chapter 27 – magnetic fields and forces...2) (if magnetic monopoles) 3) (if magnetic monopoles) 4)...
TRANSCRIPT
Electromagnetic Waves
Visible light, radio waves, cell communications, UV tanning, HDTV – these
are all examples of EM waves
Maxwell's Equations of ElectromagnetismNo magnetic Monopoles
Changing B field induces E field and Changing E field induces B field3
0 01) / ( charge/volume or Coulomb/m ) /
2) 0 0
(no magnetic monopoles found yet - if they exist then (Mag monopole density)
3)
4)
enc
m
C
E E dA Q
B B dA
B
B dxE E dl B dA
t dt
x
0 0 0 0 0 0
2
0
0 0 0 0 0 0
( current density vector (amp/m ) )
Define Displacement Current as:
4) ( )
Note that
enc
C
D
D D enc
C
E dB J B dl E dA I
t dt
J I J dA
EJ
t
ExB J J J B dl I I
t
is real current density while is a ficticious current density
Note that Stokes Theorem for ANY well behaved vector field F:
D
C
J J
F dl xF dA
Maxwell's Equations of ElectromagnetismAllow existence of magnetic monopoles
3
0 0
0 0
0 0 0
1) / ( charge/volume or Coulomb/m ) /
2) (IF magnetic monopoles)
3) (IF magnetic monopoles)
4)
enc
m m enc
m m enc
C
E E dA Q
B B dA Q
B dxE J E dl B dA I
t dt
ExB
t
0 0 0
2
2
( current density (flux) vector (amp/m ) ) - can also be a mag monopole current
Force Law ( ) ( ) (right/left hand rule for elect
enc
C
e m
dJ B dl E dA I
dt
J I J dA
vF q E v x B q B x E
c
ric/magnetic charges)
Note that Divergence (Gauss' Law) for ANY well behaved vector field F:
continuity eq for flux field F / 0V
F dA F dV F t
Units – not just MKSWhat about charge? – Farad’s – Henry’s
7
0
2
2
2 2
4 10 H/m
What is a Henry (unit of inductance)?
Recall:
V=Ldi/dt
L=inductance and units are in Henry's
V has units of J/Q (Joule/ Coulomb)
di/dt has units of /s
J/Coul J-s kg
Coul/s
H Henry
Coul
CL
oulH
2
2
12
0
2 2 2
2
7 2
2
0
12 2 2 1 3
0
-mHenry
8.854
/
10 /m (Farad/m)
What is a Farad (unit of capacitance C)?
Coul -sarad
J/Coul J kg-m
4 10 kg-m-Coul
8.854 10 Coul s kg m
F
( )
( / )
*
Coul
Q Coul Coul
V
H
F
m
F m
Farad Hen
F
s
C
ry
2( ) ( ) ( ) for any well behaved vector field F( , )
Well behave means no singularities and differentiable in space and time
There is no Physics here - just mathematics
Now let F( , )b
x xF F F x t
x t
2 2
0
20 0 0
0 0 02
22
0 0 02
e the E or B field
E field
( ) ( ) ( ) ( / ) ( )
( )( )
( )
( ) ( ) /
e
m m m m
m e
x xE E E E
EJ
B xB E Jtx J xJ xJ xJt t t t t
E JE xJ
t t
0
22
0 0 2
2 2
0
2
0 0 0 0 0 0 0 0 0 0 02
22
0 0 2
(in vacuum and no mag monopoles) ( ) 0
B field
( ) ( ) ( ) ( ) ( )
( )( )
( )
m
m
EE
t
x xB B B B
JE xE Bx J xJ xJ
t t t t
BB
t
0 0 0 0
22
0 0 2
( )
(in vacuum and no mag monopoles) ( ) 0
mm
JxJ
t
BB
t
EM Wave Equation in General – Allow Mag Mono
Maxwell’s Equation in VacuumNo charges – No mag monopoles
0 0 0 0
1) 0 0
2) 0 0
3)
4)
C
C
E E dA
B B dA
B dxE E dl B dA
t dt
E dxB B dl E dA
t dt
EM Wave Equation in MaterialsNo Magnetic Monopoles
2 2
0
20 0 0
0 0 02
22
0 0 0 02
2 2
E field
( ) ( ) ( ) ( / ) ( )
( )( )
( )
( ) ( / )
B field
( ) ( ) ( ) (0) ( )
e
e
x xE E E E
EJ
B xB E Jtxt t t t t
E JE
t t
x xB B B B
x
2
0 0 0 0 0 0 0 0 02
22
0 0 02
( )( )
( )
E xE BJ xJ xJ
t t t
BB xJ
t
Wave Equations in MaterialsNo free charges or currents
Homogeneous Wave Equation2
2
2= 0
EE
t
0
0
0
0
0 0
=
free space permeability
magnetic material relative permeability
free space permittivity
dielectric material relative permittivity
1 1Wave Speed = = =
"e
m
m m
r
r e
m r m r
m r
c c
n
n
ffective index of refraction"
22
2= 0
BB
t
Solutions for Wave EquationAny well behaved function is a solution of the homogeneous wave equation i it has the following form:
f(k x t) where k is called the "wave vector" and is the angular frequency = 2 & 2 /The ho
f k
mogeneous wave function is linear so a sum of solutions is a solution.We can thus form basis sets of solutions
Any well behaved function can be written as a Fourier seriesWe can use sin/co functions as
0
0
a basis set by taking a discrete or continuous
Fourier transform of f(k x t)We can write a general solution set consisting of the sum of the following:
E(x,t)=E sin(k x t)
B(x,t)=B sin(k
0 0
0 0
0 0
22 2 2 2
0 0 0 02
x t)
Use sin(k x t) for simplicity
E ,B are amplitude vectors
No free charges solution:
1) 0 E 0 E
2) 0 B 0 B
( ) 0 /
E k E k k
B k B k k
EE k c
t
2
22
0 0 2
80 0 0 0 0 0 0 0
0 0 00 0 0 0 0
(this is called a dispersion relation where =ck)
( ) 0
3) E B B ( E ) B E & 1 3 10 /
4) B E E
BB same
tB k
xE k x k x E cB Tesla x Volt mt
ExB k x
t
0 0 0 00 2
0 0 0 0
( B ) ( B ) E B
,E ,B are all perpendicular to each other with E B ,E and B always in phase
k x k xk c k
k k x
Poynting VectorIt Points in the Direction of Energy Flow
2 2
0 0 0 0
0
23
0
Assume Free Space (J=0 - no free currents)
Some math:
( ) ( ) ( ) ( ) ( ) [ ]2 2
Add some Physics:
=Magnetic field energy density (J/m )= 2
M
B E B EE x B B x E E x B B E
t t t
B
23
0
2
0
=Electric field energy density (J/m )= 2
1Define the Poynting vector (x,t) ( (x,t) (x,t)) (units are watts/m )
(x,t) points along wave propagation is along direction of ene
E
E
S E x B
S k k S
22
0
0 0 0
0
rgy flow
1 1 1(x,t) ( (x,t) (x,t)) (x,t) (x,t) E sin (k x t)
Continuity equation (use divergence theorem):
( ) / 0M E
V V
M E
S E x B E Bc
S dA S dV dV S tt
Time Averaged Poynting Vector2
20
0 0 0
1 1 1(x,t) ( (x,t) (x,t)) (x,t) (x,t) E sin (k x t)
(x,t) oscillates at twice the frequency of the EM wave
For example for visible light at =0.55 microns (peak of human
S E x B E Bc
S
22
0
0
vision)
545 THz Poynting vector oscillates at 1090 THz
This is too fast for virtually normal detectors
Time average of Poyting vector is useful in practical measurements
1(x,t) E sin (k x
f
Sc
2 2
20 0
0 0
2
2
0 00 0
20
1 1t) E sin (k x t) E
2
sin (k x t) 1/ 2
E 2 (x,t) E 2 (x,t) 27.5 (x,t)
Example :Solar insolation (flux from sun)
1) At surface of Earth (x,t) ~ 1000 / E ~ 868
c c
c S c S S
S w m vo
20
/
2) At top of atmosphere (x,t) ~ 1350 / E ~ 1010 /
lts m
S w m volts m
Wave Equations
This is similar to a scalar wave equation where c is the speed of the wave
Solutions to the Wave Equations
The following is a solution for ANY well behaved function fK is called the wave vector – here it is a unit vector that points in the direction of wave propagation
E, B and K form a mutual orthogonal system
E, B (M) and wavelengthK = 2π/λ
E, B and K – Right handed coordinate systemE and B are “in phase”
Properties of electromagnetic waves• Maxwell’s equations imply that in an electromagnetic wave,
both the electric and magnetic fields are perpendicular to the direction of propagation of the wave, and to each other.
• In an electromagnetic wave, there is a definite ratio between the magnitudes of the electric and magnetic fields: E = cB.
• Unlike mechanical waves, electromagnetic waves require no medium. In fact, they travel in vacuum with a definite and unchanging speed:
• Inserting the numerical values of these constants, we obtain c = 3.00 × 108 m/s.
Properties of electromagnetic waves
• The direction of propagation of an electromagnetic wave is the direction of the vector product of the electric and magnetic fields.
The Poyting VectorUnits are Flux (w/m2)
Energy in electromagnetic waves• Electromagnetic waves such as
those we have described are traveling waves that transport energy from one region to another.
• The British physicist John Poynting introduced the Poynting vector
• The magnitude of the Poyntingvector is the power per unit area in the wave, and it points in the direction of propagation.
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Energy in electromagnetic waves
• The magnitude of the average value of is called the intensity. The SI unit of intensity is 1 W/m2.
• These rooftop solar panels are tilted to be face-on to the sun so that the panels can absorb the maximum amount of wave energy.
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A simple plane electromagnetic wave• To begin our study of
electromagnetic waves, imagine that all space is divided into two regions by a plane perpendicular to the x-axis.
• At every point to the left of this plane there are uniform electric field magnetic fields as shown.
• The boundary plane, which we call the wave front, moves in the +x-direction with a constant speed c.
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Gauss’s laws and the simple plane wave• Shown is a Gaussian
surface, a rectangular box, through which the simple plane wave is traveling.
• The box encloses no electric charge.
• In order to satisfy Maxwell’s first and second equations, the electric and magnetic fields must be perpendicular to the direction of propagation; that is, the wave must be transverse.
Faraday’s law and the simple plane wave
• The simple plane wave must satisfyFaraday’s law in a vacuum.
• In a time dt, the magnetic flux through the rectangle in the xy-plane increases by an amount dΦB.
• This increase equals the flux through the shaded rectangle with area ac dt; that is, dΦB = Bac dt.
• Thus dΦB/dt = Bac.
• This and Faraday’s law imply:
Ampere’s law and the simple plane wave
• The simple plane wave must satisfy Ampere’s law in a vacuum.
• In a time dt, the electric flux through the rectangle in the xz-plane increases by an amount dΦE.
• This increase equals the flux through the shaded rectangle with area ac dt; that is, dΦE = Eac dt.
• Thus dΦE/dt = Eac. This implies E=Bc:
Sinusoidal electromagnetic waves• Electromagnetic waves
produced by an oscillating point charge are an example of sinusoidal waves that are not plane waves.
• But if we restrict our observations to a relatively small region of space at a sufficiently great distance from the source, even these waves are well approximated by plane waves.
Fields of a sinusoidal wave
• We can describe electromagnetic waves by means of wave functions:
• The wave travels to the right with speed c = ω/k.
• The amplitudes must be related by:
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Electromagnetic waves in matter• Electromagnetic waves can travel in certain types of matter,
such as air, water, or glass. • When electromagnetic waves travel in nonconducting
materials—that is, dielectrics—the speed v of the waves depends on the dielectric constant of the material.
• The ratio of the speed c in vacuum to the speed v in a material is known in optics as the index of refraction n of the material.
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Electromagnetic radiation pressure• Electromagnetic waves carry momentum and can
therefore exert radiation pressure on a surface:
• At surface of Earth - SAV ~1000 w/m2
• Above atmosphere - SAV ~1361w/m2
• For example, if the solar panels on an earth-orbiting satellite are perpendicular to the sunlight, and the radiation iscompletely absorbed, the average radiation pressure is 4.7 × 10−6 N/m2.
Standing electromagnetic waves• Electromagnetic waves can be reflected by a conductor or
dielectric, which can lead to standing waves.
• As time elapses, the pattern does not move along the x-axis; instead, at every point the electric and magnetic field vectors simply oscillate.
Standing waves in a cavity
• A typical microwave oven sets up a standing electromagnetic wave with λ = 12.2 cm, a wavelength that is strongly absorbed by the water in food.
• Because the wave has nodes spaced λ/2 = 6.1 cm apart, the food must be rotated while cooking.
• Otherwise, the portion that lies at a node—where the electric-field amplitude is zero—will remain cold.
Visible light – Red-Green-Blue
Visible light• Visible light is the segment of the
electromagnetic spectrum that we can see.
• Visible light extends from the violet end (400 nm) to the red end (700 nm).
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Eye Response to Colors and BWRods (Scotopic - BW – night vision ~ 120 million ) and Cones (Photopic – Color - ~ 6 million)
Modern Electronic Focal Plane Arrays Now Exceed Human Resolution Central fovea (retina) is primarily populated by cones – NOTE Bind spot
Three Types of Cones for Color Vision
Retina Distribution of Rods and Cones
Ultraviolet Light and VisionOur colors looks quite different
• Many insects and birds can see ultraviolet wavelengths that humans cannot.
• As an example, the left-hand photo shows how black-eyed Susans look to us.
• The right-hand photo (in false color), taken with an ultraviolet-sensitive camera, shows how these same flowers appear to the bees that pollinate them.
• Note the prominent central spot that is invisible to humans.
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Infrared VisionSanta Barbara is the IR Capital (Industrial Base) of the US
IR Imaging is very importantRemote sensing for weather, crop health, industrial, medical, astronomy
Thermal IR (8-12 microns) is Common Now (Microbolometers)
Night Vision Systems in Cars for example
Thermal IR Imaging (8-12 microns)
Impedance of the VacuumOur modern view of the vacuum is that it is
a sea of all things at negative energy. Thus it is NOT NOTHING.
It has an impedance
Electromagnetic Energy Density and Flux and Quanta
Blackbody Radiation – Planck FunctionBlackbody = Body that Absorbs all RadiationThe Universe is a Nearly Perfect Blackbody
Dispersion in materials – polarization depends on frequency