EEE 498/598 Overview of Electrical Engineering

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EEE 498/598Overview of Electrical Engineering

• Lecture 9 Faradays Law Of Electromagnetic
  Induction Displacement Current Complex
  Permittivity and Permeability

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Lecture 9 Objectives

• To study Faradays law of electromagnetic
  induction displacement current and complex
  permittivity and permeability.

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Fundamental Laws of Electrostatics

• Integral form

• Differential form

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Fundamental Laws of Magnetostatics

• Integral form

• Differential form

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Electrostatic, Magnetostatic, and
Electromagnetostatic Fields

• In the static case (no time variation), the
  electric field (specified by E and D) and the
  magnetic field (specified by B and H) are
  described by separate and independent sets of
  equations.
• In a conducting medium, both electrostatic and
  magnetostatic fields can exist, and are coupled
  through the Ohms law (J sE). Such a situation
  is called electromagnetostatic.

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Electromagnetostatic Fields

• In an electromagnetostatic field, the electric
  field is completely determined by the stationary
  charges present in the system, and the magnetic
  field is completely determined by the current.
• The magnetic field does not enter into the
  calculation of the electric field, nor does the
  electric field enter into the calculation of the
  magnetic field.

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The Three Experimental Pillars of Electromagnetics

• Electric charges attract/repel each other as
  described by Coulombs law.
• Current-carrying wires attract/repel each other
  as described by Amperes law of force.
• Magnetic fields that change with time induce
  electromotive force as described by Faradays law.

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Faradays Experiment
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Faradays Experiment (Contd)

• Upon closing the switch, current begins to flow
  in the primary coil.
• A momentary deflection of the compass needle
  indicates a brief surge of current flowing in the
  secondary coil.
• The compass needle quickly settles back to zero.
• Upon opening the switch, another brief deflection
  of the compass needle is observed.

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Faradays Law of Electromagnetic Induction

• The electromotive force induced around a closed
  loop C is equal to the time rate of decrease of
  the magnetic flux linking the loop.

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Faradays Law of Electromagnetic Induction
(Contd)

• S is any surface bounded by C

integral form of Faradays law
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Faradays Law (Contd)
Stokess theorem
assuming a stationary surface S
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Faradays Law (Contd)

• Since the above must hold for any S, we have

differential form of Faradays law (assuming a
stationary frame of reference)
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Faradays Law (Contd)

• Faradays law states that a changing magnetic
  field induces an electric field.
• The induced electric field is non-conservative.

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Lenzs Law

• The sense of the emf induced by the time-varying
  magnetic flux is such that any current it
  produces tends to set up a magnetic field that
  opposes the change in the original magnetic
  field.
• Lenzs law is a consequence of conservation of
  energy.
• Lenzs law explains the minus sign in Faradays
  law.

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Faradays Law

• The electromotive force induced around a closed
  loop C is equal to the time rate of decrease of
  the magnetic flux linking the loop.
• For a coil of N tightly wound turns

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Faradays Law (Contd)

• S is any surface bounded by C

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Faradays Law (Contd)

• Faradays law applies to situations where
• (1) the B-field is a function of time
• (2) ds is a function of time
• (3) B and ds are functions of time

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Faradays Law (Contd)

• The induced emf around a circuit can be separated
  into two terms
• (1) due to the time-rate of change of the B-field
  (transformer emf)
• (2) due to the motion of the circuit (motional
  emf)

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Faradays Law (Contd)
transformer emf
motional emf
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Moving Conductor in a Static Magnetic Field

• Consider a conducting bar moving with velocity v
  in a magnetostatic field

• The magnetic force on an electron in the
  conducting bar is given by

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Moving Conductor in a Static Magnetic Field
(Contd)

• Electrons are pulled toward end 2. End 2 becomes
  negatively charged and end 1 becomes charged.
• An electrostatic force of attraction is
  established between the two ends of the bar.

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Moving Conductor in a Static Magnetic Field
(Contd)

• The electrostatic force on an electron due to the
  induced electrostatic field is given by
• The migration of electrons stops (equilibrium is
  established) when

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Moving Conductor in a Static Magnetic Field
(Contd)

• A motional (or flux cutting) emf is produced
  given by

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Electric Field in Terms of Potential Functions

• Electrostatics

scalar electric potential
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Electric Field in Terms of Potential Functions
(Contd)

• Electrodynamics

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Electric Field in Terms of Potential Functions
(Contd)

• Electrodynamics

vector magnetic potential

• both of these potentials are now functions of
  time.

scalar electric potential
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Amperes Law and the Continuity Equation

• The differential form of Amperes law in the
  static case is
• The continuity equation is

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Amperes Law and the Continuity Equation (Contd)

• In the time-varying case, Amperes law in the
  above form is inconsistent with the continuity
  equation

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Amperes Law and the Continuity Equation (Contd)

• To resolve this inconsistency, Maxwell modified
  Amperes law to read

displacement current density
conduction current density
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Amperes Law and the Continuity Equation (Contd)

• The new form of Amperes law is consistent with
  the continuity equation as well as with the
  differential form of Gausss law

qev
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Displacement Current

• Amperes law can be written as

where
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Displacement Current (Contd)

• Displacement current is the type of current that
  flows between the plates of a capacitor.
• Displacement current is the mechanism which
  allows electromagnetic waves to propagate in a
  non-conducting medium.
• Displacement current is a consequence of the
  three experimental pillars of electromagnetics.

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Displacement Current in a Capacitor

• Consider a parallel-plate capacitor with plates
  of area A separated by a dielectric of
  permittivity e and thickness d and connected to
  an ac generator

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Displacement Current in a Capacitor (Contd)

• The electric field and displacement flux density
  in the capacitor is given by
• The displacement current density is given by

• assume fringing is negligible

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Displacement Current in a Capacitor (Contd)

• The displacement current is given by

conduction current in wire
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Conduction to Displacement Current Ratio

• Consider a conducting medium characterized by
  conductivity s and permittivity e.
• The conduction current density is given by
• The displacement current density is given by

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Conduction to Displacement Current Ratio (Contd)

• Assume that the electric field is a sinusoidal
  function of time
• Then,

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Conduction to Displacement Current Ratio (Contd)

• We have
• Therefore

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Conduction to Displacement Current Ratio (Contd)

• The value of the quantity s/we at a specified
  frequency determines the properties of the medium
  at that given frequency.
• In a metallic conductor, the displacement current
  is negligible below optical frequencies.
• In free space (or other perfect dielectric), the
  conduction current is zero and only displacement
  current can exist.

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Conduction to Displacement Current Ratio (Contd)
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Complex Permittivity

• In a good insulator, the conduction current (due
  to non-zero s) is usually negligible.
• However, at high frequencies, the rapidly varying
  electric field has to do work against molecular
  forces in alternately polarizing the bound
  electrons.
• The result is that P is not necessarily in phase
  with E, and the electric susceptibility, and
  hence the dielectric constant, are complex.

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Complex Permittivity (Contd)

• The complex dielectric constant can be written as
• Substituting the complex dielectric constant into
  the differential frequency-domain form of
  Amperes law, we have

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Complex Permittivity (Contd)

• Thus, the imaginary part of the complex
  permittivity leads to a volume current density
  term that is in phase with the electric field, as
  if the material had an effective conductivity
  given by
• The power dissipated per unit volume in the
  medium is given by

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Complex Permittivity (Contd)

• The term we?? E2 is the basis for microwave
  heating of dielectric materials.
• Often in dielectric materials, we do not
  distinguish between s and we??, and lump them
  together in we?? as

• The value of seff is often determined by
  measurements.

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Complex Permittivity (Contd)

• In general, both e? and e?? depend on frequency,
  exhibiting resonance characteristics at several
  frequencies.

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Complex Permittivity (Contd)

• In tabulating the dielectric properties of
  materials, it is customary to specify the real
  part of the dielectric constant (e? / e0) and the
  loss tangent (tand) defined as

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Complex Permeability

• Like the electric field, the magnetic field
  encounters molecular forces which require work to
  overcome in magnetizing the material.
• In analogy with permittivity, the permeability
  can also be complex

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Maxwells Equations in Differential Form for
Time-Harmonic Fields in Simple Medium
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Maxwells Curl Equations for Time-Harmonic Fields
in Simple Medium Using Complex Permittivity and
Permeability
complex permeability
complex permittivity