrent

=densityofelectriccharge

Maxwell’sEquationsD=ElectricdisplacementE=ElectricfieldB=MagneticinductionH=MagneticfieldJ=ElectricCurrent

𝜀 = 𝑝𝑒𝑟𝑚𝑖𝑡𝑡𝑖𝑣𝑖𝑡𝑦𝑜𝑓𝑠𝑝𝑎𝑐𝑒𝜇 = 𝑚𝑎𝑔𝑛𝑒𝑡𝑖𝑐𝑝𝑒𝑟𝑚𝑒𝑎𝑏𝑖𝑙𝑖𝑡𝑦𝑜𝑓𝑡ℎ𝑒𝑚𝑒𝑑𝑖𝑢𝑚

𝜎 = 𝑐𝑜𝑛𝑑𝑢𝑐𝑡𝑖𝑣𝑖𝑡𝑦𝑜𝑓𝑚𝑒𝑑𝑖𝑢𝑚

Maxwell’sEquationsD=ElectricdisplacementE=ElectricfieldB=MagneticinductionH=MagneticfieldJ=ElectricCurrent

=densityofelectriccharge

• Sinusoidalplane-wavesolutionsareparticularsolutionstotheelectromagneticwaveequation.• Thegeneralsolutionoftheelectromagneticwaveequationinhomogeneous,linear,time-independentmediacanbewrittenasalinearsuperpositionofplane-wavesofdifferentfrequenciesandpolarizations.

Maxwell’sEquationsforplanewaves

FigurefromTimGarrett



ComplexVectors(realandimaginaryparts)Ifyouneedarefresher:watchKhanvideo

https://www.youtube.com/watch?v=kpywdu1afas


ComplexVectors(realandimaginaryparts)

Constantcomplexvectors

Constantcomplexwave vectors(hereonlyconsiderrealpart)

WaveVector

• Thewavevector,k,countsthewavenumberinaparticulardirection.

• Spatialfrequencyofawave• Forexample,thewavenumberofthebluelineisthenumberofnodes(reddots)youseeperunitdistancetimes π.

• Count2nodesin4unitsofdistance,or3nodesin6unitsofdistance½nodesperunitdistance.

• Wavenumberisthenumberofradiansperunitofdistance

k=1/2π


Ifk’’=0,thentheamplitudeofthewaveisconstantThenthemediumisnonabsorbing!

• Indexofrefraction(alsocalledrefractiveindex)- N



Inavacuum:

k’’=0

PHASESPEED


RefractiveIndex

• Therefractiveindexofamaterialiscriticalindeterminingthescatteringandabsorptionoflight,withtheimaginarypartoftherefractiveindexhavingthegreatesteffectonabsorption.

RealworlduseofIndexofRefractions

Fig.3

CitationFredericE.Volz,"InfraredRefractiveIndexofAtmosphericAerosolSubstances,"Appl.Opt. 11, 755-759(1972);https://www.osapublishing.org/ao/abstract.cfm?uri=ao-11-4-755

Image©1972OpticalSocietyofAmericaandmaybeusedfornoncommercialpurposesonly.Reportacopyrightconcernregardingthisimage.

EasiertomeasureRefractiveIndexintheIRthaninthevisible

Fig.2

CitationFredericE.Volz,"InfraredRefractiveIndexofAtmosphericAerosolSubstances,"Appl.Opt. 11, 755-759(1972);https://www.osapublishing.org/ao/abstract.cfm?uri=ao-11-4-755

Image©1972OpticalSocietyofAmericaandmaybeusedfornoncommercialpurposesonly.Reportacopyrightconcernregardingthisimage.

A variety of values for the refractive index of BC has been used in global climate models including the OpticalPropertiesofAerosolsandClouds(OPAC)value of 1.74 − 0.44i [Hess et al., 1998]. As reviewed by Bond and Bergstrom [2006], reported values of the refractive index of light absorbing carbon vary widely; the real part, n, appears to vary from that of water to that of diamond, and the imaginary part, k, varies from that of negligibly absorbing material to that of graphite. Bond and Bergstrom [2006] hypothesize that strongly absorbing carbon with a single refractive index exists and that some of the variation in reported values results from void fractions in the material. Based on agreement between measured real and imaginary parts of the refractive index of light absorbing carbon, Bond and Bergstrom [2006] recommended a value of 1.95 − 0.79i at 550 nm. They caution, however, that this value may not represent void-free carbon. Stier et al. [2007] found that this value led to better agreement with observed atmospheric absorption, compared with the OPAC value. Motekiet al. [2010] found that the refractive index of ambient BC in the Tokyo urban area was about 2.26 − 1.26i at 1064 nm. The OPAC assumption for imaginary refractive index is not taken from combustion-generated particles, is lower than either of the latter two recommendations, and would lead to a MAC prediction about 30% lower if all other factors were equal.

• Indexofrefraction- N• Flux– F(W/m2)


Flux- Poynting vector

• Theinstantaneousfluxdensityoftheelectricfieldinthedirectionofthewaveis

• Foraharmonicwave(averageovercompletecycle)

𝐹 = 12 𝑐𝜀=𝐸

?

𝑆 = 𝐸×𝐻

• Indexofrefraction- N• Flux– F(W/m2)• Absorptioncoefficient– 𝛽E


AbsorptionConsiderscalaramplitudeofthewave

Nowgobacktoourdefinitionofflux

Recallthatourimaginarypartofwavevectorisresponsibleforabsorption

Absorption

𝛽E = 4π𝑛H/λ

KLM=distancerequiredforthewave’senergytobeattenuatedtoe-1 (about37%)

Absorption

Polarizationparameterapplyingtotransversewavesthatspecifiesthegeometricalorientationoftheoscillations

the"polarization"ofelectromagneticwavesreferstothedirectionoftheelectricfield.

Polarization

(a) (b) (c)

(d) (e) (f )

Linearpolarization

Circularpolarization

Ellipticalpolarization(HybridofLinearandCircular)

QuantumPropertiesofRadiation

Wavevs.Particle

• WaveNatureofRadiationmatters• ComputingScatteringandreflectionpropertiesofatmosphericparticles

• Airmolecules,aerosol,clouddroplets,raindrops• AndSurfaces

• QuantizedNatureofradiation• AbsorptionandEmissionofradiationbyINDIVIDUALAtomsandmolecules

• Photochemicalreactions

• BroadbandRadiation• Widerangeoffrequencies

• MonochromaticRadiation Radiation• Singlefrequency(onecolor)• Morecommonlyuse:quasimonochromatic(rangeoffrequencies)

• Coherent• Perfectsynchronization• ArtificialSource– radar,lidar,microwave

• IncoherentRadiation• Notphaselocked• Nosynchronization• Naturalradiationintheloweratmosphere

Monochromatic Flux

Broadband Flux

Wm-2 um-1

Wm-2

SphericalCoordinates

ElevationorZenith

SphericalCoordinates

ΩΩz = cos θ

Ω x = sin θ cos φ

Ωy = sin θ sin φ

x

y

z

θ

φ

Steradian

dθ

θ

φ

dφ

z

x y

θ

dω = sinθ dθ dφ

Intensity

dA ∝ cos 𝜃 = 𝑛U V ΩX

RelationshipbetweenFluxandIntensityUpwardFlux– Integrateoverupperhemisphere

DownwardFlux– Integrateoverlowerhemisphere

RelationshipbetweenFluxandIntensity

RelationshipbetweenFluxandIntensity

Documents

ATMOS 5140 Lecture 3 –Chapter 2 - University of Utah