General form of Faradays Law

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General form of Faradays Law
So the electromotive force around a closed path
is
And Faradays Law becomes
A changing magnetic flux produces an electric
field.
This electric field is necessarily
non-conservative.
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E produced by changing B
How about outside ro ?
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Problems with Amperes Law
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But what if..
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Maxwells correction to Amperes Law
Called displacement current, Id
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Maxwells Equations

• The two Gausss laws are symmetrical, apart from
  the absence of the term for magnetic monopoles in
  Gausss law for magnetism
• Faradays law and the Ampere-Maxwell law are
  symmetrical in that the line integrals of E and B
  around a closed path are related to the rate of
  change of the respective fluxes

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• Gausss law (electrical)
• The total electric flux through any closed
  surface equals the net charge inside that surface
  divided by eo
• This relates an electric field to the charge
  distribution that creates it
• Gausss law (magnetism)
• The total magnetic flux through any closed
  surface is zero
• This says the number of field lines that enter a
  closed volume must equal the number that leave
  that volume
• This implies the magnetic field lines cannot
  begin or end at any point
• Isolated magnetic monopoles have not been
  observed in nature

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• Faradays law of Induction
• This describes the creation of an electric field
  by a changing magnetic flux
• The law states that the emf, which is the line
  integral of the electric field around any closed
  path, equals the rate of change of the magnetic
  flux through any surface bounded by that path
• One consequence is the current induced in a
  conducting loop placed in a time-varying B
• The Ampere-Maxwell law is a generalization of
  Amperes law
• It describes the creation of a magnetic field by
  an electric field and electric currents
• The line integral of the magnetic field around
  any closed path is the given sum

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The Lorentz Force Law

• Once the electric and magnetic fields are known
  at some point in space, the force acting on a
  particle of charge q can be calculated
• F qE qv x B
• This relationship is called the Lorentz force law
• Maxwells equations, together with this force
  law, completely describe all classical
  electromagnetic interactions

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Maxwells Equations in integral form
Gausss Law
Gausss Law for Magnetism
Faradays Law
Amperes Law
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Maxwells Equations in free space (no charge or
current)
Gausss Law
Gausss Law for Magnetism
Faradays Law
Amperes Law
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Hertzs Experiment

• An induction coil is connected to a transmitter
• The transmitter consists of two spherical
  electrodes separated by a narrow gap
• The discharge between the electrodes exhibits an
  oscillatory behavior at a very high frequency
• Sparks were induced across the gap of the
  receiving electrodes when the frequency of the
  receiver was adjusted to match that of the
  transmitter
• In a series of other experiments, Hertz also
  showed that the radiation generated by this
  equipment exhibited wave properties
• Interference, diffraction, reflection, refraction
  and polarization
• He also measured the speed of the radiation

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Implication

• A magnetic field will be produced in empty space
  if there is a changing electric field.
  (correction to Ampere)
• This magnetic field will be changing.
  (originally there was none!)
• The changing magnetic field will produce an
  electric field. (Faraday)
• This changes the electric field.
• This produces a new magnetic field.
• This is a change in the magnetic field.

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An antenna
Hook up an AC source
We have changed the magnetic field near the
antenna
An electric field results! This is the start of
a radiation field.
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Look at the cross section
Called Electromagnetic Waves
Accelerating electric charges give rise to
electromagnetic waves
E and B are perpendicular (transverse) We say
that the waves are polarized. E and B are in
phase (peaks and zeros align)
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Angular Dependence of Intensity

• This shows the angular dependence of the
  radiation intensity produced by a dipole antenna
• The intensity and power radiated are a maximum in
  a plane that is perpendicular to the antenna and
  passing through its midpoint
• The intensity varies as
• (sin2 ?) / r2

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Active Figure 34.3
(SLIDESHOW MODE ONLY)
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Harmonic Plane Waves
At t 0
l spatial period or wavelength
x
l
phase velocity
At x 0
t
T temporal period
T
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Applying Faraday to radiation
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Applying Ampere to radiation
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Fields are functions of both position (x) and
time (t)
Partial derivatives are appropriate
This is a wave equation!
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The Trial Solution

• The simplest solution to the partial differential
  equations is a sinusoidal wave
• E Emax cos (kx ?t)
• B Bmax cos (kx ?t)
• The angular wave number is k 2p/?
• ? is the wavelength
• The angular frequency is ? 2p
• is the wave frequency

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The trial solution
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The speed of light (or any other electromagnetic
radiation)
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3. The speed of an electromagnetic wave
traveling in a transparent nonmagnetic substance
is , where ? is the dielectric constant of the
substance. Determine the speed of light in water,
which has a dielectric constant at optical
frequencies of 1.78.
5. Figure 34.3 shows a plane electromagnetic
sinusoidal wave propagating in the x direction.
Suppose that the wavelength is 50.0 m, and the
electric field vibrates in the xy plane with an
amplitude of 22.0 V/m. Calculate (a) the
frequency of the wave and (b) the magnitude and
direction of B when the electric field has its
maximum value in the negative y direction. (c)
Write an expression for B with the correct unit
vector, with numerical values for Bmax, k, and ?,
and with its magnitude in the form
6. Write down expressions for the electric and
magnetic fields of a sinusoidal plane
electromagnetic wave having a frequency of 3.00
GHz and traveling in the positive x direction.
The amplitude of the electric field is 300 V/m.
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The electromagnetic spectrum
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(No Transcript)
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Another look
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Energy in Waves
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Poynting Vector

• Poynting vector points in the direction the wave
  moves
• Poynting vector gives the energy passing through
  a unit area in 1 sec.
• Units are Watts/m2

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Intensity

• The wave intensity, I, is the time average of S
  (the Poynting vector) over one or more cycles
• When the average is taken, the time average of
  cos2(kx - ?t) ½ is involved

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11. How much electromagnetic energy per cubic
meter is contained in sunlight, if the intensity
of sunlight at the Earths surface under a fairly
clear sky is 1 000 W/m2?
16. Assuming that the antenna of a 10.0-kW radio
station radiates spherical electromagnetic waves,
compute the maximum value of the magnetic field
5.00 km from the antenna, and compare this value
with the surface magnetic field of the Earth.
21. A lightbulb filament has a resistance of 110
O. The bulb is plugged into a standard 120-V
(rms) outlet, and emits 1.00 of the electric
power delivered to it by electromagnetic
radiation of frequency f. Assuming that the bulb
is covered with a filter that absorbs all other
frequencies, find the amplitude of the magnetic
field 1.00 m from the bulb.
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Radiation Pressure
(Absorption of radiation by an object)
Maxwell showed
What if the radiation reflects off an object?
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Pressure and Momentum

• For a perfectly reflecting surface,
• p 2U/c and P 2S/c
• For a surface with a reflectivity somewhere
  between a perfect reflector and a perfect
  absorber, the momentum delivered to the surface
  will be somewhere in between U/c and 2U/c
• For direct sunlight, the radiation pressure is
  about 5 x 10-6 N/m2

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26. A 100-mW laser beam is reflected back upon
itself by a mirror. Calculate the force on the
mirror.
27. A radio wave transmits 25.0 W/m2 of power
per unit area. A flat surface of area A is
perpendicular to the direction of propagation of
the wave. Calculate the radiation pressure on it,
assuming the surface is a perfect absorber.
29. A 15.0-mW heliumneon laser (? 632.8 nm)
emits a beam of circular cross section with a
diameter of 2.00 mm. (a) Find the maximum
electric field in the beam. (b) What total energy
is contained in a 1.00-m length of the beam? (c)
Find the momentum carried by a 1.00-m length of
the beam.
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Background for the superior mathematics student!
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Harmonic Plane Waves
In general, We will only be concerned with the
real part of the complex phasor representation
of a plane wave. Using Eulers formula
(kx-wt) phase
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Phase Velocity - Another View
Since the plane waves remain plane waves, the
phase on a plane does not change with time
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Vector Calculus Theorems
Gauss Divergence Theorem
Stokes Theorem
And an Important Identity
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Maxwells Equations In Differential Form
Gausss Law
Gausss Law for Magnetism
Faradays Law
Amperes Law