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EEE 498/598 Overview of Electrical Engineering. Lecture 9: Faraday’s Law Of Electromagnetic Induction; Displacement Current; Complex Permittivity and Permeability. Lecture 9 Objectives.

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eee 498 598 overview of electrical engineering

EEE 498/598Overview of Electrical Engineering

Lecture 9: Faraday’s Law Of Electromagnetic Induction; Displacement Current; Complex Permittivity and Permeability

1

lecture 9 objectives
Lecture 9 Objectives
  • To study Faraday’s law of electromagnetic induction; displacement current; and complex permittivity and permeability.

2

electrostatic magnetostatic and electromagnetostatic fields
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 Ohm’s law (J = sE). Such a situation is called electromagnetostatic.

5

electromagnetostatic fields
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.

6

the three experimental pillars of electromagnetics
The Three Experimental Pillars of Electromagnetics
  • Electric charges attract/repel each other as described by Coulomb’s law.
  • Current-carrying wires attract/repel each other as described by Ampere’s law of force.
  • Magnetic fields that change with time induce electromotive force as described by Faraday’s law.

7

faraday s experiment

toroidal iron

core

switch

compass

battery

secondary

coil

primary

coil

Faraday’s Experiment

8

faraday s experiment cont d
Faraday’s Experiment (Cont’d)
  • Upon closing the switch, current begins to flow in the primary coil.
  • A momentary deflection of the compassneedle 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.

9

faraday s law of electromagnetic induction

S

C

Faraday’s 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.”

10

faraday s law of electromagnetic induction cont d
Faraday’s Law of Electromagnetic Induction (Cont’d)
  • S is any surface bounded by C

integral form of Faraday’s law

11

faraday s law cont d
Faraday’s Law (Cont’d)

Stokes’s theorem

assuming a stationary surface S

12

faraday s law cont d13
Faraday’s Law (Cont’d)
  • Since the above must hold for any S, we have

differential form of Faraday’s law (assuming a stationary frame of reference)

13

faraday s law cont d14
Faraday’s Law (Cont’d)
  • Faraday’s law states that a changing magnetic field induces an electric field.
  • The induced electric field is non-conservative.

14

lenz s law
Lenz’s 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.”
  • Lenz’s law is a consequence of conservation of energy.
  • Lenz’s law explains the minus sign in Faraday’s law.

15

faraday s law
Faraday’s 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

16

faraday s law cont d17

S

C

Faraday’s Law (Cont’d)
  • S is any surface bounded by C

17

faraday s law cont d18
Faraday’s Law (Cont’d)
  • Faraday’s 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

18

faraday s law cont d19
Faraday’s Law (Cont’d)
  • 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)

19

faraday s law cont d20
Faraday’s Law (Cont’d)

transformer emf

motional emf

20

moving conductor in a static magnetic field

2

-

B

v

+

1

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

21

moving conductor in a static magnetic field cont d

2

-

B

v

+

1

Moving Conductor in a Static Magnetic Field (Cont’d)
  • 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.

22

moving conductor in a static magnetic field cont d23
Moving Conductor in a Static Magnetic Field (Cont’d)
  • The electrostatic force on an electron due to the induced electrostatic field is given by
  • The migration of electrons stops (equilibrium is established) when

23

moving conductor in a static magnetic field cont d24
Moving Conductor in a Static Magnetic Field (Cont’d)
  • A motional(or “flux cutting”) emfis produced given by

24

electric field in terms of potential functions
Electric Field in Terms of Potential Functions
  • Electrostatics:

scalar electric potential

25

electric field in terms of potential functions cont d27
Electric Field in Terms of Potential Functions (Cont’d)
  • Electrodynamics:

vector magnetic potential

  • both of these potentials are now functions of time.

scalar electric potential

27

ampere s law and the continuity equation
Ampere’s Law and the Continuity Equation
  • The differential form of Ampere’s law in the static case is
  • The continuity equation is

28

ampere s law and the continuity equation cont d
Ampere’s Law and the Continuity Equation (Cont’d)
  • In the time-varying case, Ampere’s law in the above form is inconsistent with the continuity equation

29

ampere s law and the continuity equation cont d30
Ampere’s Law and the Continuity Equation (Cont’d)
  • To resolve this inconsistency, Maxwell modified Ampere’s law to read

displacement current density

conduction current density

30

ampere s law and the continuity equation cont d31
Ampere’s Law and the Continuity Equation (Cont’d)
  • The new form of Ampere’s law is consistent with the continuity equation as well as with the differential form of Gauss’s law

qev

31

displacement current
Displacement Current
  • Ampere’s law can be written as

where

32

displacement current cont d
Displacement Current (Cont’d)
  • 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.

33

displacement current in a capacitor

z

A

ic

+

z = d

e

id

z = 0

-

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:

34

displacement current in a capacitor cont d
Displacement Current in a Capacitor (Cont’d)
  • The electric field and displacement flux density in the capacitor is given by
  • The displacement current density is given by
  • assume fringing is negligible

35

displacement current in a capacitor cont d36
Displacement Current in a Capacitor (Cont’d)
  • The displacement current is given by

conduction current in wire

36

conduction to displacement current ratio
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

37

conduction to displacement current ratio cont d
Conduction to Displacement Current Ratio (Cont’d)
  • Assume that the electric field is a sinusoidal function of time:
  • Then,

38

conduction to displacement current ratio cont d40
Conduction to Displacement Current Ratio (Cont’d)
  • 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.

40

complex permittivity
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 Pis not necessarily in phase with E, and the electric susceptibility, and hence the dielectric constant, are complex.

42

complex permittivity cont d
Complex Permittivity (Cont’d)
  • The complex dielectric constant can be written as
  • Substituting the complex dielectric constant into the differential frequency-domain form of Ampere’s law, we have

43

complex permittivity cont d44
Complex Permittivity (Cont’d)
  • 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

44

complex permittivity cont d45
Complex Permittivity (Cont’d)
  • 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.

45

complex permittivity cont d46
Complex Permittivity (Cont’d)
  • In general, both e and e depend on frequency, exhibiting resonance characteristics at several frequencies.

46

complex permittivity cont d47
Complex Permittivity (Cont’d)
  • 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

47

complex permeability
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

48

slide50
Maxwell’s Curl Equations for Time-Harmonic Fields in Simple Medium Using Complex Permittivity and Permeability

complex

permeability

complex

permittivity

50

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