Electromagnetic Wave Propagation

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CHAPTER 4
ELECTROMAGNETIC FIELDS THEORY

• Electromagnetic Wave Propagation

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Power and Poynting Vector

• The dimension of E and H are V/m and A/m
  respectively. Hence, the vector ExH has VA/m2 as
  dimension, which is power density W/m2.
• The cross product ExH results in a vector along
  the direction of propagation that is the
  direction of the energy flow.
• The vector ExH is known as the power density or
  Poynting vector.
• Poynting vector P E x H
• which is directed along the direction of the
• electromagnetic energy flow

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Power and Poynting Vector

• Poynting Theorem states that
• The net power flowing out of a given volume v is
  equal to the time rate of decrease in the energy
  stored between v minus the conduction losses

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Power and Poynting Vector

• We will start our discussion by considering
  fields in the time domain.
• From Maxwells equation for a time-varying field,
• If we take the dot product of both sides of
  Maxwells curl of H equation with E, we obtain

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Power and Poynting Vector

• Rearranging eq (2), we have
• Also, from Maxwells curl of E equation, we have
• Substituting this into the former eq (3), we
  obtain

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Power and Poynting Vector

• If the medium is simple, we can also write
• Substituting (5) and (6) into this into (4) we
  obtain

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Power and Poynting Vector

• Integrating eq (7) over an arbitrary volume v, we
  get
• Finally, we can use the divergence theorem to
  express the left-hand side as a surface integral
  over the surface S that bounds v

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Power and Poynting Vector

• The ohmic power dissipated inside v
• The rate of decrease in energy stored in electric
  and magnetic fields

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Power and Poynting Vector

• Poynting vector using the E(z,t) and H(z,t) can
  be determined through
• P (z,t) E(z,t) x H(z,t) (9)
• Time-average poynting vector can determined by
  integrating eq (9) over the period T,

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Power and Poynting Vector

• Total time-average power crossing a given surface
  is given by

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Example 1

• In a free space medium,
• Find
• (a) H field
• (b) The time-average power carried by the wave
• (c) The total power crossing of
  plane

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Exercise 1

• In a free space medium,
• Find the total power passing through
• (a) A square plate of side 10 cm on plane
• (b) A circular disk of radius 5 cm on plane