short version : 29. maxwell’s equations & em waves
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Short Version : 29. Maxwell’s Equations & EM Waves. 29.1. The Four Laws of Electromagnetism. 4 Laws of EM (incomplete). How q produces E ; E lines begin & end on q’s. Gauss for E. No magnetic monopole; B lines form loops. Gauss for B. Changing B gives emf. Faraday. Ampere - PowerPoint PPT PresentationTRANSCRIPT
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Short Version : 29. Maxwell’s Equations & EM Waves
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29.1. The Four Laws of Electromagnetism
Law Mathematical Statement
Physical Meaning
Gauss for E0
qd
E A How q produces E;E lines begin & end on q’s.
Gauss for B
Faraday
Ampere(Steady I only)
4 Laws of EM (incomplete)
0d B A
Bddd t
E r
0d IB r
No magnetic monopole;B lines form loops.
Changing B gives emf.
Moving charges give B.
Note E-B asymmetry between the Faraday & Ampere laws.
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29.2. Ambiguity in Ampere’s Law
B in a RC circuit.
Ampere’s law: 0Cd IB r
I is current through any open surface S bounded by C.
Current flows through surfaces 1,2,& 4. But not 3. Ampere’s law fails ( for non-steady current ).
Maxwell’s modification:
0 0 0E
C
dd Id t
B r
0 Displacement curre ntEdd t
Changing E gives I , which in turn gives B.
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29.3. Maxwell’s Equations
Law Mathematical Statement
Physical Meaning
Gauss for E0
qd
E A How q produces E;E lines begin & end on q’s.
Gauss for B
Faraday
Ampere-Maxwell
0d B A
Bddd t
E r
No magnetic monopole;B lines form loops.
Changing B gives emf.
Moving charges & changing E give B.0 0 0
Edd Id t
B r
Maxwell’s Eqs (1864).Classical electromagnetism.
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Maxwell’s Equations in Vacuum
Gauss for E 0d E A
Gauss for B
Faraday
Ampere-Maxwell
0d B A
Bddd t
E r
0 0Edd
d t
B r
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29.4. Electromagnetic Waves
Electromagnetic (EM) waves
Faraday’s law:
Ampere-Maxwell’s law:
changing B gives E.
changing E gives B.
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Plane Electromagnetic Wave
EM wave in vacuum is transverse: E B k (direction of propagation).
For uniqueness, see Prob 46
Sinusoidal plane waves going in x-direction:
ˆ, ,yx t E x tE j
ˆ, ,zx t B x tB k
ˆ k E B Right-hand rule
ˆsinpE kx t j
ˆsinpB kx t k
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Gauss’s Laws
ˆsinpE kx t j
Both E & B field lines are straightlines,
so their flux over any closed surfaces vanish identically.
Hence the Gauss’s laws are satisfied.
Plane wave : ˆ, ,yx t E x tE j
ˆ, ,zx t B x tB k ˆsinpB kx t k
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Faraday’s Law
For loop at x of height h & width dx :
, ,d E x t h E x dx t h E r
, , EE x t h E x t d x hx
E d x hx
B B h dx Bd B h dxd t t
Faraday’s Law : E Bx t
Faraday’s law expressed as a differential eq :t
BE
Bddd t
E r
ˆEE y
ˆBB z
y zE Bx t
x y z
x y zV V V
i j k
V
d E A dt
B A
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Ampere-Maxwell Law
0 0E
C
ddd t
B r
C
d B x h B x dx h B r
I = 0
For loop at x of height h & width dx :
, , BB x t h B x t d x hx
B d x hx
E E h dx Ed E h dxd t t
Ampere-Maxwell Law : 0 0B Ex t
Ampere-Maxwell law expressed as a differential eq :
0 0 t
EB in vacuum
ˆBB z
ˆEE y
0 0yz EB
x t
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Conditions on Wave Fields
Faraday’s Law : E Bx t
Ampere-Maxwell Law : 0 0B Ex t
For E = E(x,t) j & B = B(x,t) k,
For a plane wave ˆ, sinpx t E kx t E j ˆ, sinpx t B kx t B k
cos cosp pk E k x t B k x t Faraday’s Law : p pk E B
Ampere-Maxwell Law : 0 0cos cosp pk B k x t E k x t
0 0p pk B E
x y z
x y zV V V
i j k
V
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29.5. Properties of Electromagnetic Waves
p pk E B 0 0p pk B E
2 20 0k
0 0
1k
speed of wave =
7 2 2 29
114 10 / /
4 9 10N A C N m
83 10 /m s = speed of light in vacuum = c
Maxwell: light is EM wave.
1983: meter is defined so that c is exactly 299,792,458 m/s.
Hence, 0 = 1 / (4 c2 107 ) C2/Nm2, where c = 299,792,458.
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Example 29.2. Laser LightA laser beam with wavelength 633 nm is propagating through air in the +z direction.
Its electric field is parallel to the x axis and has magnitude 6.0 kV/m. Find
(a) the wave frequency,
(b) the amplitude of the magnetic field, and
(c) the direction of the magnetic field.
cf
(a)8
9
3.0 10 /633 10
m sm
144.7 10 Hz
EBc
(b)
(c) y axis.
3
8
6.0 10 /3.0 10 /
V mm s
52.0 10 T
20 T
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PolarizationPolarization // E.
Radiation from antennas are polarized.
E.g., radio, TV, ….
Light from hot sources are unpolarized.
E.g., sun, light bulb, …
Reflection from surfaces polarizes.E.g., light reflecting off car hoods is partially polarized in horizontal direction.
Transmission through crystal / some plastics polarizes.E.g., Polaroid sunglasses, …
Law of Malus :
Only component of E // preferred direction e is transmitted.
2 2 2costrans inc E E
ˆ ˆtrans inc E E ε ε ˆcosincE ε = angle between Einc & .
2costrans incS S or
ˆˆ ε j
ˆˆ ε k
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20 cosS S
2 polarizers with mutually perpendicular transmission axes.
No light gets through where they overlap.
Polarization of EM wave gives info about its source & the medium it passes through.
Applications: astronomy, geological survey, material stress analysis, …
Liquid crystal display (LCD)
Vertical polarizer passes only Ev .
Unpolarized light
LC molecules rotate polarization to horizontal direction.
Horizontal polarizer passes light.
Applied V aligns molecules; polarization not rotated.
Horizontal polarizer blocks light.
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Conceptual Example 29.1. Crossed Polarizers
Unpolarized light shines on a pair of polarizers with their transmission axes perpendicular, so no light gets through the combination.
What happens when a third polarizer is sandwiched in between, with its transmission axes at 45 to the others?
1st & middle polarizers not so some light passes through.
Passed light’s polarization not to axis of last polarizer so some light passes through.
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Making the Connection
How does the intensity of light emerging from this polarizer “sandwich” compare with the intensity of the incident unpolarized light?
2costrans incS S
Intensity of light emerging from 1st polarizer :
22
10
1 cos2incS S d
12 incS ( polarized along axis of 1st polarizer )
Intensity of light emerging from middle polarizer :
22 1 cos 45S S 1
12S
14 incS
Intensity of light emerging from ensemble :
23 2 cos 45S S 2
12S
18 incS
( polarized along axis of middle polarizer.)
( polarized along axis of 3rd polarizer.)
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29.6. The Electromagnetic Spectrum
Earth’s atmosphere:
Transparent to: most radio, visible light.
Opaque to: most IR, upper UV, X-rays, rays.
UV is absorbed by ozone layer
IR by green house gases.
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29.7. Producing Electromagetic Waves
Any changing E or B will create EM waves.
Any accelerated charge produces radiation.
Radio transmitter: e’s oscillate in antenna driven by LC circuit.
X-ray tube: accelerated e’s slammed into target.
MW magnetron tube: e’s circle in B.
EM wave :f = f of q motionMost efficient: ~ dimension of emitter / reciever
Waves emit / receive axis of dipole.
Source replenishes radiated energy
LC oscillator drives I in antenna
Outgoing EM waves
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29.8. Energy & Momentum in Electromagetic Waves
Consider box of thickness dx, & face A k of EM wave.
20
12Eu EEnergy densities:
2
0
12Bu
B
Energy in box:
E BdU u u A dx 2 20
0
1 12
A dx
E B
Rate of energy moving through box:/
dU d Udt dx c
2 20
0
1 12
A c
E B
Intensity S = rate of energy flow per unit area 2 20
0
1 12
S c
E B
Plane waves:
EBc
00
1 12
S c E B cc
2
0 00
1 12
c E B
0
1 E B
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0
1S E B
Plane waves:
In general:0
1
S E B Poynting vector see Prob 64
Average intensity for plane waves :
0
1S E B
0
12
pk pkE B
E, B in phase
2
0
12pkEc
2
0
12pkBc
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Example 29.3. Solar Energy
The average intensity of noontime sunlight on a clear day is about 1 kW/m2.
(a) What are the peak electric & magnetic fields in sunlight ?
(b) At this intensity, what area of 40% efficient solar collectors would you need to replace a
4.8-kW water heater ?
62.9 10 T
(a)2
0
12pkBS c
02
pkSB
c
7 3 2
8
2 4 10 / 10 /3 10 /H m W m
m s
2.9 T
pk pkE c B 28.7 10 /V m 0.87 /kV m
(b)Area needed is 2
2
4.8 1240% 1.0 /
kW mkW m
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Waves from Localized Sources
Afar from localized source, wave is spherical :
24PSr
Intensity = power / area
2 2,S E B 1,E Br
wave fields dominates static fields away from the sources.
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Example 29.4. Cell Phone Reception
A cell phone’s typical average power output is about 0.6 W.
If the receiver at a cell tower can handle signals with peak electric fields as weak as 1.2 mV/m,
what is the maximum allowable distance from cell phone to tower ?
24PSr
02
24 pk
c PrE
2
0
12pkEc
7 8
23
2 4 10 / 3 10 / 0.6
4 1.2 10 /
H m m s W
V m
5 km
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Application: Cell Phones
Hexagonal cell area 25 km2.
~ circle of radius
6 225 10 mr
2.8 km
Transmission & reception are at different frequencies.
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Momentum & Radiation Pressure
Maxwell :
1 ˆrad U
cp S
1rad c
P S
radiation momentum
radiation pressureon absorbing surface
12rad cP S
radiation pressureon reflecting surface
Cosmos 1, a solar light-sailing spacecraft, failed at launch in 2005.