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Chapter 6 Time-Varying Field and Maxwell’s Equations 6-1 Overview 6-2 Faraday’s Law of Electromagnetic Induction 6-3 Maxwell’s Equations 6-4 Potential Functions 6-5 Time-Harmonic Fields 6-1 Overview ¾ Fundamental governing equations for electrostatic and magnetostatic models: JG ∇ ⋅ D = ρv JG ∇× E = 0 For linear and isotropic media: JG JG D=εE JG ∇⋅B = 0 JJG JG ∇× H = J JG JJG B = μH ¾ Static charges are the source of an electric field; Moving charges produce a current, which gives rise to a magnetic field. However, these fields are static fields, which do not give rise to waves. ¾ We wish to have waves, which may propagate and carry energy and information. ¾ How to generate wave? and how wave propagates? 1 6-2 Faraday’s Law of Electromagnetic Induction ¾ Fundamental postulate for electromagnetic induction is: Integral form: Differential form: JG JG ∂B ∇×E = − ∂t • • JG JG G ∂ B JG vC∫ E ⋅ dl = − ∫S ∂t ⋅ d S Point-function relationship. E in a region of time-varying B is non-conservative, and ≠ -∇V V = JG G v∫ E ⋅ d l (V) Electromotive force - emf C Transformer emf: How do we generate emf ? emf: tr Vemf m Motional emf / flux-cutting emf: Vemf tr Vemf Section 6-2.1: A stationary circuit in a time-varying magnetic field: Section 6-2.3: A moving conductor in a static magnetic field: Section 6-2.4: A moving circuit in a time-varying magnetic field: m Vemf tr m Vemf + Vemf (Pls ignore section 6-2.2: transformers) 6-2.1 A stationary circuit in a time-varying magnetic field: Re-write equation: JG JG G ∂ B JG d JG JG vC∫ E ⋅ dl = −∫S ∂t ⋅ d S = − dt ∫S B ⋅ d S JG JG Φ = ∫ B ⋅ d S (Web) Magnetic flux crossing surface S S ¾ Faraday’s Law of EM Induction tr =− Vemf dΦ dt (V) Electromotive force ( Emf ) induced in a stationary closed circuit is equal to the negative rate of increase of magnetic flux Φ linking the circuit Transformer emf: The emf induced in a stationary loop caused by a time-varying magnetic field is called a transformer emf. ¾ Lenz’s Law: The negative sign is an assertion that induced emf will cause a current to flow in the closed loop in such a direction as to oppose the change in the linking Φ. 2 Faradays’ Law of Induction ¾ Two experiments: An ammeter registers a current in the wire loop when moving magnet with respect to the loop An ammeter registers a current just when switch is closed or opened Induction Inducted current Induced emf Φ is varying with time : dΦ ≠ 0 dt ¾ Faradays’ Law of Induction: The magnitude of the emf induced in a conducting loop is equal to the rate of change of the magnetic flux through that loop E=− dΦB dt or E= −N dΦB (coil of N turns) dt ¾ An emf will be induced if any one (or more ) of the following vary with time: • B, A, θ Φ B = BA cos θ Lenz’s Law ¾ Lenz’s Law is for determining the direction of an induced current in a loop. ¾ Lenz’s Law : An induced current has a direction such that the magnetic field due to the current opposes the change in the magnetic flux ( ΔΦ ) that induces the current. - Opposition to Pole Movement An Induced Field is created which attempts to negate the applied field - Opposition to Flux Change The Induced current is such as to OPPOSE the CHANGE in applied field. 3 6-2.1 A stationary circuit in a time-varying magnetic field: Example 6-1: p231 A circular loop of N turns of conducting wire lies in the xy-plane with its center JG G at the origin of a magnetic field specified by B = a z Bo cos(π r / 2b) sin(ωt ) , where b is the radius of the loop and ω is the angular frequency. Find the emf induced in the loop. 6-2.3 A moving conductor in a static magnetic field: JG JG G JG F = qE + qu × B ¾ Total electromagnetic force: Electric force Motional electric field / Induced electric field: Lorentz’s force equation Magnetic force (see p170.) JG JG F m G JG Em = = u×B q Motional emf / flux-cutting emf: G 2 G JG G 2 JG m Vemf = V21 = ∫ E ⋅ dl = ∫ (u × B ) ⋅ dl 1 1 G JG G If moving conductor is a m Vemf = v∫ (u × B ) ⋅ dl part of a closed circuit C C -- Negative charge ++ Positive charge 4 6-2.3 A moving conductor in a static magnetic field: Example 6-2 (p236): A metal bar slides over a pair of conducting rails in a uniform magnetic field B = az B0 with a constant velocity u, as shown in Fig. 6-4. 1) Determine the open-circuit voltage Vo that appear across terminals 1 and 2 6-2.3 A moving conductor in a static magnetic field: 5 6-2.4 A moving circuit in a time-varying magnetic field: Vemf = V tr emf tr emf V +V m emf m Vemf JG ∂ B JG = −∫ ⋅dS ∂t S G JG G = v∫ (u × B) ⋅ dl C ¾ General form of Faraday’s law: Vemf = V tr emf +V m emf JG G JG G ∂ B JG = −∫ ⋅ d S + v∫ (u × B ) ⋅ dl ∂t S C (V) ¾ Another form of Faraday’s law: ' =− Vemf d JG JG dΦ B⋅dS = − ∫ dt S dt (V) 6-2.4 A moving circuit in a time-varying magnetic field: Example 6-5 p241 Angular speed: ω Linear speed: v = r ω α =ωt 6 6-2.4 A moving circuit in a time-varying magnetic field: Useful Applications A-C generator A-C motor 6-3 Maxwell’s Equations JG JG ∂B ∇× E = − ∂t JJG JG ∇× H = J JG ∇ ⋅ D = ρv JG ∇⋅B = 0 So far, we have following equations: Obvious, time-varying B gives rise to E. Question is: What about time-varying E, do you think it could induce B ? How ? Recall the equation of continuity: JG ∂ρ ∇⋅J = − v ∂t JG JJG JG ∂ D ∇× H = J + ∂t JG JJG JG ∂ρ JG ∂ D ∇ ⋅ (∇ × H ) = 0 = ∇ ⋅ J + v = ∇ ⋅ ( J + ) ∂t ∂t ¾ Introduce displacement current density Jd: (Time rate of change of D) Jd = JG ∂D ∂t 7 6-3 Maxwell’s Equations JG JJG JG ∂ D ∇× H = J + ∂t J : Conduction current density Jd : Displacement current density ¾ Displacement current density Jd: ¾ Displacement current Id: I d = ∫ J d ⋅ dS = ∫ JG ∂D ⋅ dS ∂t ¾ Displacement current is a result of time-varying electric field ¾ Without the displacement current, electromagnetic waves do not exist ¾ Displacement current flows through free space, or time-varying electric fields produce magnetic fields 6-3.1 Integral Form of Maxwell’s Equations Time-varying B gives rise to E. Time-varying E (D) gives rise to B (H). 8 6-3.1 Integral Form of Maxwell’s Equations Example 6-6 (p246): An a-c source of amplitude Vo and angular frequency ω, Vc = V0 sin(ωt ) is connected across a parallel-plate capacitor C, as shown in Fig. 6-7. (a) Verify that the displacement current in the capacitor is the same as the conduction current in the wires (b) Determine the magnetic field intensity at a distance r from the wire 9