Lecture10 maxwells equations
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Transcript of Lecture10 maxwells equations
29. Maxwell’s Equations
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Topics
Laws of Electric & Magnetic Fields
James Clerk Maxwell
Maxwell’s Equations
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Laws of Electric & Magnetic Fields
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James Clerk Maxwell1831 – 1879
In 1865, Maxwell published a paper entitled: A Dynamical Theory of the Electromagnetic Field,Philosophical Transactions of the Royal Society of London 155, 459-512 (1865). This is one of the greatest scientific papers ever written.
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Maxwell’s Equations
inside
0Closed Surface
QE dA
ε× =∫
rr
Closed Surface
0B dA× =∫rr
Closed Loop
mdE dr
dt
ϕ× = −∫r r
0 0
Closed Loop
0eB dr I
d
dtµ µ ϕε× = +∫
r r
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Displacement Current
Maxwell realized that Ampere’s law is notvalid when the current is discontinuous asis true of the current through a parallelplate capacitor:
Encircled
Closed Lo
0
op
B dr Iµ× =∫r r
wikimedia.org
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Displacement Current
He concluded that when the charge within an enclosed surface is changing it is necessary to add to Ampere’s law another current called the displacement current: ID
inside0D
edQI
t dtd
dϕε= =
wikimedia.orgD
Closed Loop
0 ( )B dr I Iµ× = +∫r r
The 2nd Unification of Forces
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The 2nd Unification of Forces
µ0 is the magneticconstant
7 20 4 10 N/Aµ π −= ×
ε0 is the electricconstant
12 2 20 8.854 10 C /(N m )ε −= × ×
70 0
12
1
2
2 2
27 2
4 10 [ ]
8.854 10 [ ]
1.1
N/A
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C /(N m )
s10 [ ]/m
µ ε π −
−
−
= ×
× ×= ×
×
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The 2nd Unification of Forces
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0 0
12.998 10 m/s
µ ε= ×
From7
02
0211.113 1 s /m0 [ ]µ ε −= ×
we can write
which is the speed of light in vacuum!
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Light
“We can scarcely avoid the conclusion that light consists in the transverse undulations of the same medium which is the cause of electric and magnetic phenomena.” (1866)
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5th Unification?
4th Unification?
3rd Unification
2nd Unification
1st Unification
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Summary
Maxwell’s EquationsGauss’s Law for EGauss’s Law for BFaraday’s LawAmpere’s Generalized Law
Electromagnetic Waves
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Topics
Maxwell’s Wave Equations
Waves – Recap
Electromagnetic Waves
Electromagnetic Radiation
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Maxwell’s Wave Equations
2 2
2 2 2
1E E
x c t
∂ ∂=∂ ∂
r rWave equation for E
2 2
2 2 2
1B B
x c t
∂ ∂=∂ ∂
r rWave equation for B
These equations describe electric andmagnetic waves traveling in the x direction
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Maxwell’s Wave Equations
yzBE
x t
∂∂ =∂ ∂
Relationship betweenEz and By
yzEB
t x
∂∂ = −∂ ∂
Relationship betweenBz and Ey
Maxwell showed that the different components of the electric and magnetic fields are related:
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Waves – Recap
( ) sin( )Ay x kx=
( ) sin ( )y x k x tA v= −
Stationary wave
Wave traveling in x direction
Wave number
2k
λπ=
2kv
T
πω = =
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Electromagnetic Waves
p( , ) sin( )yE x t kE x tω= −
Consider an electric wave, traveling in the positive x direction, but oscillating in the y direction:
We can find Bz from
yzEB
t x
∂∂ = −∂ ∂
Ep
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Electromagnetic Waves
p( , ) sin( )zB x t kB x tω= −This leads to the result
where
p p( / )kB Eω=
p pE cB=Bp
zthat is,
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y
xz
Electromagnetic Waves
p
p
ˆsin( )
ˆsin( )
E kx t j
B kx
E
B t k
ω
ω
= −
= −
r
r
Electromagnetic waves always travel in the direction ofthe Poynting vector:
0
E BS
µ×=
r rr
Units: W/m2
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Electromagnetic Waves
But the direction of the electric and magnetic fields themselves, that is, their polarization, can change
y
xz
Linear polarization
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Polarizers
1
2
3 90o
20 cosS S θ=
Law of Malus
Only a componentEpcosθ of the electric field along the polarization axis can get through
Electromagnetic Radiation
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The ElectromagneticSpectrum
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Spectral Response
http://landsat.gsfc.nasa.gov/education/compositor
Himalyan balsam
human bee butterfly
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Electromagnetic Radiation
An electromagnetic wave carries energy and momentum.
The average power per unit area iscalled the intensity of the wave
The momentum per unit time (that is, force)per unit area is called the radiation pressure
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Electromagnetic Radiation
The radiation pressure, Prad, is given by
rad
SP
c=
where the average intensity is given by
p p
0 0
1
2
E BEBS
µ µ= =
which can be written interms of energy density:
E BS cu cu= =
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The Pressure of Sunshine
Solar Luminosity L = 3.8 x 1026 W
Astronomical Unit r = 1.5 x 1011 m
Intensity S = L / 4π r2
Pressure P = S / c
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The Pressure of Sunshine
Intensity S = L / 4π r2
= 1370 W/m2
Pressure P = S / c= 4.6 µN/m2
31 Credit: Michael Carroll, The Planetary Society
Interstellar Travel
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Summary
Maxwell’s Equations2nd Unification of forcesElectromagnetic wavesUniversal speed c = 3 x 108 m/s
Electromagnetic WavesGamma rays to radio wavesCarry energy and momentumExert pressure