Download Standing electromagnetic waves

Physics 272
April 15
Spring 2014
http://www.phys.hawaii.edu/~philipvd/pvd_14_spring_272_uhm.html
Prof. Philip von Doetinchem
[email protected]
Phys272 - Spring 14 - von Doetinchem - 291
Lunar eclipse
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Hertzian dipole
Source: http://de.wikipedia.org/wiki/Hertzscher_Dipol
Change of electric field over time
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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
–
Magnetic field has
only a z component
–
Both move in x
direction with
velocity c
We have to test if
this assumption is
consistent with
Maxwell's equation
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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
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Magnetic flux change:
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Magnetic field component in z direction is crucial to comply to Faraday's law
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Plane electromagnetic waves
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Why perpendicular?
In a very similar way:
use Ampere's law (no conduction current):
Electric and magnetic fields must be perpendicular
to fulfill Faraday's and Ampere's law
Definite propagation velocity:
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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
Magnitudes of E and B are related by the propagation velocity: E=cB
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
–
E and B fields can be superposed
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For each superposition the simple principles of the plane electromagnetic waves apply
–
Wave packets or sinusoidal waves fulfill Maxwell's equations
Phys272 - Spring 14 - von Doetinchem - 297
Key properties of electromagnetic waves
●
●
Electromagnetic waves fulfill the general wave equation for both electric and
magnetic fields (not discussed here):
Wave is transverse:
–
●
●
Electric field and magnetic field are perpendicular to the direction of the wave
Magnitudes of E and B are related by the propagation velocity: E=cB
In vacuum (or if medium is not changing) the electromagnetic wave is traveling
at a constant velocity
●
Electromagnetic waves do not need a medium
●
Simple plane waves can be generalized
–
E and B fields can be superposed
–
For each superposition the simple principles of the plane electromagnetic waves apply
–
Wave packets or sinusoidal waves fulfill Maxwell's equations
Phys272 - Spring 14 - von Doetinchem - 298
Sinusoidal electromagnetic waves
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At any instant the fields are uniform over any plane perpendicular to
the direction of propagation with speed c
The entire pattern
travels in the direction
of propagation
Electric and magnetic field
are still transverse to the
propagation direction at
any instant
In a small region of space at great distance from source
→ electromagnetic waves can be treated as plane waves
Velocity, wave length, and frequency are related:
Phys272 - Spring 14 - von Doetinchem - 299
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
Vector product of
electric field and
magnetic field always
points in the propagation direction:
Electromagnetic wave can be described by wave function
depending on location and time
k is the wave number: 2/
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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:
–
Negative sign: positive x direction
–
Positive sign: negative y 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:
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Why are magnetic and electric field in phase?
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Relation between electric and magnetic field using
Faraday's law:
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Why are magnetic and electric field in phase?
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Use our approach for the wave function assuming a
phase angle:
To make these equations equal at all times we have
to require
Phys272 - Spring 14 - von Doetinchem - 304
Electromagnetic waves in matter
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The wave speed changes in matter
(we assume 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
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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”
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Energy and momentum in electromagnetic waves
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Electromagnetic waves transport energy:
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Microwave ovens, radio transmitters, lasers for eye
surgery, …
Combined energy density of electric and magnetic
components:
In vacuum electric field and magnetic field carry
half of the total energy density
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Electromagnetic energy flow and the Poynting vector
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Energy is required to establish electric and magnetic fields
→ electromagnetic waves transport energy from one region to the other
Definition:
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energy transferred per unit time per unit cross-sectional area
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or power per unit area
In a certain time the wave moves in space with the propagation velocity
→ energy in a particular region:
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Electromagnetic energy flow and the Poynting vector
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Poynting vector is a
vector quantity:
it points in the direction
of the propagation of the wave
and depends on time
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SI unit: 1 W/m2
John Henry Poynting
(1852-1914)
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Total energy flow per unit time:
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Electromagnetic energy flow and the Poynting vector
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Frequency of electromagnetic
waves is fast → average Poynting
vector value (intensity)
For our sinusoidal wave:
zero on average
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The intensity for a sinusoidal wave is exactly half of the
maximum value:
This is what we sense when looking at, e.g., light from the
sun
→ variations are too fast to be noticeable for us
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Energy in a sinusoidal wave
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A radio station on the
earth's surface emits
a sinusoidal wave with
average total power of
50kW
Assume that transmitter radiates equally in all
directions
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Area of hemisphere:
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Intensity:
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Energy in a sinusoidal wave
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A radio station on the earth's
surface emits a sinusoidal
wave with average total power
of 50kW
Assume that transmitter radiates
equally in all directions
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Electric field amplitude:
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Magnetic field amplitude:
magnetic field associated with electromagnetic wave is extremely small compared to
what we saw before
→ it is easier to have device sensitive to the electric field
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Electromagnetic momentum flow and radiation pressure
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Electromagnetic waves also carry momentum
(→ deeper explanation requires quantum physics)
Electromagnetic wave's momentum is transferred to
a surface
Transferred momentum
per time equals the
force on the surface
Sun and stars
create radiation
pressure that effects the
surrounding material
→ star forming regions
Massive Star Forming Region DR21 in Infrared
Credit: A. Marston (ESTEC/ESA) et al., JPL, Caltech, NASA
Phys272 - Spring 14 - von Doetinchem - 316
Solar sail
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Solar sails try to make use out of the radiation
pressure as a propulsion system
Usage of sails big as football fields should catch the
radiation pressure by the sun
Concept is proven, but
not yet part of real
spacecrafts
Source: http://en.wikipedia.org/wiki/Solar_sail
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Standing electromagnetic waves
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Electromagnetic waves can be reflected on surfaces
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Dielectrics or conductors can serve as reflectors
Superposition principle of electric and magnetic fields
also applies to electromagnetic waves
Superposition of
incident and
reflected wave
forms a standing
wave
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Reminder: equipotentials and conductors
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Electric force is conservative → it is not possible to
do work on a test charge like that:
Electric field lines are perpendicular to surface of
conductor.
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Standing electromagnetic waves
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Electric field cannot have a net component parallel to the surface
Oscillating current are induced in surface that give rise to additional field that
cancels out the electric field of the incident electromagnetic wave
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This also creates the reflected wave:
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The sum of incident and reflected wave must be 0 at all times on the surface:
Phys272 - Spring 14 - von Doetinchem - 322
Standing electromagnetic waves
●
●
Electric field cannot have a net component parallel to the surface
Oscillating current are induced in surface that give rise to additional field that
cancels out the electric field of the incident electromagnetic wave
●
This also creates the reflected wave:
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The sum of incident and reflected wave must be 0 at all times on the surface:
Phys272 - Spring 14 - von Doetinchem - 323
Standing electromagnetic waves
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Position and time factorize for electric field
What does the magnetic field look like?
→ Faraday's law still applies
Integrate:
A standing wave that was reflected on a conductor shows a 90deg phase angle
between electric and magnetic field
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Standing waves in a cavity
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Insert a second conducting plane: cavity
→ example: microwave oven
On both planes the electric field has to vanish
A standing wave is created
when the electromagnetic
wave wavelength is an
integer multiple of /2
Measuring the node positions
→ measurement of
wavelength
Reflections generally also
happen on surfaces of two
materials:
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Part of the wave is transmitted and a part is reflected
Phys272 - Spring 14 - von Doetinchem - 325
Review
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Maxwell's equations predict the existence of
electromagnetic waves that propagate at the speed of
light
Electromagnetic waves are transverse:
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In matter the wave speed is reduced
–
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Electric and magnetic fields are perpendicular to
propagation direction
Electromagnetic wave cannot travel faster than the speed
of light.
The poynting vector describes the energy flow rate.
The averaged value is called the intensity
Nodal planes occur for standing electromagnetic
waves
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