Download Electromagnetic waves

Physics 272
April 14
Spring 2015
www.phys.hawaii.edu/~philipvd/pvd_15_spring_272_uhm
go.hawaii.edu/KO
Prof. Philip von Doetinchem
[email protected]
PHYS272 - Spring 15 - von Doetinchem - 85
Resonance in alternating-current circuits
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It is important to understand how LRC circuits
depend on the angular frequency 
Example: radio signal of a certain frequency
produces greatest signal when the circuit is tuned to
this frequency
→ circuit is in resonance
Connect AC source with constant voltage and
variable frequency to LRC series circuit
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Current has same frequency
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Impedance depends on frequency
PHYS272 - Spring 15 - von Doetinchem - 86
Resonance in alternating-current circuits
minimum of impedance
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A minimum in impedance exists where X L and XC
cancel
PHYS272 - Spring 15 - von Doetinchem - 87
Circuit behavior at resonance
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Position of minimum impedance is called resonance
frequency
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Capacitive and inductive reactance cancel:
This is the same frequency of an ideal LC circuit
Inductive and capacitive voltages have a 180deg
phase angle
→ cancel at all times at resonance frequency
Voltage across resistor is equal to source voltage at
resonance
(circuit behaves like there are no inductors or
capacitors)
PHYS272 - Spring 15 - von Doetinchem - 88
Tailoring an AC circuit
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Resonance frequency can be
changed by changing L
and/or C
Tuning knob in old radios was
moving capacitor plates
→ change in capacitance
Source: http://commons.wikimedia.org/wiki/File:Autoradio_De_Wald.jpg
Modern approach: change L by moving ferrite core
PHYS272 - Spring 15 - von Doetinchem - 89
Tailoring an AC circuit
https://www.youtube.com/watch?v=ZYgFuUl9_Vs
PHYS272 - Spring 15 - von Doetinchem - 90
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Sharp increase at
resonance frequency
current
Tailoring an ac circuit
R
Value of resistance
determines how sharp
the change is
Important for
discriminating between
different radio stations
If the peak is too sharp
→ information will be
lost
3xR
9xR
frequency
PHYS272 - Spring 15 - von Doetinchem - 91
Tuning a radio
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Resonance frequency:
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Reactances:
PHYS272 - Spring 15 - von Doetinchem - 92
Tuning a radio
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RMS current:
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Voltages:
VL, VC are both significantly larger than V R
→ they cancel at resonance (180deg phase angle)
PHYS272 - Spring 15 - von Doetinchem - 93
Transformers
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Advantage of AC
over DC electric
power distribution:
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Stepping voltage
levels up and down
is easy
Long distance transmission:
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high voltages (500kV)
→ lower i2R losses at the same energy throughput
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Thinner cables for high voltages
Safety requires that the normal user operates at
much lower voltages
PHYS272 - Spring 15 - von Doetinchem - 95
How transformers work
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AC source causes an alternating current in the primary
coil (the one that is powered)
Alternating magnetic flux in core (made of material with
high permeability)
→ induced emf in primary coil
Same alternating flux goes through secondary coil
→ also induces emf at same frequency as primary coil
PHYS272 - Spring 15 - von Doetinchem - 96
How transformers work
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Magnetic flux through both coils is the same:
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Induced emf is equal to terminal voltage if windings have zero resistance
PHYS272 - Spring 15 - von Doetinchem - 97
How transformers work
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Choice of windings determines the voltage output of
the secondary coil
Wall charger transforms 110V AC
to, e.g., 5V DC for USB
(also using a diode)
PHYS272 - Spring 15 - von Doetinchem - 98
Energy considerations for transformers
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Power delivered to the primary coil is the same that is taken
out by the secondary to power a device
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Not only voltages are transformed, but also the impedance
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Real transformers have losses due to
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non-zero resistance (transformer gets warm)
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Hysteresis in the core
PHYS272 - Spring 15 - von Doetinchem - 99
Energy considerations for transformers
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Real transformers have losses due to
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Magnetic field is constantly changing
→ eddy currents build up
→ energy wasted
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Use laminated cores to narrow
path for eddy-currents
→ smaller radius
→ lower opposing magnetic field
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Magnetic flux in each path is much
smaller
Transformers typically reach
efficiencies of greater than 90%
PHYS272 - Spring 15 - von Doetinchem - 100
Electromagnetic waves
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What is light?
→ electromagnetic wave
→ electromagnetism is needed
(not the complete story → QFT)
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Time varying magnetic field creates electric field
Time varying electric field creates magnetic field
→ sustain each other and create an
electromagnetic wave that propagates through
space
PHYS272 - Spring 15 - von Doetinchem - 103
Electromagnetic waves
Source: http://en.wikipedia.org/wiki/Light
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Electromagnetic waves carry energy and momentum
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Electric and magnetic fields in sinusoidal waves have defined frequency
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Infrared, visible light, UV, X-ray, -ray, etc. all follow the same principle,
but at different wavelength
Electromagnetic waves do not require medium (like mechanical waves)
PHYS272 - Spring 15 - von Doetinchem - 104
Electricity, magnetism, and light
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Maxwell's equation relate magnetic and electric field
Moving charges produce both electric and magnetic
fields
An electromagnetic wave is formed when a charge
accelerates
We never spoke about how fast a magnetic field can
be measured at a certain distance after a charge
starts moving:
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Electromagnetic waves do not travel with infinite speed
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Generating electromagnetic radiation
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Look at oscillating L-C circuit
Transformation of the LC circuit into a conducting
rod:
→ this is still an oscillating LC circuit
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Generating electromagnetic radiation
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What do the electric and magnetic fields look like:
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In initial configuration fields are contained in capacitor and inductor
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In the rod configuration the electric and magnetic field overlap
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Current is moving up and down the rod
→ charge in “capacitor” and current in “inductor” are changing with
time
→ electric and magnetic field change with time
→ electric and magnetic field propagate with finite velocity
→ electromagnetic wave
PHYS272 - Spring 15 - von Doetinchem - 107
Hertzian dipole
http://de.wikipedia.org/wiki/Hertzscher_Dipol#/media/File:DipoleRadiation.gif
Change of electric field over time
PHYS272 - Spring 15 - von Doetinchem - 108
Plane electromagnetic waves
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Electromagnetic waves are transverse waves with an electric
and a magnetic component
→ similar to waves on a string
Make the following assumption:
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Electric and magnetic field configuration with wave-like behavior
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Electric field has only a y component
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Magnetic field has
only a z component
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Both move in x
direction with
velocity c
We have to test if
this assumption is
consistent with
Maxwell's equation
different instances
in time
PHYS272 - Spring 15 - von Doetinchem - 109
Plane electromagnetic waves
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Why perpendicular?
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Faraday's law:
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Integration path
perpendicular to electric
field: no contribution
(1)
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Magnetic flux change:
Propagation velocity
(2)
(1) and (2)
area increase
in time dt
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Magnetic field component in z direction is crucial to comply to Faraday's law
PHYS272 - Spring 15 - von Doetinchem - 110
Plane electromagnetic waves
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In a very similar way:
use Ampere's law (no conduction current):
any components of E and B not perpendicular to each
other and not perpendicular to the direction do not play a
role to fulfill Maxwell's equation
→ Electric and magnetic fields must be perpendicular
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To fulfill Maxwell's equations → wave must also fulfill:
Propagation velocity
PHYS272 - Spring 15 - von Doetinchem - 111
Key properties of electromagnetic waves
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Electromagnetic waves fulfill the general wave equation for both electric
and magnetic fields (not discussed here):
Wave is transverse:
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Electric field and magnetic field are perpendicular to the direction of the wave
In vacuum (or if medium is not changing) the electromagnetic wave is
traveling at a constant velocity
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Electromagnetic waves do not need a medium
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Simple plane waves can be generalized
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E and B fields can be superposed
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For each superposition the simple principles of the plane electromagnetic waves
apply
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Also wave packets or sinusoidal waves fulfill Maxwell's equations
PHYS272 - Spring 15 - von Doetinchem - 112
Sinusoidal electromagnetic waves
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Sinusoidal waves are essential, e.g., to understand light,
X-rays, or radio communication
The entire pattern travels in the direction of propagation
Electric and magnetic field are still transverse to the
propagation direction at any instant
snapshot at particular instant of time
PHYS272 - Spring 15 - von Doetinchem - 113
Fields of a sinusoidal waves
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Electric and magnetic fields oscillate in phase: the fields are at
the same time at maximum, zero, and minimum
Electromagnetic wave can be described by wave function
depending on location and time
k is the wave number: 2/
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Vector product of electric field and magnetic field always points
in the propagation direction
PHYS272 - Spring 15 - von Doetinchem - 114
Fields of a sinusoidal waves
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The electric field and magnetic field amplitudes are
related by:
The sign in front of t denotes the direction of the
wave:
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Negative sign: positive x direction
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Positive sign: negative x direction
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Why are magnetic and electric field in phase?
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Assume arbitrary phase angle:
Relation between electric and magnetic field using
Faraday's law:
snapshot at
particular
instant of time
PHYS272 - Spring 15 - von Doetinchem - 116
Why are magnetic and electric field in phase?
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Applying Faraday's law:
assumption: variation of BZ between
location x and x+x is small at any
particular instance in time
calculate magnetic
flux through area
PHYS272 - Spring 15 - von Doetinchem - 117
Why are magnetic and electric field in phase?
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Use our approach for the wave function assuming a
phase angle:
=
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To make these equations equal at all times we have
to require
PHYS272 - Spring 15 - von Doetinchem - 118
Electromagnetic waves in matter
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The wave speed changes in matter (we assume a non-conducting dielectric)
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We have to use the dielectric constant and the permittivity of the material:
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Electromagnetic waves can never travel faster than the speed of light in
vacuum
Definition of index of refraction n of a material
(→ optics)
Careful:
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dielectric “constant” depends on frequency
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for high frequencies materials cannot be polarized as fast as for constant
electric fields → reduces dielectric “constant”
PHYS272 - Spring 15 - von Doetinchem - 119