Electromagnetic Waves

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Chapter 34

• Electromagnetic Waves

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Maxwells Theory

• Electricity and magnetism were originally thought
  to be unrelated
• Maxwells theory showed a close relationship
  between all electric and magnetic phenomena and
  proved that electric and magnetic fields play
  symmetric roles in nature
• Maxwell hypothesized that a changing electric
  field would produce a magnetic field

• He calculated the speed of light 3x108 m/s
  and concluded that light and other
  electromagnetic waves consist of fluctuating
  electric and magnetic fields

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Maxwells Theory

• Stationary charges produce only electric fields
• Charges in uniform motion (constant velocity)
  produce electric and magnetic fields
• Charges that are accelerated produce electric and
  magnetic fields and electromagnetic waves
• A changing magnetic field produces an electric
  field

• A changing electric field produces a magnetic
  field
• These fields are in phase and, at any point, they
  both reach their maximum value at the same time

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Modifications to Ampères Law

• Ampères Law is used to analyze magnetic fields
  created by currents
• But this form is valid only if any electric
  fields present are constant in time
• Maxwell modified the equation to include
  time-varying electric fields and added another
  term, called the displacement current, Id
• This showed that magnetic fields are produced
  both by conduction currents and by time-varying
  electric fields

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Maxwells Equations

• In his unified theory of electromagnetism,
  Maxwell showed that the fundamental laws are
  expressed in these four equations

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Maxwells Equations

• Gauss Law relates an electric field to the
  charge distribution that creates it
• The total electric flux through any closed
  surface equals the net charge inside that surface
  divided by eo

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Maxwells Equations

• Gauss Law in magnetism the net magnetic flux
  through a closed surface is zero
• The number of magnetic field lines that enter a
  closed volume must equal the number that leave
  that volume
• If this wasnt true, there would be magnetic
  monopoles found in nature

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Maxwells Equations

• Faradays Law of Induction describes the creation
  of an electric field by a time-varying magnetic
  field
• The emf (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

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Maxwells Equations

• Ampère-Maxwell Law describes the creation of a
  magnetic field by a changing electric field and
  by electric current
• The line integral of the magnetic field around
  any closed path is the sum of mo times the net
  current through that path and eomo times the rate
  of change of electric flux through any surface
  bounded by that path

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Maxwells Equations

• Once the electric and magnetic fields are known
  at some point in space, the force acting on a
  particle of charge q can be found
• Maxwells equations with the Lorentz Force Law
  completely describe all classical electromagnetic
  interactions

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Maxwells Equations

• In empty space, q 0 and I 0
• The equations can be solved with wave-like
  solutions (electromagnetic waves), which are
  traveling at the speed of light
• This result led Maxwell to predict that light
  waves were a form of electromagnetic radiation

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Electromagnetic Waves

• From Maxwells equations applied to empty space,
  the following relationships can be found
• The simplest solutions to these partial
  differential equations are sinusoidal waves
  electromagnetic waves
• The speed of the electromagnetic wave is

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Plane Electromagnetic Waves

• The vectors for the electric and magnetic fields
  in an em wave have a specific space-time behavior
  consistent with Maxwells equations
• Assume an em wave that travels in the x direction
• We also assume that at any point in space, the
  magnitudes E and B of the fields depend upon x
  and t only
• The electric field is assumed to be in the y
  direction and the magnetic field in the z
  direction

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Plane Electromagnetic Waves

• The components of the electric and magnetic
  fields of plane electromagnetic waves are
  perpendicular to each other and perpendicular to
  the direction of propagation
• Thus, electromagnetic waves are transverse waves
• Waves in which the electric and magnetic fields
  are restricted to being parallel to a pair of
  perpendicular axes are said to be linearly
  polarized waves

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Poynting Vector

• Electromagnetic waves carry energy
• As they propagate through space, they can
  transfer that energy to objects in their path
• The rate of flow of energy in an em wave is
  described by a vector, S, called the Poynting
  vector defined as
• Its direction is the direction of propagation and
  its magnitude varies in time
• The SI units J/(s.m2) W/m2
• Those are units of power per unit area

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Poynting Vector

• Energy carried by em waves is shared equally by
  the electric and magnetic fields
• 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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Energy Density

• The energy density, u, is the energy per unit
  volume
• It can be shown that
• The instantaneous energy density associated with
  the magnetic field of an em wave equals the
  instantaneous energy density associated with the
  electric field and in a given volume this energy
  is shared equally by E and B
• The total instantaneous energy density is the sum
  of the energy densities associated with each
  field
• u uE uB eoE2 B2 / µo

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Energy Density

• When this is averaged over one or more cycles,
  the total average becomes
• uavg eo(E2)avg ½ eoE2max B2max / 2µo
• The intensity of an em wave equals the average
  energy density multiplied by the speed of light
• I Savg cuavg
• Electromagnetic waves transport linear momentum
  as well as energy
• As this momentum is absorbed by some surface,
  pressure is exerted on the surface

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Chapter 34Problem 6

• An electron moves through a uniform electric
  field E (2.50 i 5.00 j) V/m and a uniform
  magnetic field B (0.400 k) T. Determine the
  acceleration of the electron when it has a
  velocity v 10.0 i m/s.

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Hertzs Experiment

• In 1887 Hertz was the first to experimentally
  generate and detect electromagnetic waves
• An induction coil was connected to two large
  spheres forming a capacitor
• Oscillations were initiated by short voltage
  pulses by the coil
• As the air in the gap is ionized, it becomes a
  better conductor
• At a very high frequencies the discharge between
  the electrodes exhibited an oscillatory behavior

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Hertzs Experiment

• The inductor and capacitor formed the
  transmitter, equivalent to an LC circuit from a
  circuit viewpoint
• Several meters away from the transmitter was the
  receiver (a single loop of wire connected to two
  spheres) with its own inductance and capacitance
• When the resonance frequencies of the transmitter
  and receiver matched, energy transfer occurred
  between them

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Hertzs Results

• Hertz hypothesized the energy transfer was in the
  form of waves (now known to be electromagnetic
  waves)
• Hertz confirmed Maxwells theory by showing the
  waves existed and had all the properties of light
  waves (with different frequencies and
  wavelengths)
• Hertz measured the speed of the waves from the
  transmitter (used the waves to form an
  interference pattern and calculated the
  wavelength)
• The measured speed was very close to 3 x 108 m/s,
  the known speed of light, which provided evidence
  in support of Maxwells theory

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Electromagnetic Waves Produced by an Antenna

• Neither stationary charges nor steady currents
  can produce electromagnetic waves
• The fundamental mechanism responsible for this
  radiation when a charged particle undergoes an
  acceleration, it must radiate energy in the form
  of electromagnetic waves
• Electromagnetic waves are radiated by any circuit
  carrying alternating current
• An alternating voltage applied to the wires of an
  antenna forces the electric charge in the antenna
  to oscillate

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Electromagnetic Waves Produced by an Antenna

• Half-wave antenna two rods are connected to an
  ac source, charges oscillate between the rods (a)
• As oscillations continue, the rods become less
  charged, the field near the charges decreases and
  the field produced at t 0 moves away from the
  rod (b)
• The charges and field reverse (c) and the
  oscillations continue (d)

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Electromagnetic Waves Produced by an Antenna

• Because the oscillating charges in the rod
  produce a current, there is also a magnetic field
  generated
• As the current changes, the magnetic field
  spreads out from the antenna
• The magnetic field lines form concentric circles
  around the antenna and are perpendicular to the
  electric field lines at all points
• The antenna can be approximated by an oscillating
  electric dipole

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The Spectrum of EM Waves

• Types of electromagnetic waves are distinguished
  by their frequencies (wavelengths) c ƒ ?
• There is no sharp division between one kind of em
  wave and the next note the overlap between
  types of waves

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The Spectrum of EM Waves

• Radio waves are used in radio and television
  communication systems
• Microwaves (1 mm to 30 cm) are well suited for
  radar systems microwave ovens are an
  application
• Infrared waves are produced by hot objects and
  molecules and are readily absorbed by most
  materials

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The Spectrum of EM Waves

• Visible light (a small range of the spectrum from
  400 nm to 700 nm) part of the spectrum detected
  by the human eye
• Ultraviolet light (400 nm to 0.6 nm) Sun is an
  important source of uv light, however most uv
  light from the sun is absorbed in the
  stratosphere by ozone

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The Spectrum of EM Waves

• X-rays most common source is acceleration of
  high-energy electrons striking a metal target,
  also used as a diagnostic tool in medicine
• Gamma rays emitted by radioactive nuclei, are
  highly penetrating and cause serious damage when
  absorbed by living tissue

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Chapter 34Problem 11

• In SI units, the electric field in an
  electromagnetic wave is described by Ey 100 sin
  (1.00 107 x ?t). Find (a) the amplitude of
  the corresponding magnetic field oscillations,
  (b) the wavelength ?, and (c) the frequency f.

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Answers to Even Numbered Problems Chapter 34
Problem 10 733 nT
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Properties of em Waves, 3

• Electromagnetic waves obey the superposition
  principle