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1 EEE 498/598 Overview of Electrical Engineering Lecture 9: Faradays Law Of Electromagnetic Induction; Displacement Current; Complex Permittivity and Permeability

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

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Lecture 9 3 Fundamental Laws of Electrostatics Integral form Integral form Differential form

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Lecture 9 4 Fundamental Laws of Magnetostatics Integral form Integral form Differential form

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Lecture 9 5 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 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 = E ). Such a situation is called electromagnetostatic. In a conducting medium, both electrostatic and magnetostatic fields can exist, and are coupled through the Ohms law ( J = E ). Such a situation is called electromagnetostatic.

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Lecture 9 6 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. 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. 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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Lecture 9 7 The Three Experimental Pillars of Electromagnetics Electric charges attract/repel each other as described by Coulombs law. 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. 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. Magnetic fields that change with time induce electromotive force as described by Faradays law.

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Lecture 9 8 Faradays Experiment battery switch toroidal iron core compass primary coil secondary coil

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Lecture 9 9 Faradays Experiment (Contd) Upon closing the switch, current begins to flow in the primary coil. 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. 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. The compass needle quickly settles back to zero. Upon opening the switch, another brief deflection of the compass needle is observed. Upon opening the switch, another brief deflection of the compass needle is observed.

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Lecture 9 10 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. 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

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Lecture 9 11 Faradays Law of Electromagnetic Induction (Contd) S is any surface bounded by C integral form of Faradays law

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Lecture 9 12 Faradays Law (Contd) Stokess theorem assuming a stationary surface S

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Lecture 9 13 Faradays Law (Contd) Since the above must hold for any S, we have 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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Lecture 9 14 Faradays Law (Contd) Faradays law states that a changing magnetic field induces an electric field. Faradays law states that a changing magnetic field induces an electric field. The induced electric field is non- conservative. The induced electric field is non- conservative.

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Lecture 9 15 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. 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 is a consequence of conservation of energy. Lenzs law explains the minus sign in Faradays law. Lenzs law explains the minus sign in Faradays law.

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Lecture 9 16 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. 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 For a coil of N tightly wound turns

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Lecture 9 17 S is any surface bounded by C Faradays Law (Contd) C S

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Lecture 9 18 Faradays Law (Contd) Faradays law applies to situations where Faradays law applies to situations where (1) the B -field is a function of time (1) the B -field is a function of time (2) ds is a function of time (2) ds is a function of time (3) B and ds are functions of time (3) B and ds are functions of time

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Lecture 9 19 Faradays Law (Contd) The induced emf around a circuit can be separated into two terms: 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 ) (1) due to the time-rate of change of the B- field ( transformer emf ) (2) due to the motion of the circuit ( motional emf ) (2) due to the motion of the circuit ( motional emf )

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Lecture 9 20 Faradays Law (Contd) transformer emf motional emf

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Lecture 9 21 Moving Conductor in a Static Magnetic Field Consider a conducting bar moving with velocity v in a magnetostatic field: Consider a conducting bar moving with velocity v in a magnetostatic field: B v 2 1 + - The magnetic force on an electron in the conducting bar is given by

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Lecture 9 22 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. B v 2 1 + -

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Lecture 9 23 Moving Conductor in a Static Magnetic Field (Contd) The electrostatic force on an electron due to the induced electrostatic field is given by The electrostatic force on an electron due to the induced electrostatic field is given by The migration of electrons stops (equilibrium is established) when The migration of electrons stops (equilibrium is established) when

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Lecture 9 24 Moving Conductor in a Static Magnetic Field (Contd) A motional (or flux cutting) emf is produced given by A motional (or flux cutting) emf is produced given by

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Lecture 9 25 Electric Field in Terms of Potential Functions Electrostatics: Electrostatics: scalar electric potential

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Lecture 9 26 Electric Field in Terms of Potential Functions (Contd) Electrodynamics: Electrodynamics:

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Lecture 9 27 Electric Field in Terms of Potential Functions (Contd) Electrodynamics: Electrodynamics: scalar electric potential vector magnetic potential both of these potentials are now functions of time.

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Lecture 9 28 The differential form of Amperes law in the static case is The differential form of Amperes law in the static case is The continuity equation is The continuity equation is Amperes Law and the Continuity Equation

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Lecture 9 29 Amperes Law and the Continuity Equation (Contd) In the time-varying case, Amperes law in the above form is inconsistent with the continuity equation In the time-varying case, Amperes law in the above form is inconsistent with the continuity equation

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Lecture 9 30 Amperes Law and the Continuity Equation (Contd) To resolve this inconsistency, Maxwell modified Amperes law to read To resolve this inconsistency, Maxwell modified Amperes law to read conduction current density displacement current density

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Lecture 9 31 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 The new form of Amperes law is consistent with the continuity equation as well as with the differential form of Gausss law q ev

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Lecture 9 32 Displacement Current Amperes law can be written as Amperes law can be written as where

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Lecture 9 33 Displacement Current (Contd) Displacement current is the type of current that flows between the plates of a capacitor. 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 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. Displacement current is a consequence of the three experimental pillars of electromagnetics.

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Lecture 9 34 Displacement Current in a Capacitor Consider a parallel-plate capacitor with plates of area A separated by a dielectric of permittivity and thickness d and connected to an ac generator: Consider a parallel-plate capacitor with plates of area A separated by a dielectric of permittivity and thickness d and connected to an ac generator: + - z = 0 z = d icic A idid z

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Lecture 9 35 Displacement Current in a Capacitor (Contd) The electric field and displacement flux density in the capacitor is given by The electric field and displacement flux density in the capacitor is given by The displacement current density is given by The displacement current density is given by assume fringing is negligible

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Lecture 9 36 Displacement Current in a Capacitor (Contd) The displacement current is given by The displacement current is given by conduction current in wire

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Lecture 9 37 Conduction to Displacement Current Ratio Consider a conducting medium characterized by conductivity and permittivity. Consider a conducting medium characterized by conductivity and permittivity. The conduction current density is given by The conduction current density is given by The displacement current density is given by The displacement current density is given by

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Lecture 9 38 Conduction to Displacement Current Ratio (Contd) Assume that the electric field is a sinusoidal function of time: Assume that the electric field is a sinusoidal function of time: Then, Then,

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Lecture 9 39 Conduction to Displacement Current Ratio (Contd) We have We have Therefore Therefore

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Lecture 9 40 Conduction to Displacement Current Ratio (Contd) The value of the quantity at a specified frequency determines the properties of the medium at that given frequency. The value of the quantity 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 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. In free space (or other perfect dielectric), the conduction current is zero and only displacement current can exist.

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Lecture 9 41 good conductor good insulator Conduction to Displacement Current Ratio (Contd)

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Lecture 9 42 Complex Permittivity In a good insulator, the conduction current (due to non-zero ) is usually negligible. In a good insulator, the conduction current (due to non-zero ) 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. 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. 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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Lecture 9 43 Complex Permittivity (Contd) The complex dielectric constant can be written as The complex dielectric constant can be written as Substituting the complex dielectric constant into the differential frequency-domain form of Amperes law, we have Substituting the complex dielectric constant into the differential frequency-domain form of Amperes law, we have

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Lecture 9 44 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 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 The power dissipated per unit volume in the medium is given by

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Lecture 9 45 Complex Permittivity (Contd) The term E 2 is the basis for microwave heating of dielectric materials. The term E 2 is the basis for microwave heating of dielectric materials. Often in dielectric materials, we do not distinguish between and, and lump them together in as Often in dielectric materials, we do not distinguish between and, and lump them together in as The value of eff is often determined by measurements.

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Lecture 9 46 Complex Permittivity (Contd) In general, both and depend on frequency, exhibiting resonance characteristics at several frequencies. In general, both and depend on frequency, exhibiting resonance characteristics at several frequencies.

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Lecture 9 47 Complex Permittivity (Contd) In tabulating the dielectric properties of materials, it is customary to specify the real part of the dielectric constant ( / 0 ) and the loss tangent ( tan ) defined as In tabulating the dielectric properties of materials, it is customary to specify the real part of the dielectric constant ( / 0 ) and the loss tangent ( tan ) defined as

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Lecture 9 48 Complex Permeability Like the electric field, the magnetic field encounters molecular forces which require work to overcome in magnetizing the material. 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 In analogy with permittivity, the permeability can also be complex

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Lecture 9 49 Maxwells Equations in Differential Form for Time-Harmonic Fields in Simple Medium

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Lecture 9 50 Maxwells Curl Equations for Time-Harmonic Fields in Simple Medium Using Complex Permittivity and Permeability complex permittivity complex permeability

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