WHAT ARE ELECTROMAGNETIC WAVES? A “wave” is a periodic disturbance propagating through a medium EM Waves Convey Undulations in EM Fields: Electric field E

wave velocity 0 z Magnetic field H

Electric energy density null

Magnetic z energy density Electric and Magnetic Fields are Useful Fictions: Explain all classical electrical experiments with simple equations

4 Maxwell’s equations plus the Lorentz force law

(quantum effects are separate) L2-1 WHAT ARE ELECTRIC AND MAGNETIC FIELDS?

=+×µ[] Lorentz Force Law: f q() E vo H Newtons

E Electric field[] volts/meter; V/m H Magnetic field[] amperes/meter; A/m f Mechanical force[] newtons; N q Charge on a particle[] coulombs; C v Particle velocity vector[] meters/second; m/sec −6 µ×o Vacuum permeability 1.26 10 Henries Electric and Magnetic Fields are what Produce Force f: f== qE when v 0, defining E via an observable

f=×µ qvo H when E = 0, defining H via an observable +++ +++ force f -- +++ L2-2 MAXWELL’S EQUATIONS Differential Form: 1Tesla = Webers/m2 = 104 Gauss Faraday's Law:∇× E =−∂ B ∂ t Gauss's Laws ∇• D =ρ Ampere's Law:∇× H = J +∂ D ∂ t ∇• B = 0

E Electric field[] volts/meter; V/m H Magnetic field[] amperes/meter; A/m B Magnetic flux density[] Tesla; T 1  D Electric displacment coulombs/m22 ; C/m  J Electric current density amperes/m22 ; A/m  ρ Electric charge density coulombs/m33 ; C/m Integral Form: Eds•=−∂∂•() Bt da Dda •=ρ dv ∫∫cA ∫ AV ∫ Hds•=() J +∂∂• Dt da Bda0 •= ∫∫cA ∫ A

Stoke’s A Gauss’ V da da Theorem Law A ds L2-3 VECTOR OPERATORS ∇, ×, •

“Del” (∇) Operator: Gradient of φ: ∇=xxyyzzˆˆˆ ∂∂ + ∂∂ + ∂∂ ∇φ =∂xxyyzzˆˆˆφ ∂+∂φ ∂+∂φ ∂ “Vector Cross Product”: =++ AxAyAzAˆˆˆxyz

xyzˆˆˆ =−+−+xABˆˆ()yz AB zy yAB() zx AB xz AB×= detAxyz A A zABˆ ()xy− AB yx BBBxyz

“Vector Dot Product”: A•= B ABxx + AB yy + AB zz

“Divergence of A”: ∇•AAxAyAz =∂xyz ∂ +∂ ∂ +∂ ∂ xyzˆˆˆ “Curl of A”: ∇×=Adetx ∂∂∂∂∂∂ y z AAAxyz

“Laplacian Operator”: ∇•() ∇φ =∇2222222φ =∂() ∂xyz +∂∂ +∂∂ φ L2-4 MAXWELL’S EQUATIONS: VACUUM SOLUTION Maxwell’s Equations: Faraday's Law:∇× E =−∂ B ∂ t Gauss's Laws ∇• D =ρ Ampere's Law:∇× H = J +∂ D ∂ t ∇• B = 0 0 0 BH= µ EM Wave Equation: o =ε () () DEo Eliminate H :∇× ∇× E =−µo () ∂∂ t ∇× H Use identity:∇×() ∇× A =∇() ∇• A −∇2 A ()222() Yields:∇∇• E −∇ E =−µooo()∂∂ t ∇× H =−µ ε∂ E ∂ t 0 12 22 EM Wave Equation∇− Eµooε∂ E ∂ t = 0 Since: Second derivative in space = const × second derivative in time,

Solution is any f(r,t) with identical space and time dependences

1 Homogeneous Vector Helmholz Equation L2-5 WAVE EQUATION SOLUTION Wave Equation has Many Solutions! 222 ∇−µε∂∂=EEt0oo Where ∇2φ =∂() 222222 ∂xyz +∂∂ +∂∂ φ Example: () Try:EyEz,t= ˆ y [no x,y dependence, “UPW”] ()22 2 2 ∂∂zEyooy −µε∂ E ∂ t = 0 () () Ey+ z,t=− E+ () t z c , where E . is an arbitrary function ()−2 Test: −−=cE"tzc++()µooε−⇒ E"tzc()c1= ()µooε Generally: c≅× 3 108 [] m/s in vacuum () Ey z,t=−++ E+−() t zc E() t zc more generally

E+(t – z/c) propagation z = t = 0, arg = 0 The position where arg = 0 arg = 0 moves at velocity c z z = ct L2-6 UNIFORM PLANE WAVE IN Z-DIRECTION

Electric Fields (Example): Ey(z,t) = E+(t - z/c) [V/m] ω ω More specifically, let: Ey(z,t) = E+ cos (t – z/c) = E+ cos ( t – kz), where k = ω/c = ω µooε

Magnetic Fields: Use Faraday’s Law: ∇×EBt = −∂ ∂

xyzˆˆˆ ∇×=EdetxyzxEcostkzz ∂∂∂∂∂∂=−∂ˆ + () ω−∂ 0 0 EEExyz 0 0 =−xkEˆ + sin() ω t − kz () ()[] H z,t=µ xˆ () 1o ∫ kE+ sin ω− t kz dt A/m () () =−xkEˆˆ()++ ωµoo cos ω t − kz =− xE() η cos ω t − kz  k1==µ εη=; µ ε ω c oo o o o

L2-7 UNIFORM PLANE WAVE EM FIELDS

EM Wave in z direction: () ()() () E z,t=ω− yEˆˆ++ t kz , H z,t =−η x() Eo cos ω− t kz

x Hz,0()

Ez,0() z

y Electric energy density Magnetic energy density z

L2-8 ACOUSTIC WAVES A “wave” is a periodic disturbance propagating through a medium Acoustic Waves: Velocity air molecules

Pressure

wave velocity 0 z Particle velocity

Kinetic ∆ Potential energy energy

z

Generally, two types of energy are interchanged

L2-9