Intro Physics II
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Intro Physics II Physics 11b Lecture 20
!
HW #9 due tomorrow (4/26)
!
Final Exam !
Thurs. May 15, 9:15, Science Center C
E & M in Free Space EM Waves Light Crisis!
What We Did Last Time
E and B in Free Space r
!
Displacement current !
!
!
Draw connections between electrostatic summary and magnetism summary ! Induction ! Displacement Current
Gauss’ Law
r r B ⋅ da = 0
No monopoles
ClosedSurface
Last piece, demanded by Ampere’s Law
Maxwell Equations (complete)
r
∫∫ E ⋅ da = 0
∫∫r
ClosedSurf ace
r dΦ B E ⋅ dl = − ∫ dt ClosedPath r r dΦ E B ⋅ d l = µ 0ε 0 dt ∫
Closed Path
r r F( )
r r r r r E( ) B( )
Energy conservation (Kirchoff) + Faraday’s Law Ampere’s Law + Displacement Current Lorenz Force Law
Q=I=0 Simple Form
How particles respond to fields
1
Hmmm…
ANY E and B in Free Space !We will now show a very curious fact; if ! !
!Line integral of E around rectangle in xy plane
Ey=E0sin(kx-ωt) Bz=B0sin(kx-ωt)
!
Ey=E0sin(kx-ωt) !Bz=B0sin(kx-ωt) !
! !
Flux of B through loop
!
Displacement sez
!
These fields satisfy our equations of E & M !
Bdy-(B+dB)∆y=-dB∆y
!
dΦE/dt=(dE/dt)dx∆y !
Even in free space No sources necessary
B B+dB
-dB∆y=µ0ε0(dE/dt)dx∆y
∆y
− µ 0ε 0
dE dB = dt dx
dx
ANY E and B in Free Space
Putting it all together
!Line integral of B around rectangle in xz plane !
Flux of E through loop
!
Faraday sez
dΦB/dt=(dB/dt)dx∆z !
dx
∆z E
(E+dE)∆z-E∆z=dE∆z
!
!
E+dE
−
dE∆z= −(dB/dt)dx∆z
dE
=−
dB
−
∂ ∂E
=
∂E ∂B = ∂x ∂t
∂ ∂B ∂ ∂E
=−
− µ 0ε 0
∂ ∂B
∂ ∂B
∂ ∂E
= − µ 0ε 0 −
2
2
∂ E2
=µε
0 0
∂ E 2
∂E ∂B = ∂t ∂x
1 0 0
∂ ∂B ∂ ∂E = ∂ ∂ ∂ ∂t 2
2
∂ B2
=µε
0 0
∂ B 2
2
Hmmm…
The Velocity k E0sin(kx-ωt)=µ0ε0ω2 E0sin(kx-ωt) !Only works if ω2/k2=1/µ0ε0 !sin(kx-wt)=sin[k(x-w/kt)] ! 2
!
1 sin k x − t µ 0ε 0
!
Ey=E0sin(kx-ωt) !Bz=B0sin(kx-ωt)
Exactly the same happens for B !
!
∂2E ∂2E = µ 0ε 0 2 2 ∂t ∂x
∂2B ∂2B µ ε = 0 0 ∂x 2 ∂t 2
Recognize form of sin as a traveling wave from 11a!
What is value of 1/Sqrt(µ0ε0)? !
1 sin k x − t µ 0ε 0 !
Traveling wave
!
Ey=E0sin(kx-ωt) !Bz=B0sin(kx-ωt)
!
=3x108 m /s
!
This is the speed of light
!
E, B fields propagate through free space (no charges) at speed of light
!
!
Velocity
ε0=8.85x10-12 C2/(Nm2) µ0=4π x 10 -7 Tm/A
They ARE light
Conditions Velocity with respect to what? You can measure all the properties on earth !Or the sun !
!
Traveling wave with speed c
sin[k (x − ct )]
!
!
You get the same number (laws of physics are the same)
But you get same speed of light
!
!
Even though earth is Moving wrt the sun
Light should have the same speed regardless = !Speed of light seems internally inconsistent! !
! !
Least of our problems Let this guy take care of it
B and E perpendicular to one another !Each perpendicular to direction of motion !ExB is positive direction of motion !B/E=1/c !
! !
Ey=E0sin(kx-ωt) Bz=B0sin(kx-ωt)
!
!
From
dE
=−
dB
ω is free parameter !
Can be anything
3
Energy in Light
Poynting Vector !We already know how to calculate
Add uB and uE
!
!
!What is energy transport per unit time per unit area? !Can show that it is
Energy density
r 1 r r S = (E × B ) µ0
1 1 B2 u = ε0E2 + 2 2 µ0 ! !
Ey=E0sin(kx-ωt) Bz=B0sin(kx-ωt)
!
!
Use B=E/c, 1/c2=µ0ε0
!
1 1 µ 0ε 0 E 2 = ε0E 2 u = ε0E 2 + 2 2 µ0
If charge is stationary for a long time, field gets established !Imagine it starts moving with velocity v !After time t, field lines far away (outside circle of size ct) still static !
Cannot send any message faster than speed of light
Nearby fields (inside circle) change !Field lines cannot end !
ct
Lines inside and outside of circle must be connected along the circle
“Kink” in fields is EM wave; moves out to large distances !Accelerated charges produce EM waves
!
P=
2
2q a
Visible Light !
!
!
!
~100 MHz Wavelength~3 m
Microwaves ! !
!
400-700 nm wavelength ! c=fλ=2πω/λ ! f=750 THz to 420 THz
Radio waves !
!
2
E RMS BRMS µ0
The Spectrum of Light
!
vt
!Vector tells you direction of transport !Time-average of sines
S=
How Do We Get Light? !
Ey=E0sin(kx-ωt) Bz=B0sin(kx-ωt)
~1 MHz Wavelength ~cm
X-rays !
Wavelengths ~pm ! “Picometers”
3
4
Energy in Two Waves !The energy in one wave is proportional to E02
!
If you have two waves !
!
!
E-fields superimpose
Interference Cartoon Waves of amplitude E propagate from sources 1 and 2
!
!
E’=E1+E2
! !
Energy adds
!
!
There is the problem of interference
U=U1+U2=E12+E12 ! Or? U=E’2=(E1+E2)2=E12+E22+2E1E2
!
!
Along solid lines ! ! !
!
! !
!
d
= sin θ !
E1=-E2 out of phase U α (E1+E2)2 = 0 “Destructive Interference”
Sunrise on Beebe Lake
Along solid lines !
λ
E1=E2; in phase U α (E1+E2)2 = 4E2 “Constructive interference”
Along dashed lines !
Interference Cartoon
U1 α E2 U2 α E2 U1+U2 α 2E2
! !
E1=E2; in phase U α (E1+E2)2 = 4A “Constructive interference”
Along dashed lines ! ! !
E1=-E2 out of phase U α (E1+E2)2 = 0 “Destructive Interference”
5
Waves !All the usual wave equations apply
Accelerating Charges P=
dE 2q 2a 2 = dt 3c 3
a=
v2 r
!Accelerating charge should radiate energy, as we saw !Suppose an electron orbits an atom !
Constantly radiates power and spirals inwards toward nucleus
!According to this, atoms are not stable!
Line Frequencies !When you heat atoms of various kinds ! !
Out comes light At a particular frequency
!Frequencies are continuous in E & M !
!
Integral and differential equations give CONTINUOUS equations
What causes this?
6
More on Energy
Blackbody Radiation
!Suppose you shine colored lights on a metal !
!
Summary !
E & M Waves
!
Light Crises! ! ! ! !
P=
Ultraviolet Catastrophe !
Thermodynamics is able to predict spectrum from blackbody radiation !
!
Recall StefanBoltzmann only gives total energy
Spectrum is totally wrong at high frequency (UV)!
∂2E ∂2E = µ 0ε 0 2 2 ∂t ∂ 2 B ∂x ∂ 2B = µ 0ε 0 2 2 ∂x ∂t 2
!
Blue light: electrons are ejected from metal surface ! Binding energy is overcome Red light: no electrons emitted ! Binding energy never overcome ! No matter HOW MUCH energy is in the E-fields
!
2
2q a 3
r 1 r r S = (E × B ) sin[k (x − ct )] µ 0
Atomic Stability Emission lines Photoelectric effect Blackbody radiation
7
Intro Physics II Physics 11b Lecture 20
!
HW #9 due tomorrow (4/26)
!
Final Exam !
Thurs. May 15, 9:15, Science Center C
E & M in Free Space EM Waves Light Crisis!
What We Did Last Time
E and B in Free Space r
!
Displacement current !
!
!
Draw connections between electrostatic summary and magnetism summary ! Induction ! Displacement Current
Gauss’ Law
r r B ⋅ da = 0
No monopoles
ClosedSurface
Last piece, demanded by Ampere’s Law
Maxwell Equations (complete)
r
∫∫ E ⋅ da = 0
∫∫r
ClosedSurf ace
r dΦ B E ⋅ dl = − ∫ dt ClosedPath r r dΦ E B ⋅ d l = µ 0ε 0 dt ∫
Closed Path
r r F( )
r r r r r E( ) B( )
Energy conservation (Kirchoff) + Faraday’s Law Ampere’s Law + Displacement Current Lorenz Force Law
Q=I=0 Simple Form
How particles respond to fields
1
Hmmm…
ANY E and B in Free Space !We will now show a very curious fact; if ! !
!Line integral of E around rectangle in xy plane
Ey=E0sin(kx-ωt) Bz=B0sin(kx-ωt)
!
Ey=E0sin(kx-ωt) !Bz=B0sin(kx-ωt) !
! !
Flux of B through loop
!
Displacement sez
!
These fields satisfy our equations of E & M !
Bdy-(B+dB)∆y=-dB∆y
!
dΦE/dt=(dE/dt)dx∆y !
Even in free space No sources necessary
B B+dB
-dB∆y=µ0ε0(dE/dt)dx∆y
∆y
− µ 0ε 0
dE dB = dt dx
dx
ANY E and B in Free Space
Putting it all together
!Line integral of B around rectangle in xz plane !
Flux of E through loop
!
Faraday sez
dΦB/dt=(dB/dt)dx∆z !
dx
∆z E
(E+dE)∆z-E∆z=dE∆z
!
!
E+dE
−
dE∆z= −(dB/dt)dx∆z
dE
=−
dB
−
∂ ∂E
=
∂E ∂B = ∂x ∂t
∂ ∂B ∂ ∂E
=−
− µ 0ε 0
∂ ∂B
∂ ∂B
∂ ∂E
= − µ 0ε 0 −
2
2
∂ E2
=µε
0 0
∂ E 2
∂E ∂B = ∂t ∂x
1 0 0
∂ ∂B ∂ ∂E = ∂ ∂ ∂ ∂t 2
2
∂ B2
=µε
0 0
∂ B 2
2
Hmmm…
The Velocity k E0sin(kx-ωt)=µ0ε0ω2 E0sin(kx-ωt) !Only works if ω2/k2=1/µ0ε0 !sin(kx-wt)=sin[k(x-w/kt)] ! 2
!
1 sin k x − t µ 0ε 0
!
Ey=E0sin(kx-ωt) !Bz=B0sin(kx-ωt)
Exactly the same happens for B !
!
∂2E ∂2E = µ 0ε 0 2 2 ∂t ∂x
∂2B ∂2B µ ε = 0 0 ∂x 2 ∂t 2
Recognize form of sin as a traveling wave from 11a!
What is value of 1/Sqrt(µ0ε0)? !
1 sin k x − t µ 0ε 0 !
Traveling wave
!
Ey=E0sin(kx-ωt) !Bz=B0sin(kx-ωt)
!
=3x108 m /s
!
This is the speed of light
!
E, B fields propagate through free space (no charges) at speed of light
!
!
Velocity
ε0=8.85x10-12 C2/(Nm2) µ0=4π x 10 -7 Tm/A
They ARE light
Conditions Velocity with respect to what? You can measure all the properties on earth !Or the sun !
!
Traveling wave with speed c
sin[k (x − ct )]
!
!
You get the same number (laws of physics are the same)
But you get same speed of light
!
!
Even though earth is Moving wrt the sun
Light should have the same speed regardless = !Speed of light seems internally inconsistent! !
! !
Least of our problems Let this guy take care of it
B and E perpendicular to one another !Each perpendicular to direction of motion !ExB is positive direction of motion !B/E=1/c !
! !
Ey=E0sin(kx-ωt) Bz=B0sin(kx-ωt)
!
!
From
dE
=−
dB
ω is free parameter !
Can be anything
3
Energy in Light
Poynting Vector !We already know how to calculate
Add uB and uE
!
!
!What is energy transport per unit time per unit area? !Can show that it is
Energy density
r 1 r r S = (E × B ) µ0
1 1 B2 u = ε0E2 + 2 2 µ0 ! !
Ey=E0sin(kx-ωt) Bz=B0sin(kx-ωt)
!
!
Use B=E/c, 1/c2=µ0ε0
!
1 1 µ 0ε 0 E 2 = ε0E 2 u = ε0E 2 + 2 2 µ0
If charge is stationary for a long time, field gets established !Imagine it starts moving with velocity v !After time t, field lines far away (outside circle of size ct) still static !
Cannot send any message faster than speed of light
Nearby fields (inside circle) change !Field lines cannot end !
ct
Lines inside and outside of circle must be connected along the circle
“Kink” in fields is EM wave; moves out to large distances !Accelerated charges produce EM waves
!
P=
2
2q a
Visible Light !
!
!
!
~100 MHz Wavelength~3 m
Microwaves ! !
!
400-700 nm wavelength ! c=fλ=2πω/λ ! f=750 THz to 420 THz
Radio waves !
!
2
E RMS BRMS µ0
The Spectrum of Light
!
vt
!Vector tells you direction of transport !Time-average of sines
S=
How Do We Get Light? !
Ey=E0sin(kx-ωt) Bz=B0sin(kx-ωt)
~1 MHz Wavelength ~cm
X-rays !
Wavelengths ~pm ! “Picometers”
3
4
Energy in Two Waves !The energy in one wave is proportional to E02
!
If you have two waves !
!
!
E-fields superimpose
Interference Cartoon Waves of amplitude E propagate from sources 1 and 2
!
!
E’=E1+E2
! !
Energy adds
!
!
There is the problem of interference
U=U1+U2=E12+E12 ! Or? U=E’2=(E1+E2)2=E12+E22+2E1E2
!
!
Along solid lines ! ! !
!
! !
!
d
= sin θ !
E1=-E2 out of phase U α (E1+E2)2 = 0 “Destructive Interference”
Sunrise on Beebe Lake
Along solid lines !
λ
E1=E2; in phase U α (E1+E2)2 = 4E2 “Constructive interference”
Along dashed lines !
Interference Cartoon
U1 α E2 U2 α E2 U1+U2 α 2E2
! !
E1=E2; in phase U α (E1+E2)2 = 4A “Constructive interference”
Along dashed lines ! ! !
E1=-E2 out of phase U α (E1+E2)2 = 0 “Destructive Interference”
5
Waves !All the usual wave equations apply
Accelerating Charges P=
dE 2q 2a 2 = dt 3c 3
a=
v2 r
!Accelerating charge should radiate energy, as we saw !Suppose an electron orbits an atom !
Constantly radiates power and spirals inwards toward nucleus
!According to this, atoms are not stable!
Line Frequencies !When you heat atoms of various kinds ! !
Out comes light At a particular frequency
!Frequencies are continuous in E & M !
!
Integral and differential equations give CONTINUOUS equations
What causes this?
6
More on Energy
Blackbody Radiation
!Suppose you shine colored lights on a metal !
!
Summary !
E & M Waves
!
Light Crises! ! ! ! !
P=
Ultraviolet Catastrophe !
Thermodynamics is able to predict spectrum from blackbody radiation !
!
Recall StefanBoltzmann only gives total energy
Spectrum is totally wrong at high frequency (UV)!
∂2E ∂2E = µ 0ε 0 2 2 ∂t ∂ 2 B ∂x ∂ 2B = µ 0ε 0 2 2 ∂x ∂t 2
!
Blue light: electrons are ejected from metal surface ! Binding energy is overcome Red light: no electrons emitted ! Binding energy never overcome ! No matter HOW MUCH energy is in the E-fields
!
2
2q a 3
r 1 r r S = (E × B ) sin[k (x − ct )] µ 0
Atomic Stability Emission lines Photoelectric effect Blackbody radiation
7
