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Transcript
21.2 Faraday’s Law of Induction and Lenz’s Law Magnetic Flux - similar to electric flux Imagine a coil of wire (area, A) in a magnetic field Zero flux Faraday’s Law of Induction • Changing Magnetic flux induces an EMF • Lenz’s Law • Induced EMF in a Moving Conductor; Eddy Currents • Faraday generalized: Changing Magnetic Field induces an Electric Field • Electric Generators • Transformers • Self Inductance and Inductors • Energy Stored in a Magnetic Field • LR Circuit 21.2 Faraday’s Law of Induction and Lenz’s Law B θ Φ Β = BAcos45 θ Φ Β = BA Max flux Units: weber (Wb), 1 Wb = 1 T·m2 (21-1) number of lines passing through the coil ∝ Φ B through the coil What is the total magnetic flux through any closed surface? 21.2 Faraday’s Law of Induction; Lenz’s Law Experiments by Faraday and others showed that… Minus sign in Faraday’s Law tells us that the induced emf opposes the original change, ie. if ΦB changes though a coil of wire, an emf is induced and The current produced by an induced emf moves in a direction such that its magnetic field opposes the original change in flux. [Lenz’s Law] the induced emf is proportional to the rate of change of magnetic flux, φB through the coil. BIND Number of loops Examples IIND [Faraday’s Law] IIND IIND S N No IIND N Induced Emf Rate of change of magnetic flux through coil Pull the loop out of the South magnetic pole North magnetic pole Magnetic field increases magnetic field which moving toward loop moving toward loop in the plane of the page points out of the page into the page into the page 21.2 Faraday’s Law of Induction; Lenz’s Law 21.2 Faraday’s Law of Induction; Lenz’s Law Problem Solving using Lenz’s Law 1. Magnetic flux, ΦB: The magnetic flux will also change if the area of the loop changes… What ‘direction’? Is Φ B increasing, decreasing, or constant? 2. Induced magnetic field tries to keep the flux constant. If ΦB is increasing, the induced magnetic field, BIND, points in the opposite direction. .. or if the angle between the loop and the field changes. Induced current If ΦB is decreasing, the induced magnetic field, BIND, points in the same direction. 3. Direction of the induced current can be determined using RHR-1. 4. Remember that the external field and the field due to the induced current are different. 1 Question... Question... A very long straight wire carries a steady current down. A loop of wire is moved towards the current. What is the magnetic flux through the wire loop ? Here is another way to induce an emf in a conductor… A conducting rod moves to the right with velocity, v, perpendicular to a magnetic field, B. What is the direction of the induced current in the wire loop? IIND BIND 21.3 (Motional) EMF in a Moving Conductor (21-3 and Ex 21-8) l I v Area, A1 A) Counter clockwise B) Clockwise C) There is no induced current 21.3 A “Slide generator" Wire loop What happens? An emf is induced in the rod of magnitude: A) BA1 B) BA2 C) B(A1 - A2) ε = Blv Fe A 737 is flying at 200 m/s through a region where the Earth’s magnetic field is 5 x 10-5 T and pointing DOWN. A F Now there is a continuous path for the electrons and the induced emf causes a current to flow. Because the conductor is moving this is sometimes referred to as motional emf Question... (21-3 and Ex 21-8) Rest the moving rod on a U-shaped conductor… (21-3) How much potential difference is created across the 35 m wingspan ? I v B But … the induced current interacts with the magnetic field, producing a drag force (F=ILB) that resists the motion of the rod. F F Note, is different from the upward force e, on the electrons that produced the initial current. 21.6 Eddy Currents Induced currents (Eddy currents) can flow in any shaped conductor. Drag forces associated with eddy currents can dramatically slow a conductor moving into or out of a magnetic field. Drag force resists motion of wheel 1. 2. 3. Zero because there is no closed circuit for a current to flow. 0.35 V with wing A positively charged 0.35 V with wing A negatively charged 21.5 Electric Generators Rotating metal wheel Eddy currents The axle is rotated by an external force e.g. falling water or steam. As it turns a sinusoidal emf, is induced: Generator eqn. area of each turn B points into page here Note, Faraday’s law can be generalized to: A changing magnetic field induces an electric field. - regardless of whether there are conductors around or not. Axle Mechanical A generator transforms MECHANICAL energy into ELECTRICAL energy. (21-5) number of turns on coil Peak voltage, B Electrical Output voltage of generator Angular frequency (radians/s) ω = 2πf, f = frequency (Hz) = When sine function = +1 When the generator is connected to a circuit, an ac current flows. This creates a drag force (counter torque) which resists the motion of the motor. 2 21.5 Electric Generators 21.7 Transformers and Transmission of Power A generator has a coil of wire rotating in a magnetic field. The rotation rate INCREASES. Up till now we have been changing the magnetic flux through a coil and inducing a current. What happens to the maximum output voltage of the generator? It.. 1. increases 2. decreases 3. varies sinusoidally 4. stays the same If instead we pass a changing I through a coil, the magnetic flux it produces also changes. Note, Mechanical Energy - pass that changing flux through a 2nd coil, an emf can be induced in the 2nd coil. Electrical Energy This is the basis of a transformer An electric generator can be used as a motor and vice versa. 21.7 Transformers and Transmission of Power (changing) 21.9 Self Inductance and Inductors A Transformer is a device for increasing or decreasing ac voltage. What is an inductor? • primary and secondary coil: interwoven or linked by an iron core. - its just a coil of wire • Nearly all magnetic flux produced by primary coil passes through secondary coil. When an ac voltage is applied to the primary coil an ac voltage of the same frequency is induced in the secondary coil. (21-6) Can show: [rms or peak values] (21-6) transformer requires ac STEP-UP transformer increases the voltage (NS > NP) STEP-DOWN transformer decreases the voltage (NP > NS) Transformers play an important role in the transmission of electricity 21.9 Self Inductance (21-9) - + Induced emf - + Induced emf If I changes in a single coil, then φB changes and an emf is induced in that same coil. This is known as self inductance 21.9 Self Inductance The induced emf is proportional to the rate of change of the current and it opposes the change (Lenz’s Law): L = self-inductance Units: henry, H. 1 H = 1 V·s/A = 1 Ω·s. When this coil of wire is put in a circuit it has interesting effects because of Faraday’s Law Induced emf tries to prevent the current from increasing as it enters the inductor at A Self inductance, L depends on the size and shape of the coil and the presence of an iron core (which increases L). L can be calculated for an empty coil using this formula: N2A L = µ0 l A N loops l Induced emf tries to prevent the current from decreasing. An Inductor resists any change in the current. 3 * a pure inductor + resistor in series could represent a real coil of wire or an electromagnet Question… 21.11 LR Circuit The current through a 220 mH inductor increases from 0.4 to 1.6 Amperes in 640 ms. What happens when a pure inductor and resistor* are connected to a battery and switch? Switch ON (position 1): I increasing 220 mH What is the induced emf across the inductor? a) b) c) d) e) -0.13 V -0.41 V 0V +2.7V +8.4 V - + Initially: I increases rapidly A large emf develops across L to oppose the increasing current. Most of the voltage drop is across the inductor 2 - + + - 1 With time: I increases less rapidly. Eventually: All the voltage drop is across R. I Current flowing in the circuit Induced emf prevents current rising immediately to max value. 21.11 LR Circuit 21.10 Energy Stored in a Magnetic Field Switching OFF (position 2): Current in the circuit We saw in section 17-9, that energy can be stored in an electric field ( uE = 12 ε0 E2 ). Energy can also be stored in a magnetic field, for example in an inductor or solenoid. 2 The energy density of the magnetic field is given by: When the battery is removed from the circuit the current gradually dies away PE per unit volume Units: J/m3 = 1 2 (21-10) LR circuit similar to RC circuit but time constant is inversely proportional to R. where What is the energy density of the Earth’s magnetic field? Induced emf across inductor prevents I dropping immediately to zero Summary of Chapter 21 • Magnetic flux: • Changing magnetic flux induces an emf: • Induced emf opposes the original flux change. • Changing magnetic field induces an electric field • Electric generator converts mechanical energy to electrical energy. Changing magnetic flux in the coils induce an emf, which drives an alternating current through an external circuit. • Self inductance: •Transformer changes the magnitude of an ac voltage: • Energy density stored in magnetic field: = 1 2 4