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Chung-Ang University Field & Wave Electromagnetics CH 7. Time-Varying Fields and Maxwell’s Equations

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Chung-Ang University Field & Wave Electromagnetics 7.1 Introduction Fundamentl Relations Electrostatic Model Magnetostatic Model Governing equations Constitutive relations (linear and isotropic media) In the static case, electric field vectors and and magnetic field vectors and form separate and independent pairs. Fundamental Relations for Electrostatic and Magnetostatic Models In a conducting medium, static electric and magnetic fields may both exist and form an electromagnetostatic field.

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Chung-Ang University Field & Wave Electromagnetics 7.1 Introduction A static electric field in a conducting medium causes a steady current to flow that, In turn, gives rise to a static magnetic field. The electric field can be completely determined from the static electric charges or potential distributions. The magnetic field is a consequence; it does not enter into the calculation of the electric field. In this chapter we will see that a changing magnetic field gives rise to an electric field, and vice versa.

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction Michael Faraday, in 1831, discovered experimentally that a current was induced in a conducting loop when the magnetic flux linking the loop changed. The quantitative relationship between the induced emf and the rate of change of flux linkage, based on experimental observation, is known as Faraday’s law. Fundamental Postulate for Electromagnetic Induction (7 - 1) Equation 7-1 expresses a point-function relationship; that is, it applies to every point in space, whether it be in free space or in a material medium. The electric field intensity in a region of time-varying magnetic flux density is therefore nonconservative and cannot be expressed as the gradient of a scalar potential.

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction Taking the surface integral of both sides of Eq.(7-1) over an open surface and applying Stokes’s theorem, we obtain (7 - 2) 7–2.1 A STATIONARY CIRCUIT IN A TIME-VARYING MAGNETIC FIELD Equation (7-2) is valid for any surface S with a bounding contour C, whether or not a physical circuit exists around C. For a stationary circuit with a contour C and surface S, Eq(7 - 2) can be written as (7 - 3)

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction If we define= emf induced in circuit with contour C (V) (7 – 4) = magnetic flux crossing surface S (Wb), (7 – 5) then Eq.(7 - 3) becomes(V). (7 – 6) Equation (7 – 6) states that the electromotive force induced in a stationary closed circuit is equal to the negative rate of increase of the magnetic flux linking the circuit. This is a statement of Faraday’s law of electromagnetic induction.

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction TRANSFORMERS A transformer is an alternating-current (a-c) device that transforms voltage, currents, and impendances. (a)Schematic diagram of a transformer. FIGURE 7-1 +_+_ +_+_ For the closed path in the magnetic circuit In Fig.7-1(a) traced by magnetic flux, we have, from Eq.(6-101), ( Eq.(6-101), ) (7 – 7) (7 – 8) ( core of length l, cross-sectional area S, permeability ) Substituting Eq.(7-8) in Eq.(7-7) (7 – 9) back

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction (b) An equivalent circuit. FIGURE 7-1 Ideal transformer +_+_ + _ a) Ideal transformer. For an ideal transformer we assume that, and Eq.(7 - 9)becomes(7 – 10)

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction Eq.(7 - 10) states that the ratio of the currents in the primary and secondary windings of an ideal transformer is equal to the inverse ratio of the numbers of turns. Faraday’s law tells us that (7 - 11) and (7 – 12) From Eqs. (7 – 11) and (7 – 12) we have (7 – 13) Thus, the ratio of the voltages across the primary and secondary windings of an ideal transformer is equal to the turns ratio.

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction When the secondary winding is terminated in a load resistance, as shown in Fig. 7-1(a) click, the effective load seen by the source connected to primary winding isclick or (7 – 14a) For a sinusoidal source and a load impedance, it is obvious that the effective load seen by the source is, an impedance transformation. We have (7 – 14b) b) Real transformer. Referring to Eq. (7 – 9), we can write the magnetic flux linkages of the primary and secondary windings as

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction (7 – 15) (7 – 16) Using Eqs. (7 - 15) and (7 - 16) in Eqs. (7 - 11) and (7 - 12), we obtain (7 – 17)(7 – 18) where (7 – 19) the self-inductance of the primary winding. (7 – 20) the self-inductance of the secondary winding. (7 – 21) the mutual inductance between the primary and secondary windings.

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction For an ideal transformer there is no leakage flux, and For real transformers, k < 1, (7 -22) Where k is called the coefficient of coupling. For real transformers we have the following real-life conditions. the existence of leakage flux ( k < 1 ), noninfinite inductances, nonzero winding resistances, the presence of hysteresis and eddy-current losses. The nonlinear nature of the ferromagnetic core further compounds the difficulty of an exact analysis of real transformers.

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction Eddy currents. When time-varying magnetic flux flows in the ferromagnetic core, an induced emf will result in accordance with Faraday’s law. This induced emf will produce local currents in the conducting core normal to the magnetic flux. These currents are called eddy currents. Eddy currents produce ohmic power loss and cause local heating. This is the principle of induction heating. In transformers, eddy-current power loss is undesirable and can be reduced by using core materials that have high permeability but low conductivity (high and low ). For low-frequency, high-power applications an economical way for eddy-current power loss is to use laminated cores.

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction A MOVING CONDUCTOR IN A STATIC MAGNETIC FIELD. ☉☉ ☉ ☉ ☉ ☉ ☉ ☉ ☉ ☉ ☉ ☉ 2121 B Figure 7-2 A conducting bar moving in a magnetic field. A force will cause the freely movable electrons in the conductor to drift toward one end of the conductor and leave the other end positively charged. This separation of the positive and negative charges creates a Coulombian force of attr- action. The charge-separation process continues until the electric and magnetic forces balance each other and a state of equilibri- um is reached.

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction To an observer moving with the conductor there is no apparent motion, and the magnetic force per unit charge can be interpreted as an induced electric field acting along the conductor and producing a voltage (7 – 23) If the moving conductor is a part of a closed circuit C, then the emf generated around the circuit is (V). (7 – 24) This is referred to as a flux cutting emf or a motional emf. Obviously, only the part of the circuit that moves in a direction not parallel to the magnetic flux will contribute in Eq. (7 – 24).

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction A MOVING CIRCUIT IN A TIME-VARYING MAGNETIC FIELD. Lorentz’s force equation (7 – 31) The force on q can be interpreted as caused by an electric field, where (7 – 32)(7 – 33) Hence, when a conducting circuit with contour C and surface S moves with a velocity in a field, we use Eq. (7 – 33) in Eq. (7 – 2) to obtain (V).(7 – 34) ( Eq. 7-2 ) Eq. (7 – 34) is the general form of Faraday’s law for a moving circuit in a time-varying magnetic field. or

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction Let us consider a circuit with contour that moves from at time t to at time in a changing magnetic field. FIGURE 7-5 A moving circuit in a time-varying magnetic field. The time-rate of change of magnetic flux through the contour is (7 – 35)

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction in Eq. (7 - 35) can be expanded as a Taylor’s series: (7 – 36) Substitution of Eq. (7 – 36) in Eq. (7 – 35) yields (7 – 37) An element of the side surface is (7 – 38) Apply the divergence theorem for at time t to the region sketched in Fig. 7 – 5 : (7 – 39)

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction Using Eq. (7 – 38) in Eq. (7 – 39) and noting that, we have Combining Eqs. (7 – 37) and (7 – 40), we obtain (7 – 40) which can be identified as the negative of the right side of Eq. (7 – 34). If we designate emf induced in circuit C measured in the moving frame, (7 – 42) (7 – 41) Eq. (7 – 34) can be written simply as (V), (7 – 43)

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Chung-Ang University Field & Wave Electromagnetics 7.2 Faraday’s Law of Electromagnetic Induction Eq. (7 – 43) is of the same form as Eq. (7 – 6). If a circuit is not in motion, reduces to, and Eqs. (7 – 43) and (7 – 6) are exactly the same. Faraday’s law that the emf induced in a closed circuit equals the negative time-rate of increase of the magnetic flux linking a circuit applies to a stationary circuit as well as a moving one.

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Chung-Ang University Field & Wave Electromagnetics 7.3 Maxwell’s Equations The fundamental postulate for electromagnetic induction assures us that a time- varying magnetic field gives rise to an electric field. Time-varying case: The revised set of two curl and two divergence equations from Table 7 – 1: (7 – 47a)(7 – 47b) (7 – 47c)(7 – 47d) The mathematical expression of charge conservation is the equation of continuity : (7 – 48) Divergence of Eq. (7 – 47b) :(7 – 49) (null identity) since Eq. (7 – 48) asserts that does not vanish in a time-varying situation, Eq. (7 – 49) is, in general, not true.

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Chung-Ang University Field & Wave Electromagnetics 7.3 Maxwell’s Equations How should Eqs. (7 – 47a, b, c, d) be modified so that they are consistent with Eq. (7 – 48)? First of all, a term must be added to the right side of Eq. (7 – 49) : Using Eq. (7 – 47c) in Eq. (7 – 50), we have which implies that Eq. (7 – 52) indicates that a time-varying electric field will give rise to a magnetic field, even in the absence of a current flow. The additional term is necessary to make Eq. (7 – 52) consistent with the principle of conservation of charge. (7 – 50) (7 – 51) (7 – 52) The term is called displacement current density.

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Chung-Ang University Field & Wave Electromagnetics 7.3 Maxwell’s Equations In order to be consistent with the equation of continuity in a time varying situation, both of the curl equations in Table 7 – 1 must be generalized. The set of four consistent equations to replace the inconsistent equations, Eqs. (7 – 47a, b, c, d), are (7 – 53a) (7 – 53b) (7 – 53c) (7 – 53d) They are known as Maxwell’s equations.

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Chung-Ang University Field & Wave Electromagnetics 7.3 Maxwell’s Equations INTEGRAL FORM OF MAXWELL’S EQUATIONS. The four Maxwell’s equations in (7 – 53a, b, c, d) are differential equations that are valid at every point in space. In explaining electromagnetic phenomena in a physical environment we must deal with finite objects of specified shapes and boundaries. It is convenient to convert the differential forms into their integral-form equivalents. We take the surface integral of both sides of the curl equations in Eqs. (7 – 53a, b) over an open surface S with contour C and apply Stokes’s theorem to obtain (7 – 54a) (7 – 54b)

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Chung-Ang University Field & Wave Electromagnetics 7.3 Maxwell’s Equations Taking the volume integral of both sides of the divergence equations in Eqs. (7 – 53c, d) over a volume V with a closed surface S and using divergence theorem, we have (7 – 54c) (7 – 54d) The set of four equations in (7 – 54a, b, c, d) are the integral form of Maxwell’s equations.

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Chung-Ang University Field & Wave Electromagnetics 7.3 Maxwell’s Equations Maxwell’s Equations Differential FormIntegral FormSignificance Faraday’s law Ampere’s circuital law Gauss’s law No isolated magnetic charge

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