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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 498/598Overview of Electrical Engineering

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

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• To study Faraday’s law of electromagnetic induction; displacement current; and complex permittivity and permeability.

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Differential form

Fundamental Laws of Electrostatics

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Differential form

Fundamental Laws of Magnetostatics

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• 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.

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• 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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• 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.

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toroidal iron Fields

core

switch

compass

battery

secondary

coil

primary

coil

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• 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.

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S Fields

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.”

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• S is any surface bounded by C

integral form of Faraday’s law

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Faraday’s Law (Cont’d) Fields

Stokes’s theorem

assuming a stationary surface S

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Faraday’s Law (Cont’d) Fields

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

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

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Faraday’s Law (Cont’d) Fields

• Faraday’s law states that a changing magnetic field induces an electric field.

• The induced electric field is non-conservative.

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Lenz’s Law Fields

• “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.

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• “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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S Fields

C

• S is any surface bounded by C

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Faraday’s Law (Cont’d) Fields

• 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

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Faraday’s Law (Cont’d) Fields

• 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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Faraday’s Law (Cont’d) Fields

transformer emf

motional emf

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2 Fields

-

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

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2 Fields

-

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.

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• 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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• A motional(or “flux cutting”) emfis produced given by

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• Electrostatics:

scalar electric potential

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• Electrodynamics:

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• Electrodynamics:

vector magnetic potential

• both of these potentials are now functions of time.

scalar electric potential

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• The differential form of Ampere’s law in the static case is

• The continuity equation is

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

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• To resolve this inconsistency, Maxwell modified Ampere’s law to read

displacement current density

conduction current density

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• 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

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Displacement Current Fields

• Ampere’s law can be written as

where

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• 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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z Fields

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:

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• 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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• The displacement current is given by

conduction current in wire

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

• Then,

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• We have

• Therefore

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• 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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good Fields

conductor

good insulator

Conduction to Displacement Current Ratio (Cont’d)

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Complex Permittivity Fields

• 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.

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• 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

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

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• 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 Fields

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

complex

permeability

complex

permittivity

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