Wave PhenomenaPhysics 15c

Lecture 17Radiation of EM Waves

(H&L Chapter 10)

What We Did Last Time

! Studied reflection and refraction! Derived Snell’s law three times

! Using Huygens’ principle! Using Fermat’s principle

! Light “chooses” the fastest path between two points

! Solving boundary-condition problem! Fresnel coefficients " intensities of reflected/refracted light

! Brewster’s angle " Reflection is polarized

2 1

1 2

sinsin

nn

θθ

=

1

2

ZZ

β ≡

2

1

coscos

a θθ

≡2

VR α βα β

−= +

2

4( )

VT αβα β

=+

211

HR αβαβ

−= +

2

4( 1)

HT αβαβ

=+

IE

IB

TETB

RERB1θ

2θ

Reflectivity and Transmittivity

! R + T = 1 for both cases

2VR α β

α β −

= + 2

4( )

VT αβα β

=+

211

HR αβαβ

−= +

2

4( 1)

HT αβαβ

=+

Vertical polarization Horizontal polarization

Brewster’sangle

For air " glass, β = 1.5

VR

VT

HR

HT

Slow " Fast Transition

! What if n1 > n2?

! R = 1, T = 0 beyond the critical angle

For glass " air, β = 1/1.5

Total internal reflection

Total internal reflection

VR

VT

HR

HT

Total Internal Reflection

! Angle of refraction is determined by the difference between wavelengths! Wavelength along the surface

satisfy

! What if we choose θ2 so thatis shorter than λ1?

! Can’t match the wavefrontsno matter what θ1 we use

! No EM waves above water

1λ

2λ

1θ

2θ

sλ1 2

1 2sin sinsλ λλ

θ θ= =

2 2sinsλ λ θ=1λ

2λ2θ

sλ

Boundary Condition

! Examine boundary conditions! No EM Field above water!

!

!

!

! Only solution is! Something went wrong…

θ θIE

IB RERB

1 2E E=! !

1 1 2 2E Eε ε⊥ ⊥=

1 2H H=! !

1 2B B⊥ ⊥=

( ) cos 0I RE E θ+ =

2 ( )sin 0I RE Eε θ− =

0I RH H− =

Nothing0I RE E= =

Oops!No waves anywhere

1 1,ε µ

2 2,ε µ

Imaginary Waves

! There must be EM field above water! Boundary conditions cannot be satisfied without it! But it cannot be usual EM waves either

! Solution: above water

! Wave equation

! Wavelength along the surface is linked to

( ) ( )0 0

i kx i z t z i kx te e eκ ω κ ω+ − − −= =E E E2 2

2 12 2

nc t

∂∇ =

∂EE

22 2 21

2

nkc

κ ω− =

2s kλ π=

2

2

2sin sins

cn

λ πλθ ω θ

= = 2 sinnkc

ω θ=

2 2 22 1sinn n

cωκ θ= −

Imaginary Waves

! EM “waves” above water satisfies wave equation and boundary conditions! It shrinks as e–κz " Not traveling waves

! “Range” of the imaginary waves is

! Comparable to

! Total internal reflection creates “leakage field” that extends a few wavelengths! You can detect it by having two

boundaries close enough to each other

( )0

z i kx te eκ ω− −=E E

2 2 22 1

1sin

cn nκ ω θ

=−

11

2 cn

πλω

=

Brewster’s Angle

! OK, so we know reflected light is polarized, so what?! Is there anything deeper than solving equations?

! Look at the condition α = β again! Assume

! A little trig gives us

2 1 2 2 1

1 2 1 1 2

sinsin

nn

ε µ ε θβε µ ε θ

≡ = = =2

1

coscos

θαθ

≡

1 2µ µ=

2 1

1 2

cos sincos sin

θ θθ θ

=

1 2sin 2 sin 2θ θ= 1 2θ θ= 1 2 2πθ θ+ =or

This isn’t the case This must be

2VR α β

α β −

= +

Goals for This Week

! At Brewster’s angle,! Reflected light and refracted light are

perpendicular to each other! Is this a coincidence?

! Seems farfetched! If not, what is the significance?

! We need to dig into the origin of refraction! What makes matter dielectric?

! We did this for coaxial cable in Lecture #12! How does it cause light to slow down?

1 2 2πθ θ+ =

1θ 1θ

2θ

Creating EM Waves

! Look at the “full” version of Maxwell’s equations

! We need electric charge to create EM waves

! The charge must be moving! Stationary charge create only static E field! Constant current create only static B field

t∂

∇× = −∂BE

0∇ ⋅ =B0

ρε

∇ ⋅ =E

0 00 tµε µ ∂

∇× =∂

+EB J

SI

Charge density Current density

! Let’s move a point charge +q with constant velocity v! H&L Section 10.2 calculates E(r) and B(r)! Poynting vector

! No power is radiated outward

! We shouldn’t be surprised! Think about Relativity! There is a coordinate system in which +q is at rest

! No energy is radiated by EM waves in that frame! Same should be true in any coordinate system

Moving Charge

vq+

EB

Sr⊥S r

Constant velocity " No radiation

inertial frame

E&M Puzzle

! Charge +q creates E at position r! Exactly which direction does E point?

! Parallel to r, right?

! EM field travels at speed c! Field at r is not determined by

where +q is now, but T = r/c ago! In the meantime, +q moved by! So…

! Which point does the tail of E point?! Where +q is now, or where it was at t = −T ?

vq+

E

r

TvvrvTc

= nowt = -T

E&M Puzzle

! Answer: where +q is now! See your E&M textbook for a real explanation

! Sloppy explanation:! Consider two charges +q and −q

moving in parallel! Let –q fall toward +q

vq+

E

r

vq+

vq−

E

F

E

F

“now”

“past”

Hit

Miss

# Watch this in a frame where v = 0" They will collide

# “Now” hypothesis is correct

How Does It Know?

! But how does E at r know where the charge is now?! It can’t – Special Relativity!

! Sounds odd, but that’s the only Relativistically-Correct way

! E doesn’t necessarily point back to the actual location of the charge if it accelerated (or decelerated)! We are on to something now…

vq+

E

rE(r, t) is determined by what q was doing at t – r/c, in such a way that it points back to where q should be now if v has beenconstant

Accelerating Charge

! Consider this picture:! Charge +q is at rest until t = 0! It accelerates for ∆t by a = dv/dt! Keep constant velocity v after ∆t

! Want to know E at large distance r! Easy for t < 0 and t > ∆t! What happens during the

acceleration?q+

v

t = 0 t = ∆t

r

E

Before/After Acceleration

A B C

: 0A t =:B t t= ∆:C t t T= ∆ +

acceleratebetween

now

Radius c(∆t + T) from A# Outside this circle,

acceleration has not happened

# E points outward from A

Radius cT from B# Inside this circle, the

acceleration has finished# E points outward from C

How can we connect them?

During Acceleration

! E field lines must becontinuous! They start and end

only where there ischarge

! So we just connectthem like "

! Acceleration causes“kinks” in E field! Let’s looks at it

more closely

A B C

Radiated Field

! We are interested in E at large distance r! Radial component is

usual Coulomb field! Geometry gives us the ratio

! ET proportional to accelerationA B C0t = t∆ t T∆ +

c(∆t

+ T)

cT

θ

0ETE

0 204

qErπε

=

0

sinTE vTE c t

θ≈

∆v c"

t T∆ "assuming

20

20

sin4

sin4

Tq vTE

r c tqa

c r

θπε

θπε

=∆

=

c t∆sinvT θ

vat

=∆

rTc

=

Radiated Field

! At large r,! i.e., E becomes transverse! This is the EM radiation!

! We like to write it this way:! E is transverse! Proportional to acceleration a

! Direction opposite to a! Stronger radiation at 90°! Decreases with 1/r

00

24qE

rπε= 2

0

sin4Tqa

rE

cθ

πε=

0TE E#

0 sin ( )4

rT c

qE E a tr

µ θπ

= = −

20 0 1/ cµ ε =

Delayed by r/c due to propagation speed

a

Radiated Power

! Poynting vector is

! Intensity of radiation is not isotropic, but! It goes with square of q

! Sign of charge doesn’t matter! It goes with square of a

! Deceleration radiates just the same! It goes with 1/r2

! As it should for spherical waves

0 sin ( )4

rT c

qE a tr

µ θπ

= − 0 sin ( )4

T rT c

qEB a tc c r

µ θπ

= = −

2 220

2 20

sin ( )16

T rT c

qBS E a tc r

µ θµ π

= = −

2sin θ∝a

EM Waves in Matter

! Matter is made of charged particles! Electrons and protons, in particular

! EM waves in matter make them oscillate! They radiate EM waves! EM waves in matter = original + radiated waves! This is why EM waves propagate differently in matter

IncomingRadiated

Brewster’s Angle

! Why is there reflection in the first place?! Because! Electrons in medium 2 is responsible

! Which way do the electrons oscillate?! Parallel to ET

! They radiate EM waves as

! At Brewster’s angle, reflection isparallel to ET " θ = 0 " No reflection!

1θ 1θ

2θ

IE

IB RERB

TE

TB

1 2Z Z≠

TE2 220

2 2

sin ( )16

rc

qS a tc r

µ θπ

= −Angle of radiation

relative to ET

Larmor Formula

! What’s the total power radiated by a moving charge?! Integrate S over the surface

of a sphere at r

! Accelerating charge loses its energy at a rate proportional to acceleration squared! Think about an atom as electrons circling around the nucleus

! Circular motion " acceleration " radiation " loss of energy " electrons fall into the nucleus

! Paradox stimulated early development of QM

Joseph Larmor(1857–1942)

2 22 2 300 0

00

2 2

sin sin8 6q a q aP d Sr

cd d

cπ π πµφ θ θ θ θ

ππµ

= = =∫ ∫ ∫

2 220

2 2

sin ( )16

rc

qS a tc r

µ θπ

= −

Larmorformula

Summary

! Reviewed reflection and refraction! Total internal reflection is more subtle than it looks

! Imaginary waves extend a few λ beyond the surface! Studied how to create EM waves

! Accelerated charge radiates EM waves! Power proportional to (acceleration)2

! Polarization parallel to the acceleration! Why reflection is polarized at Brewster’s angle

! Next: why light travels slower in matter?

0 sin ( )4

rT c

qE a tr

µ θπ

= −

2 20

6q aP

cµ

π=