Today in Physics 218: review IIdmw/phy218/Lectures/Lect_83b.pdf28 April 2004 Physics 218, Spring...

28 April 2004 Physics 218, Spring 2004 1

Today in Physics 218: review II

Here’s a laundry-list-like reminder of the contents of the second half of the course:

Retarded potentials and radiation by time-variable charge distributionsPathlength differences and diffractionElectrodynamics and the special theory of relativity

Left and right panels from “The Garden of Earthly Delights,” Hieronymus Bosch, c. 1504.


Generally useful math facts

( )

( )

3

32 2

2

1

ˆ ˆ1 , 4

ˆ1 1 1ˆ , ,

dδ τ

πδ

′= − = ⇒

= =

⎛ ⎞ ⎛ ⎞ ⎛ ⎞′= = − = −⎜ ⎟ ⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠ ⎝ ⎠

∫r r , rV

r

r rr

r r rr

r rr r rr

◊ ◊— —

— — — —

( )cos cos cos sin sin :α β α β α β± = ∓

2

0 0

sin , , 1

sin 4

d d d

d dπ π

θ θ φ πα α

θ θ φ π2

Ω = Ω ≅

Ω = =∫ ∫

Divergence and delta function

Trig identities

Solid angle


Generally useful math facts (continued)

First-order approximations

( )( )

( )( )

( )( )

( )

( ) ( )

∞ +

=∞

=+ + +∞

=∞ +

=∞

=

= − = − + − ≅+

= − = − + − ≅

−= = + + + ≅

+

= − = − + − ≅+

= = + + + ≅ +

+ = −

∑

∑

∑

∑

∑

…

…

…

…

…

2 1 3 5

02 2 4

02 2 2 2 2 1 3 5

02 1 3 5

02

0

sin 12 1 ! 6 120

cos 1 1 12 ! 2 120

2 2 1 2tan2 2 ! 3 15

arctan 12 1 3 5

1 1! 2

ln 1 1

ii

ii

i

ii i i

i

ii

i

ii

x

ii

i

x x xx x xi

x x xxi

B x x xx x xi

x x xx x xi

x xe x xi

xx

( )( )

( )

∞ +

=∞

=

= − + − ≅+

−+ = = + + + ≅ +

−

∑

∑

…

…

1 2 3

0

0

1 2 3

1!1 1 1! ! 2

i

n i

i

x xx xi

n nnx x nx x nxi n i


Generally useful math facts (continued)

Fourier transforms, 2-D

Rayleigh’s theorem

Bessel functions

( ) ( )

( ) ( ) ( )

( ) ( )

( ) ( )

( )

( ) ( )

2 2

2cos

0

1

10

1( , ) ,2

1, ,2

, ,

2

( )

i xs yt

i xs yt

mi mv u v

m

m mm m

um m

m m

f x y F s t e dsdt

F s t f x y e dxdy

f x y dxdy F s t dsdt

iJ u e dv

d u J u u J udu

u J u v J v dv

π

π

π

π

∞ ∞− +

−∞ −∞∞ ∞

+

−∞ −∞∞ ∞ ∞ ∞

−∞ −∞ −∞ −∞

−+

−

−

=

=

=

=

⎡ ⎤ =⎣ ⎦

=

∫ ∫

∫ ∫

∫ ∫ ∫ ∫

∫

∫


Retarded potentials and radiation

Retarded potentials and retarded timeRetarded potentials and the Lorentz gaugeRetarded potentials as solutions to the inhomogeneous wave equation

( ) ( )

( ) ( )

,,

,1,

t c dV t

t c dt

c

ρ τ

τ

′ ′−=

′ ′−=

∫

∫

rr

J rA r

V

V

r

r

r

r

rt t c= − r


Retarded potentials and radiation (continued)

Retarded potentials for an oscillating electric dipole

The far field

( )

( )

02

0

near rad

0rad

0

cos2 cos

2cos

2 sin2

sin sin ˆ

cos sin ˆ

p rV tcr

p rtrc c

V Vp t r c

rcp t r c

rc

θω

ω θω

ω θ ω

ω θ ω

⎛ ⎞= −⎜ ⎟⎝ ⎠

⎛ ⎞− −⎜ ⎟⎝ ⎠

= +

−=

−−

A

r

θ

Far field: r dλ



Radiated fields and intensity for an oscillating electric dipole

Total scattering cross section of a dielectric sphere

20

rad 2

20

rad 2

2 220

2

2 40

3

3 6 2 4

scattered 4

sinˆ cos

sinˆ cos

sin ˆ ˆ8

34 , 23

esc I sc

p rtcrc

p rtr cc

pc Irc

pP

caP I

c

ω θω

ω θ ω

ω θπ

ω

χ ωπσ σ

⎛ ⎞= − −⎜ ⎟⎝ ⎠

⎛ ⎞= − −⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

=

⎛ ⎞= = ⎜ ⎟⎝ ⎠

E

B

S r r

θ

φ



The color and polarization of the sky; reddening in sunsets and interstellar cloudsDemonstration of the wavelength and polarization dependence of Rayleigh scatteringMagnetic dipole radiation

( )

( )

0 0

20

2

20

2

2 220

2

2mag

elec

Electric dipole with , , = magnetic dipole:

1 sin ˆ, cos

sin ˆ, cos

sin ˆ ˆ8

1

p m

m rt tc t r cc

m rt tr cc

mc Irc

bc

ω θ ω

ω θ ω

ω θπ

ω

↔↔ ↔ −

∂ ⎛ ⎞= − = −⎜ ⎟∂ ⎝ ⎠

⎛ ⎞= = − −⎜ ⎟⎝ ⎠

⎛ ⎞ ⎛ ⎞= =⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎝ ⎠⎝ ⎠

⎛ ⎞= ⎜ ⎟⎝ ⎠

E B B E

AE r

B r A

S r r

S

S

¥

φ

θ—



Multipole expansion for the potentials in radiating systemsRadiation field in the dipole approximationRadiation by accelerating charges: the Larmorformula

( ) ( ) ( )

( )

( )

2

rad 2

rad 2

2 22

3 2 3

ˆ ˆ,

1,

sin ˆ,

sin ˆ

1 sin 2ˆ , 4 3

t r c t r cQV tr rcr

trc

pt

rcp

rcp p

Pc r c

θ

θ

θπ

− −≅ + +

≅

=

=

= =

p pr r r

A r p

E r

B

S r

◊ ◊

θ

φ



Problems with moving charges Motion, snapshots and lengthsThe Liénard-WiechertpotentialsFields from moving charges

( )

( )

( )

( )

: instead, .

,1 ˆ1

,1 ˆ1

,

rtq

V t

cq

tc

c

V tc

′≠ −

= −

=⎛ ⎞−⎜ ⎟⎝ ⎠

=⎛ ⎞−⎜ ⎟⎝ ⎠

=

r rr w

rv

vA rv

v r

r

r

r r

r r

◊

◊



Fields from moving charges.The generalized Coulomb field and the radiation field.Example: radiation by electric charge accelerating from rest, a rederivation of the Larmor formula.

( )( ) ( )2 2

3

rad

2 2 2

3 2

2 2

3

ˆ

sin ˆ423

GC

qc v

q ac

q aP

c

θπ

⎡ ⎤= − +⎢ ⎥⎣ ⎦

= +=

=

=

E u u au

E EB E

S

rr

r

r

rr

¥ ¥◊

¥

2 22

3 sin4v c

q adPd c

θπ

⎛ ⎞ =⎜ ⎟Ω⎝ ⎠



Relativistic charges and the generalized Larmorformula

Bremsstrahlung

( )( )

( )

( )

( )( )

22

5emitted

226 2

emitted 3

2 2 2

3 5emitted,B

2 26

B 3

0

5max, 8

max,

ˆ4 ˆ

23

sin4 1 cos

23

1 if 12

1 84 5

v c

v c

qdPd

qP a

cc

q adPd c

q aP

c

dP d

dP d

π

γ

θπ β θ

γ

βθ β

γ→

⎡ ⎤⎛ ⎞ ⎣ ⎦=⎜ ⎟Ω⎝ ⎠

⎡ ⎤⎛ ⎞= −⎢ ⎥⎜ ⎟⎝ ⎠⎢ ⎥⎣ ⎦

⎛ ⎞ =⎜ ⎟Ω⎝ ⎠ −

=

−≅ →

Ω ⎛ ⎞= ⎜ ⎟Ω ⎝ ⎠

u a

u

v a

r

r

¥ ¥

◊

¥



Synchrotron radiationRadiation reaction The Abraham-Lorentzformula; radiation reaction forceRadiation reaction: a fundamental inconsistency of electrodynamics.Runaway solutions and acausal“preaccelerations.”

( )

( )( )

2 2

3 3

2 2 22 2

3 5

2 24

3

2

rad 3

14 1 cos

1 sin cos

4 1 cos

23

23

q adPd c

q ac

q aP

cqc

π β θ

β θ φ

π β θ

γ

⎛ ⎞ =⎜ ⎟Ω⎝ ⎠ −

−−

−

=

= −F a


Fields as sources of radiation: Huygens’s principle.The Kirchhoff integral: “the far field is the Fourier transform of the near field.”

Diffraction

( - )( , ) i k tAF

x y dadE e ω′ ′ ′

=E r

r

( ) ( ), , ( , , ) .x yikr i k x k y

F x y NeE k k t E x y t e dx dy

rλ

∞ ∞′ ′− +

−∞ −∞

′ ′ ′ ′= ∫ ∫


Circular-aperture diffraction and the Airy patternCircular obstacles, and Poisson’s spot.

Diffraction (continued)

( )

( ) ( ) ( )

2 202 2

21

08

20

NF

F F

cE AIr

J kaI ka I

ka

πλ

θθ

θ

=

⎡ ⎤= ⎢ ⎥

⎣ ⎦

1 1.22Dλθ =


The facts about rainbows, and the short explanation of all the factsBrief survey of the history of the study of rainbowsThe geometrical optics of raindropsDispersion and the color of rainbowsBrewster’s angle and the polarization of rainbows


2

2

20

sin , cos 1

1sin sin

12 332 4

2 arcsin 4arcsin

y yr r

yn nrry n

y yr nr

θ θ

θ θ

θ θ θ π

π

= = − −

′ = =

= −

′∆ = − +

⎛ ⎞ ⎛ ⎞= − +⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠



Supernumerary arcsCaustics and diffractionAiry’s theory of the rainbow and the supernumerary arcs

( )22 32 2

302 2

drop

3 cos .4 28

cE rI w w dwhR

λ π ζπλ

⎛ ⎞⎛ ⎞ ⎜ ⎟= −⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎝ ⎠ ⎝ ⎠∫


Electrodynamics and special relativity

Relativity and the four basic areas of physics

Brief review of the basics of the special theory of relativity ( )

20

0

002

2

1 ,

1

,, ,

LvL L Lc

tt tvc

x x vty y z z

vxt tc

γ

γ

γ

γ

⊥ ⊥⎛ ⎞= − = =⎜ ⎟⎝ ⎠∆

∆ = = ∆⎛ ⎞− ⎜ ⎟⎝ ⎠

′ = −′ ′= =

⎛ ⎞′ = −⎜ ⎟⎝ ⎠


Electrodynamics and special relativity (continued)

The Lorentztransformation and four-vectorsScalar products of four-vectors, and Lorentzinvariants

0 0

1 1

2 2

3 3

0 00 0

0 0 1 00 0 0 1

x x

x x

x x

x x

γ γβγβ γ

⎛ ⎞ ⎛ ⎞−⎛ ⎞⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟−⎜ ⎟=⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟⎝ ⎠

⎝ ⎠ ⎝ ⎠

a b a bµ µµ µ=

( )

0

10 1 2 3

2

3

,

a

aa a a a a a

a

a

µµ

⎛ ⎞⎜ ⎟⎜ ⎟

= = −⎜ ⎟⎜ ⎟⎜ ⎟⎜ ⎟⎝ ⎠



The Einstein summation conventionThe Minkowski invariant intervalProper time and four-velocityFour-momentum and the relativistic energy

( )

2 2 2

2

2

2

2

02

2

0 0

22 2 2 2

1 ,

1

1

,

u

u

I x x c t d

u dxd dtdc

dd u

cdt cc cd u

cEp m mc

p p p p E p c mc

µµ

µµ

µ µµ µ

τ ητ

γτ

η γτ

η

= ∆ ∆ = − ∆ +

= − =

= = =

−

= = =

−

= = =

= = − =

x u u

p η

η



Newton’s laws in relativity

The Minkowski force

Relativistic transformation of forces

2 2

2 2

00

, 1

1

1

1 ,

dd mmdt dt u c

d ddtd d dt u c

dp dpd EK Kd d c d

F F

µµ

τ τ

τ τ τ

γ⊥ ⊥

= = =−

= = =−

= = ⇒ =

= =

pv uF p

p pK F

F F



Relativistic transformations of E and B.

( ) ( )( ) ( )

( )( )

, , ,

, , .

Or:

, ,

, .

x x y y z z z y

x x y y z z z y

E E E E B E E B

B B B B E B B E

γ β γ β

γ β γ β

γ

γ⊥ ⊥ ⊥

⊥ ⊥ ⊥

= = − = +

= = + = −

= = +

= = −

E E E E B

B B B B E

¥

¥

β

β



The electromagnetic field four-tensor.

00 01 02 03

10 11 12 13

20 21 22 23

30 31 32 33

0

0

0

0

.0

0.

0

0

x y z

x z y

y z x

z y x

x y z

x z y

y z x

z y x

E E EF F F FE B BF F F F

FE B BF F F FE B BF F F F

A AB B B

B E EG

B E E

B E E

µν

µ ν ν µ

µν

⎛ ⎞ ⎛ ⎞⎜ ⎟ ⎜ ⎟

− −⎜ ⎟ ⎜ ⎟= =⎜ ⎟ ⎜ ⎟− −⎜ ⎟ ⎜ ⎟⎜ ⎟ ⎜ ⎟− −⎜ ⎟ ⎝ ⎠⎝ ⎠

= ∂ − ∂

⎛ ⎞⎜ ⎟− −⎜ ⎟

= ⎜ ⎟− −⎜ ⎟⎜ ⎟− −⎝ ⎠



Charge and current densities, the Maxwell equations, and the Lorentzforce, in tensor formThe four-potential and gauge transformationsThe relativistic analogue of the inhomogeneous wave equation for potentials.

( )

( )2

, , 04 , 0

, , 04

J c J

F J Gc

qK F

cA V A

A A Jc

µ µµ

µν µ µνν ν

µ µνν

µ νν

µ ν µ µν

ρ

π

η

π

= ∂ =

∂ = ∂ =

=

= ∂ =

≡ ∂ ∂ = −

J

A

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Today in Physics 218: review IIdmw/phy218/Lectures/Lect_83b.pdf28 April 2004 Physics 218, Spring...

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