Document 2648968

Electromagne,c Waves All electromagne-c waves travel in a vacuum with the same speed, a speed that we now call the speed of light. Proper,es of Electromagne,c Waves Any electromagne-c wave must sa-sfy four basic condi-ons: 1.  The fields E and B and are perpendicular to the
direction of propagation vem.Thus an
electromagnetic wave is a transverse wave.
2.  E and B are perpendicular to each other in a manner
such that E × B is in the direction of vem.
3.  The wave travels in vacuum at speed vem = c
4.  E = cB at any point on the wave.
Proper,es of Electromagne,c Waves The energy flow of an electromagne-c wave is described by the Poyn,ng vector defined as The magnitude of the Poyn-ng vector is The intensity of an electromagne-c wave whose electric field amplitude is E0 is Radia,on Pressure It’s interes-ng to consider the force of an electromagne-c wave exerted on an object per unit area, which is called the radia,on pressure prad. The radia-on pressure on an object that absorbs all the light is energy absorbed
( E = pc )
c
Δp ( energy absorbed ) / Δt P
F=
=
=
Δt
c
c
where P is the power (joules per second) of the light.
Δp =
where I is the intensity of the light wave. The subscript on prad is important in this context to dis-nguish the radia-on pressure from the momentum p. Example Solar sailing Intermediate/Advanced Concepts Wave equa-ons in a medium The induced polariza-on in Maxwell’s Equa-ons yields another term in the wave equa-on: 2
2
2
2
∂
E
1
∂
E
∂
E
∂
E
− 2 2 =0
− µε 2 = 0
2
2
∂z
v ∂t
∂t
∂z
This is the Inhomogeneous Wave Equa,on. The polariza-on is the driving term for a new solu-on to this equa-on. ∂2 E
∂2 E
− µ0ε 0 2 = 0
2
∂z
∂t
∂2 E 1 ∂2 E
− 2 2 =0
2
∂z
c ∂t
Homogeneous (Vacuum) Wave Equa,on E ( z, t ) = Re{E0 ei( kz −ωt ) }
= 12 {E0 ei( kz −ωt ) + E*0 e − i( kz −ωt ) }
=| E0 | cos ( kz − ωt )
2
c
µε
2
n = 2=
v
µ0ε 0
c
=n
v
Propaga-on of EM Waves Polariza-on and Propaga-on Energy and Intensity S = E×H
•  Poyn,ng vector describes flows of E-­‐M power •  Power flow is directed along this vector (usually parallel to k) •  Intensity is average energy transfer (i.e. the -me averaged Poyning vector: I=<S>=P/A, where P is the power (energy transferred per second) of a wave that impinges on area A. sin 2 ( kx − ω t )
= cos 2 ( kx − ω t ) =
1239.85
cε 0 2 cε 0
2
2
S = I ≡| E ( t ) × H ( t ) |=
E =
Ex + E y ) hω[eV ] =
(
λ[nm]
2
2
cε 0 ≈ 2.654 ×10−3 A / V
h = 1.05457266 × 10 Js
example E = 1V / m
−34
I = ? W / m2
1
2
Polariza-on & Plane of Polariza-on Linear versus Circular Polariza-on Linear polariza-on (frozen -me) Linear polariza-on (fixed space) Circular polariza-on (linear components) Circular polariza-on (frozen -me) Circular polariza-on (fixed space) Polariza-on: Summary ŷ
!
ŷ E

iδ
E = E x eiδ x x̂ + E y e y ŷ
x̂
x̂
linear polariza-on y-­‐direc-on right circular polariza-on Phase difference δ = δ x − δ y
Phase difference = 00 r
Ex
r
Ex
Phase difference è 90 0 (π/2, λ/4) ẑ
ẑ or t
!
Ey
leU circular polariza-on (+: posi-ve helicity ) !
Ey
ẑ
ẑ
leU ellip-cal polariza-on Phase difference è 180 0 (π, λ/2) r
Ex
ẑ
!
Ey
ẑ
Polariza-on Applets •  Polariza-on Explora-on h_p://webphysics.davidson.edu/physlet_resources/dav_op-cs/Examples/polariza-on.html •  3D View of Polarized Light h_p://fipsgold.physik.uni-­‐kl.de/soUware/java/polarisa-on/index.html Quarter wave plate Half wave plate Quiz for the Lab – Bonus Credit 0.2 pts Methods for genera-ng polarized light h_p://hyperphysics.phy-­‐astr.gsu.edu/hbase/phyopt/polar.html Polariza-on by Reflec-on h_p://hyperphysics.phy-­‐astr.gsu.edu/hbase/phyopt/polar.html A Polarizing Filter Malus’s Law Suppose a polarized light wave of intensity I0 approaches a polarizing filter. θ is the angle between the incident plane of polariza-on and the polarizer axis. The transmi_ed intensity is given by Malus’s Law: If the light incident on a polarizing filter is unpolarized, the transmi_ed intensity is In other words, a polarizing filter passes 50% of unpolarized light and blocks 50%. Malus’s Law Polarized sunglasses Brewster Angle Polariza-on by sca_ering (Rayleigh sca_ering/Blue Sky) Circularly polarized light in nature Morphology and microstructure of cellular pa_ern of C. gloriosa Reflec-on and Transmission @ dielectric interface Beyond Snell’s Law: Polariza-on? Reflec-on and Transmission (Fresnel’s equa-ons) Can be deduced from the applica,on of boundary condi,ons of EM waves. An online calculator is available at hJp://hyperphysics.phy-­‐astr.gsu.edu/hbase/phyopt/freseq.html Reflec-on and Transmission of Energy @ dielectric interfaces Reflec-on and Transmission (Fresnel’s equa-ons) Can be deduced from the applica,on of boundary condi,ons of EM waves. Reflec-on and Transmission of Energy @ dielectric interfaces Energy Conserva-on Normal Incidence Reflectance and Transmi_ance @ dielectric interfaces