Wave Phenomena Physics 15c Lecture 15 lectromagnetic Waves (H&L Sections 9.5 7)
What We Did Last Time! Studied spherical waves! Wave equation of isotropic waves! Solution e! Intensity decreases with! Doppler shift i( kx±ωt)! Doppler shift for sound and for light! Special Relativity matters for light! Shock waves and Mach cones! Object flying faster than sound! Čerenkov effect for light in material r 1 r ( rξ (,) r t ) w c sinθ v f c c ( ξ (,)) c r r t r sound sound f v v observer source f c+ v f c v
Goals For Today! lectromagnetic waves in free space! Maxwell s equations " Wave equation " Plane waves! Power density is given by the Poynting vector! Polarization! lectromagnetic waves in matter! Insulators! Reflection at the boundary! Conductors! Skin depth! verything is Physics 15b
Maxwell s quations! In vacuum, t 1 c SI We use SI CGS 1 c 1 c! Try eliminating t 1 c ( ) ( ) 1 ( ) c Do you know the AC-CA rule?
Wave quations! Using this rule,! We get A ( C) ( A C) C( A ) ( ) ( ) ( ) c! Similarly, we can derive c! They look just like the 3-D wave equation from weeks ago! We know the solutions Wave equations for the M waves in free space Very useful rule Prove it!
Plane Waves! Solutions must be plane waves! and are not completely free! Must satisfy all of Maxwell s equations!!!! e i( kx ωt) k t 1 c i( kx ωt) e ω ck k k ω ω k c and are perpendicular to k and are perpendicular to each other
Transverse Waves! M waves in free space is transverse k k k! From k ω and ω ck! If you want H, H µ cµ Z Vacuum impedance (377Ω) c if we were using CGS
Poynting Vector! Poynting vector is defined by 1 S H µ k S Direction is same as k! Using H Z H! Units: in V/m, Z in ohms " S in W/m S Z
Power Density! Poynting vector tells us:! nergy is flowing in the direction of k! nergy flow density is Z! If we average over time, cos ( k x ωt) S Z Z i( kx ωt) S e Average energy flow density of M waves in free space
Polarization! For a given k, there are two possible directions of! is in the x- or y-plane! Direction of is called the polarization! They are independent solutions of the wave eqn! Linear combinations make all the possibilities
Impedance! ε, µ and c, Z are related to each other 1 Z ε µ cz c! Only two of them are independent! Impedance Z is the least popular in textbooks! You don t really need it! ut it s quite useful " You ll see later today c 1 µ Z ε µ ε
Linear Polarization! Looking up from downstream (+z) y y y x x Add x Horizontally polarized Vertically polarized e i( kx ωt)! Can make any angle from the horizontal and vertical waves! Can do more Linearly polarized
Circular Polarization! We can use complex numbers for! Try, for example, e e ie! Take the real part i( kx ωt) ( i( kx ωt), i( kx ωt) )! Consider at the origin! rotates around y (, t) x ωt (, i ) Circularly polarized Re( ) (cos( k x ωt), sin( k x ωt)) Re( ) (cos ωt,sin ωt) x
lliptic Polarization! General solution is a combination of linear and circular polarizations " lliptic polarization! Linear polarization x and y are in sync! Circular polarization x and y are 9 off-phase y y y x x x
Polarization of Transverse Waves! Any transverse waves in 3-D have possible directions of oscillation " Polarization! xample: transverse waves of a string
Polarizing Filters! A polarizing filter passes only one polarization! e.g. vertically polarizing filter blocks horizontally-polarized light! Analogy: transverse waves on a string running through a narrow slit! Widely used in photography and sunglasses! Reason will be discussed in the next lecture
Polarizing Filters! If the polarization of incoming light is at an angle, y + x unit vector in y in ˆ ˆ in ( cosθ sin θ ) out ˆ inycosθ! Amplitude is cosθ " Power is! asy to see using two filters θ unit vector in x P P cos θ out in θ Malus Law # Light intensity falls as θ increases " at θ 9 # What happens with 3 filters? y x
Triple Polarizers! Result with 3 polarizing filters a bit surprising! Filters 1 + 3 block all light! Adding makes some light pass!! Nothing really to be surprised! After filter 1,! After filter,! After filter 3, 1 1ˆ y ˆ ˆ 1cos θ ( ycosθ + xsin θ) 3 ˆ 1xcosθ sinθ 1 sin θ x ˆ θ nd filter introduces x-component again Maximum intensity is 5% at 45
M Waves in Matter! Vacuum is not the most exciting medium! What happens when light passes through water?! What happens when radio waves hit a metal wall?! Distinguish insulators and conductors! They do very different things in M waves! Let s start with an insulator
Insulators! Insulators are easy! Just replace ε ε µ µ! Speed of light becomes 1 c 1 c ε µ n εµ t! Plane wave solutions looks the same e i( kx ωt)! Power density is S Z Z Index of refraction e µ ε i( kx ωt) ω N: ε and µ may be ω-dependent in dispersive medium! 1 k εµ Impedance of the material εµ
Reflection and Transmission! M waves crosses boundary between two insulators! e.g., light (in air) hits a glass window! At the boundary,! must be continuous! H must be continuous! Use the relationship H! Continuity of H becomes I + R T H + H H Note sign! I R T Z 1 1 ε 1, µ 1 ε, µ I R I R T I R T Does this look Z Z Z familiar? T
Reflection and Transmission! Problem reduced to + and! Similar to LC transmission lines Z Z1 Z R I H R Z + Z Z! Solution: T 1 Z Z + Z 1 I R T! Determined by the impedance Z Z Z I R T 1 1! Reflectivity R how much power is reflected R 1 Z1+ Z Z Z I xample: H Z T Z + Z 1 1 Z1 Z + Z 1 H H air Z 377 I I S S R T Z Z Z + Z 4ZZ 1 ( Z + Z ) 1 1 1 Ω Z glass 5Ω Rair glass.4 About 4% reflection S I S I
Index of Refraction! Index of refraction n is defined by! Or n εµ ε µ! For most insulators, µ µ c w c n 1 εµ Approximate n! People like to use this approximation! Probably because n is more familiar concept than Z!.g. reflectivity is ε ε R Z µ µ ε ε Z1 Z 1 n1 1 n n1 n Z1+ Z 1 n1+ 1 n n1+ n Z n
Conductors! lectric field in conductor causes current! Now we need the full form of Maxwell s equations ρ ε t εµ + µ J! asy to derive wave equations ε µ + µ σ Conductivity J σ Free charge can t exist in conductor µσ ε µ + µ σ New term looks like damping
Conductors! We can still find plane waves! Plug in to the wave equation ε µ + µ σ! We ve seen imaginary k before! Re-define k to split real and imaginary parts ( ) k! Waves moving toward +z look like + iκ εµω + iµσω e e κz i( kz ωt) e k e e i( kx ωt) εµω + iµσω κ z i( kz ωt) k is imaginary e i( kx ωt) xponentiallyshrinking oscillation
Skin Depth! M waves in conductor shrinks exponentially! Amplitude decreases by 1/e at! M waves penetrate metal only a few times the skin depth d 1 κ Skin depth e κ z z! Is the sign of κ right? ( ) k + iκ εµω + iµσω kκ µσω >! Signs of k and κ always the same " e κz shrinks as e i(kz ωt) moves forward
Skin Depth k! Solve for k and κ ( ) k εµ σ ω 1+ + 1 εω! κ grows with frequency! At low frequency " + iκ εµω + iµσω 1/ κ εµ σ κ ω 1+ 1 εω µσω! Skin depth gets shorter as the frequency increases! High frequency waves can be stopped by thin metal 1/
Superconductors! Perfect conductor has infinite conductivity ( ) k+ iκ εµω + iµσω σ κ d 1κ! M waves cannot enter superconductors Skin depth is zero!! Generally, current in conductor flows in the skin depth! For normal conductors, it s thinner for higher frequencies! DC current (ω ) flows the entire thickness! For superconductors, it s zero no matter what the frequency! Only the surface matters. ulk is useless! Superconducting cables are made of very thin filaments
Summary! Discussed M waves in vacuum and in matter! Maxwell s equations " Wave equation " Plane waves c c! and are transverse " Polarization! M waves in insulators propagate according to c w! M waves in conductors limited by the skin depth! Thinner for higher frequencies! Zero for perfect (super) conductors vacuum c 1 n Z Z εµ µ µ ε ε n Wave velocity Reflectivity