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Optics, IDC202
Lecture 2.
Rejish Nath
Contents
1.
2.
3.
4.
Maxwell’s equations in Vacuum.
Electro-magnetic wave equation.
Poynting vector.
Irradiance.
Literature:
1. Optics, (Eugene Hecht and A. R. Ganesan)
2. Optical Physics, (A. Lipson, S. G. Lipson and H. Lipson)
Will add more references later on.
ElectroMagne,cfields(EMF):Maxwell’sequa,ons(MEs)
Light is an electro magnetic wave.
Electro-magnetic spectrum
Maxwell’s theory of light propagation
We should get a wave equation out of Maxwell’s equations.
ElectroMagne,cfields(EMF):Maxwell’sequa,ons(MEs)
Recapofelectricityandmagne,sm.
Force exerted by the electric field E on a point charge q is
FE = qE
Force exerted by the magnetic field B on a point charge q moving with a
velocity v (Lorentz force)
FB = qv ⇥ B
Electric fields can be created by both electric charges and time-varying
magnetic fields.
Magnetic fields can be created by both electric currents and time-varying
electric fields.
ElectroMagne,cfields(EMF):Maxwell’sequa,ons(MEs)
Differen,alformsofMEsinvacuum(Nochargesandnocurrents)
@B
@t
Faraday’slawofinduc,on
r·E=0
r⇥E=
r·B=0
@E
r ⇥ B = ✏ 0 µ0
@t
Gauss’slaw
Ampere’slaw
Thesetwoequa,onsweneedtosolvetogetthe,medependentfields.
The,merateofchangeofelectricfieldisassociatedwithacurrent
calledthedisplacementcurrent,resultsinamagne2cfield.
Apartfrommagnitudes,theroleofEandBisiden,cal.Inotherwords
theequa,onsaresymmetricwithrespecttoEandB.
ElectroMagne,cfields(EMF):Maxwell’sequa,ons(MEs)
Differen,alformsofMEsinvacuum(Nochargesandnocurrents)
@B
@t
Faraday’slawofinduc,on
r·E=0
r⇥E=
r·B=0
@E
r ⇥ B = ✏ 0 µ0
@t
Gauss’slaw
Ampere’slaw
Theelectricandmagne,cfieldsmutuallyperpendiculartoeachother
andalsoperpendiculartothedirec,onofpropaga,on.Foraplane
wave,itcanbeeasilyshownthat,
r ⇥ E = ik ⇥ E
TransverseofnatureofEandBinanEM-disturbance.
ElectroMagne,cwaveequa,on
@E
r ⇥ B = ✏ 0 µ0
@t
@(r ⇥ E)
r ⇥ r ⇥ B = ✏ 0 µ0
@t
r ⇥ (r ⇥ ...) = r(r·)
r(r · B)
zero
2
r B=
r2
2
1 @ B
c2 @t2
2
1 @ B
r B= 2 2
c @t
2
Onecanalsogeneralisethis
inthepresenceofsources.
1
c =
µ0 ✏ 0
2
Similarlyoneobtainsfor
electricfield.
ElectroMagne,cwaveequa,on
2
1
@
B
2
r B= 2 2
c @t
2
1
@
E
2
r E= 2 2
c @t
Timevaryingcurrentj(x,t)
ThecrosslinkedBandE
propagateawayfrom
thesource
ElectroMagne,cwave:Poten,als
Vector and Scalar potentials provide an alternative way to describe
an Electro magnetic wave.
B=r⇥A
E=
@A
@t
rV
thepoten,alsarenot
uniquelydefined
InvacuumV=0,hencethevectorpoten,aldefinestheEM-field.
Gauge invariance
@
0
V =V
@t
0
A =A+r
Gaugetransforma,on
Poten,alsalsosa,sfyawaveequa,on
ThephysicalresultsdonotdependontheGauge.
ElectroMagne,cwave:EnergydensityandPoyn,ngvector
AnEMwavecarriesbothenergyandmomentum.
Theycancarrythisenergyforwardtomillionsofmillionsofmiles.
Poyn,ngvector:representsthedirec,onalenergyfluxdensity
1
S=
E⇥B
µ0
Direc,onofSgivesthedirec,onofenergyflux.
• MagnitudeofSgivestheamountofenergypassedthrough
unitareaperunit,me. TheunitofSisWa6/m2.
•
Theeffectofagivenenergy-densitydependsonwhichradia,onyou
aretalkingabout.Forinstance,1Wa6/m2of550nm(greenlight)might
warmourskin,butthesameof300nmUVradia2onwillseverelyburn
youinfewminutes.Thewavelengthisanimportantfactor!!!
ElectroMagne,cwave:EnergydensityandPoyn,ngvector
Let’staketheexampleofharmonicplanewave:
E = E0 cos (k · r
!t)
B = B0 cos (k · r
!t)
1
S=
E0 ⇥ B0 cos2 (k · r
µ0
Thedetectorsmeasurethe
integratedenergyfluxover
some,me.
!t)
determinesthe
instantaneousflowof
energydensity
Whensomethingisoscilla,ng,wetalkaboutaveragevalues:
(usefulwhentalkingaboutlong-,mebehaviours)
2
1
hSi =
E 0 ⇥ B0
2µ0
hcos (k · r
!t)i = 1/2
Themagnitudeof⟨S⟩ is called
irradiance (I).
ElectroMagne,cwave:Irradianceandunits
1
c✏0 2
I = |hSi| =
|E0 ⇥ B0 | =
E0
2µ0
2
2
TheunitofIisWa6/m .
Photosynthesisorchemicalreac,onsinwhichlightissupplyingasthe
energyforareac,on,theunitof“Einstein”isused.
1 Einstein is equal to
the energy of one mole of photons.
Mostofthecases“Wa\/m2”isnotofmuchuseunlessyouspecifythe
wavelength.
Thenweneed“Wa\/m2/nm”,typicalwavelengthsareorderofnm.
Some,mesweneed“photons/m2/nm”,forthatoneneedtoconvert
Wa6 no.ofphotons.How??
ElectroMagnetic wave: Irradiance and units
Watt ⇒ no. of photons. How??
divide by hc/𝜆. But you need to know the wavelength!!
Photosynthetically active radiation (PAR) detector which detects a
wavelength range from 400 nm to 700 nm. PAR is used by
photosynthetic organisms in the process of photosynthesis.
Spectra would look completely different in different units!!!
Which is more fundamental??? wavelength or frequency???
Lets consider light moving through a medium of refractive index n
⌫ = c/n
Speed of light changes in the medium,
what should change on the left side??
ElectroMagnetic wave: Irradiance and units
⌫ = c/n
Speed of light changes in the medium,
what should change on the left side??
It is the wavelength which found to be changing, and frequency is
considered more fundamental to light.
For instance, a 550 nm “green” light in vacuum has a wavelength
of 414 nm in water.
The unit of irradiance is “Watt/m2” or “Photons/s/cm2”.
In front of a fire, the warmth of your skin is proportional to irradiance.
The brightness of a white paper is proportional to the irradiance at its
surface.
ElectroMagnetic wave: Irradiance and units
Vector irradiance
light
This explains why you feel hotter in the
afternoon than in the morning or evening.
Let the irradiance be I0.
Surface
light
✓
Vector irradiance depends on the
orientation of the surface relative
to the illumination.
What is it for this case?
Surface
I = I0 cos ✓
ElectroMagnetic wave: Irradiance and units
Scalar irradiance (amount of light passing through a spherical body)
direction independent
Inverse square law
The irradiance from a point source is proportional to 1/r2.
Verify yourself using the spherical wave solution.
ElectroMagnetic wave: Radiance
sphere
Radians and steradians
✓
r
2Danalogof
angle:Solid-angle
l
✓ = l/r
⌦ = A/r2
radians
steradians
A
r
Radiance is the energy density per solid angle normal to the beam.