Maxwell

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Maxwells Equations and Electromagnetic Waves
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Content

• Changing Electric Fields Produce Magnetic
  Fields Ampères Law and Displacement Current
• Gausss Law for Magnetism
• Maxwells Equations
• Production of Electromagnetic Waves
• Electromagnetic Waves, and Their Speed, Derived
  from Maxwells Equations
• Light as an Electromagnetic Wave and the
  Electromagnetic Spectrum

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Content

• Measuring the Speed of Light
• Energy in EM Waves the Poynting Vector
• Radiation Pressure
• Radio and Television Wireless Communication

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Displacement Current
Ampères law relates the magnetic field around a
current to the current through a surface. Current
I passes through both surface 1 and 2.
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Displacement Current
In order for Ampères law to hold, it cant
matter which surface we choose. But look at a
discharging capacitor there is a current through
surface 1 but none through surface 2
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Displacement Current
Maxwell proposed a new type of current, called
the displacement current, ID. Therefore, Ampères
law is modified accordingly as
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Displacement Current
Maxwell realized that the changing electric flux
must be associated with a magnetic field. For
example, for a parallel-plate capacitor
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Displacement Current
With Maxwells modification, Amperes law now
becomes
Where the second term is called the displacement
current
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Displacement Current
Charging capacitor. A 30-pF air-gap capacitor has
circular plates of area A 100 cm2. It is
charged by a 70-V battery through a 2.0-O
resistor. At the instant the battery is
connected, the electric field between the plates
is changing most rapidly. At this instant,
calculate (a) the current into the plates, and
(b) the rate of change of electric field between
the plates. (c) Determine the magnetic field
induced between the plates. Assume E is uniform
between the plates at any instant and is zero at
all points beyond the edges of the plates.
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Gausss Law for Magnetism
Gausss law relates the electric field on a
closed surface to the net charge enclosed by that
surface. The analogous law for magnetic fields is
different, as there are no single magnetic point
charges (monopoles)
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Maxwells Equations
The complete set of equations describing electric
and magnetic fields is called Maxwells
equations. In the absence of dielectric or
magnetic materials, they are
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Maxwells Equations
In the absence of currents and charges, they are
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Electromagnetic Waves
According to Maxwells Equations in the absence
of currents and charges, the E and B fields also
satisfy Maxwells wave equations
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Electromagnetic Waves
A wave traveling along the x-axis with a speed v
satisfies the wave equation
Therefore, we see that the wave speed is
which is exactly the speed of light.
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Production of Electromagnetic Waves
Since a changing electric field produces a
magnetic field, and a changing magnetic field
produces an electric field, once sinusoidal
fields are created they can propagate on their
own. These propagating fields are called
electromagnetic waves.
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Production of Electromagnetic Waves
Oscillating charges will produce electromagnetic
waves
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Production of Electromagnetic Waves
Close to the antenna, the fields are complicated,
and are called the near field
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Production of Electromagnetic Waves
Far from the source, the waves are plane waves
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Production of Electromagnetic Waves
The electric and magnetic waves are perpendicular
to each other, and to the direction of
propagation.
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Electromagnetic Waves
This figure shows an electromagnetic wave of
wavelength ? and frequency f. The electric and
magnetic fields are given by
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Electromagnetic Waves
The electric and magnetic fields are related by
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Electromagnetic Waves
Determining E and B in EM waves. Assume a 60-Hz
EM wave is a sinusoidal wave propagating in the z
direction with E pointing in the x direction, and
E0 2.0 V/m. Write vector expressions for E and
B as functions of position and time.
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Energy and Momentum
Energy is stored in both electric and magnetic
fields, giving the total energy density of an
electromagnetic wave
Since EcB, each field contributes half the total
energy density
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Energy and Momentum
This energy is transported by the wave.
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Energy and Momentum
The energy transported through a unit area per
unit time is called the intensity
The energy floe is perpendicular to both E and B.
The Poynting vector is defined as
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Energy and Momentum
Typically we are interested in the average value
of Sav
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Energy and Momentum
E and B from the Sun. Radiation from the Sun
reaches the Earth (above the atmosphere) at a
rate of about 1350 J/sm2 ( 1350 W/m2). Assume
that this is a single EM wave, and calculate the
maximum values of E and B.
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Energy and Momentum
In addition to carrying energy, electromagnetic
waves also carry momentum
If the wave is completely absorbed, the radiation
pressure exerted on the surface is
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Radiation Pressure
If the wave is perfectly reflected, the momentum
change is doubled
and the radiation pressure exerted on the surface
is
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Radiation Pressure
Solar pressure. Radiation from the Sun that
reaches the Earths surface (after passing
through the atmosphere) transports energy at a
rate of about 1000 W/m2. Estimate the pressure
and force exerted by the Sun on your outstretched
hand.
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Radiation Pressure
A solar sail. Proposals have been made to use the
radiation pressure from the Sun to help propel
spacecraft around the solar system. (a) About how
much force would be applied on a 1 km x 1 km
highly reflective sail, and (b) by how much would
this increase the speed of a 5000-kg spacecraft
in one year? (c) If the spacecraft started from
rest, about how far would it travel in a year?
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Light as an Electromagnetic Wave and the
Electromagnetic Spectrum
The frequency of an electromagnetic wave is
related to its wavelength and to the speed of
light
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Light as an Electromagnetic Wave and the
Electromagnetic Spectrum
Electromagnetic waves can have any wavelength we
have given different names to different parts of
the wavelength spectrum.
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Light as an Electromagnetic Wave and the
Electromagnetic Spectrum
Wavelengths of EM waves. Calculate the wavelength
(a) of a 60-Hz EM wave, (b) of a 93.3-MHz FM
radio wave, and (c) of a beam of visible red
light from a laser at frequency 4.74 x 1014 Hz.
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Light as an Electromagnetic Wave and the
Electromagnetic Spectrum
Cell phone antenna. The antenna of a cell phone
is often ¼ wavelength long. A particular cell
phone has an 8.5-cm-long straight rod for its
antenna. Estimate the operating frequency of this
phone.
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Light as an Electromagnetic Wave and the
Electromagnetic Spectrum
Phone call time lag. You make a telephone call
from New York to a friend in London. Estimate how
long it will take the electrical signal generated
by your voice to reach London, assuming the
signal is (a) carried on a telephone cable under
the Atlantic Ocean, and (b) sent via satellite
36,000 km above the ocean. Would this cause a
noticeable delay in either case?
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Measuring the Speed of Light
The speed of light was known to be very large,
although careful studies of the orbits of
Jupiters moons showed that it is finite. One
important measurement, by Michelson, used a
rotating mirror
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Measuring the Speed of Light
Over the years, measurements have become more and
more precise now the speed of light is defined
to be c 2.99792458 108 m/s.
This is then used to define the meter.
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Radio and Television Wireless Communication
This figure illustrates the process by which a
radio station transmits information. The audio
signal is combined with a carrier wave.
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Radio and Television Wireless Communication
The mixing of signal and carrier can be done two
ways. First, by using the signal to modify the
amplitude of the carrier (AM)
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Radio and Television Wireless Communication
Second, by using the signal to modify the
frequency of the carrier (FM)
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Radio and Television Wireless Communication
At the receiving end, the wave is received,
demodulated, amplified, and sent to a loudspeaker.
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Radio and Television Wireless Communication
The receiving antenna is bathed in waves of many
frequencies a tuner is used to select the
desired one.
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Radio and Television Wireless Communication
A straight antenna will have a current induced in
it by the varying electric fields of a radio
wave a circular antenna will have a current
induced by the changing magnetic flux.
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Radio and Television Wireless Communication
Tuning a station. Calculate the transmitting
wavelength of an FM radio station that transmits
at 100 MHz.
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Summary

• Maxwells equations are the basic equations of
  electromagnetism

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Summary

• Electromagnetic waves are produced by
  accelerating charges the propagation speed is
  given by

• The fields are perpendicular to each other and
  to the direction of propagation.

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Summary

• The wavelength and frequency of EM waves are
  related

• The electromagnetic spectrum includes all
  wavelengths, from radio waves through visible
  light to gamma rays.
• The Poynting vector describes the energy carried
  by EM waves