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Lecture 9: Faraday’s Law Of Electromagnetic Induction; Displacement Current; Complex Permittivity and Permeability Lecture 9 1 To study Faraday’s law of electromagnetic induction; displacement current; and complex permittivity and permeability. Lecture 9 2 Integral form E dl 0 E 0 D qev C Dds q ev S Differential form dv V D E Lecture 9 3 Integral form H dl J d s C Differential form H J S B 0 Bds 0 S B H Lecture 9 4 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. Lecture 9 5 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. Lecture 9 6 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. Lecture 9 7 switch toroidal iron core compass battery secondary coil primary coil Lecture 9 8 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. Lecture 9 9 “The electromotive force induced around a closed loop Γ is equal to the time rate of decrease of the magnetic flux linking the loop.” d e dt S C Lecture 9 10 Bds • S is any surface bounded by Γ S e E d l C d E d l B d s C dt S integral form of Faraday’s law Lecture 9 11 Stokes’s theorem E d l E d s C S d B B d s ds dt S t S assuming a stationary surface S Lecture 9 12 Since the above must hold for any S, we have differential form of Faraday’s law (assuming a stationary frame of reference) B E t Lecture 9 13 Faraday’s law states that a changing magnetic field induces an electric field. The induced electric field is nonconservative. Lecture 9 14 “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. Lecture 9 15 “The electromotive force induced around a closed loop Γis equal to the time rate of decrease of the magnetic flux linking the loop.” d e dt For a coil of N tightly wound turns d e N dt 16 Lecture 9 Bds S S Γ • S is any surface bounded by Γ e E d l C Lecture 9 17 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 Lecture 9 18 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) Lecture 9 19 d e B d s dt S B ds t S v B d l C transformer emf motional emf Lecture 9 20 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 2 B v + F m ev B 1 Lecture 9 21 2 B v + 1 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. Lecture 9 22 The electrostatic force on an electron due to the induced electrostatic field is given by The migration of electrons stops e E when (equilibrium F is eestablished) F e F m E v B Lecture 9 23 A motional (or “flux cutting”) emf is produced given by 1 e v B d l 2 Lecture 9 24 Electrostatics: E 0 E scalar electric potential Lecture 9 25 Electrodynamics: B A B E A t t A A E 0 E t t Lecture 9 26 Electrodynamics: A E t vector magnetic potential • both of these potentials are now functions of time. scalar electric potential Lecture 9 27 The differential form of Ampere’s law in the static case is H J The continuity equation is qev J 0 t Lecture 9 28 In the time-varying case, Ampere’s law in the above form is inconsistent with the continuity equation J H 0 Lecture 9 29 To resolve this inconsistency, Maxwell modified Ampere’s law to read D H J c t conduction current density displacement current density Lecture 9 30 The new form of Ampere’s law is consistent with the continuity equation as well as with the differential form of Gauss’s law J c D H 0 t qev Lecture 9 31 Ampere’s law can be written as H Jc J d where D Jd displaceme nt current density (A/m 2 ) t Lecture 9 32 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. Lecture 9 33 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 A z=d z=0 ic + v ( t ) V0 cos t id Lecture 9 34 The electric field and displacement flux density in the capacitor is given by V0 v(t ) E aˆ z aˆ z cos t d d V0 D E aˆ z cos t d • assume fringing is negligible The displacement current density is given by V0 D Jd aˆ z sin t t d Lecture 9 35 The displacement current is given by id J d d s J d A S A d dv CV0 sin t C ic dt V0 sin t conduction current in wire Lecture 9 36 Consider a conducting medium characterized by conductivity s and permittivity . The conduction current density is given by J s E c The displacement current density is given by E Jd t Lecture 9 37 Assume that the electric field is a sinusoidal function of time: Then, E E0 cos t J c sE0 cos t J d E0 sin t Lecture 9 38 We have Therefore Jc max sE0 Jd max E0 J c max Jd max s Lecture 9 39 The value of the quantity s/ 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. Lecture 9 40 10 10 10 10 10 10 s 10 10 10 10 10 Humid Soil ( r = 30, s = 10-2 S/m) 6 5 4 good conductor 3 2 1 0 -1 -2 -3 good insulator -4 10 0 10 2 10 4 10 6 Frequency (Hz) 41 10 8 10 10 Lecture 9 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. Lecture 9 42 The complex dielectric constant can be written as c j Substituting the complex dielectric constant into the differential frequencydomain form of Ampere’s law, we have H s E j E E Lecture 9 43 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 s eff s The power dissipated per unit volume in the medium is given by s eff E sE E 2 2 2 Lecture 9 44 The term E2 is the basis for microwave heating of dielectric materials. Often in dielectric materials, we do not distinguish between s and , and lump them together in as s eff • The value of seff is often determined by measurements. Lecture 9 45 In general, both and depend on frequency, exhibiting resonance characteristics at several frequencies. 1 Imag Part of Dielectric Constant Real Part of Dielectric Constant 2.5 2 1.5 1 0.5 0 0 2 4 6 8 10 12 14 16 18 0.9 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0 20 Normalized Frequency 0 2 4 6 8 10 12 14 16 Normalized Frequency Lecture 9 46 18 20 In tabulating the dielectric properties of materials, it is customary to specify the real part of the dielectric constant ( / 0) and the loss tangent (tand) defined as tan d Lecture 9 47 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 c j Lecture 9 48 E j s m H K i H j s e E J i E qev H qmv Lecture 9 49 complex permeability E j H K i H j E J i complex permittivity Lecture 9 50