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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 1

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Lecture 9 Objectives To study Faraday’s law of electromagnetic induction; displacement current; and complex permittivity and permeability. 2

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**Fundamental Laws of Electrostatics**

Integral form Differential form 3

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**Fundamental Laws of Magnetostatics**

Integral form Differential form 4

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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 Ohm’s law (J = sE). Such a situation is called electromagnetostatic. 5

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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. 6

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**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

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**Faraday’s Experiment battery switch toroidal iron core compass primary**

coil secondary 8

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**Faraday’s Experiment (Cont’d)**

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. 9

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**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.” C S 10

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**Faraday’s Law of Electromagnetic Induction (Cont’d)**

S is any surface bounded by C integral form of Faraday’s law 11

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**Faraday’s Law (Cont’d)**

Stokes’s theorem assuming a stationary surface S 12

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**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

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**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

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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

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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

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**Faraday’s Law (Cont’d)**

S is any surface bounded by C 17

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**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

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**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

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**Faraday’s Law (Cont’d)**

transformer emf motional emf 20

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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 B v 2 1 + - 21

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**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. B v 2 1 + - 22

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**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

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

A motional (or “flux cutting”) emf is produced given by 24

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**Electric Field in Terms of Potential Functions**

Electrostatics: scalar electric potential 25

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**Electric Field in Terms of Potential Functions (Cont’d)**

Electrodynamics: 26

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**Electric Field in Terms of Potential Functions (Cont’d)**

Electrodynamics: vector magnetic potential scalar electric potential both of these potentials are now functions of time. 27

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**Ampere’s Law and the Continuity Equation**

The differential form of Ampere’s law in the static case is The continuity equation is 28

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**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

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**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

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**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

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Displacement Current Ampere’s law can be written as where 32

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**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

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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: + - z = 0 z = d e ic A id z 34

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**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

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**Displacement Current in a Capacitor (Cont’d)**

The displacement current is given by conduction current in wire 36

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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 37

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**Conduction to Displacement Current Ratio (Cont’d)**

Assume that the electric field is a sinusoidal function of time: Then, 38

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**Conduction to Displacement Current Ratio (Cont’d)**

We have Therefore 39

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**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

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**Conduction to Displacement Current Ratio (Cont’d)**

good conductor good insulator 41

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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. 42

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**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

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**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

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**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

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**Complex Permittivity (Cont’d)**

In general, both e and e depend on frequency, exhibiting resonance characteristics at several frequencies. 46

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**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

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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 48

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**Maxwell’s Equations in Differential Form for Time-Harmonic Fields in Simple Medium**

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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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6. Maxwell’s Equations In Time-Varying Fields

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