10/26/17 Update Last week Completed Sources of Magnetic Fields (Chapter 30) This week A. B. Kaye, Ph.D. Associate Professor of Physics (Chapter 31) Next week 30 October 3 November 2017 Chapter 32 Induction Michael Faraday Induced Fields Magnetic fields may vary in time 22 Sept 1791 25 Aug 1867 Experiments conducted by Michael Faraday and Joseph Henry in 1831 showed that an emf can be induced in a circuit by a changing magnetic field British physicist and chemist Great experimental scientist The results of these experiments led to of Induction Contributions to early electricity include: An induced current is produced by a changing magnetic field There is an induced emf associated with the induced current A current can be produced without a battery present in the circuit Faraday s law of induction describes the induced emf Invention of motor, generator, and transformer Electromagnetic induction Laws of electrolysis Faraday cage 1
EMF Produced by a Changing Magnetic Field, 1 of Induction A loop of wire is connected to a sensitive ammeter When a magnet is moved toward the loop, the ammeter deflects The direction was arbitrarily chosen to be negative Loop of wire B field lines Induced current EMF Produced by a Changing Magnetic Field, 2 EMF Produced by a Changing Magnetic Field, 3 When the magnet is held stationary, there is no deflection of the ammeter Therefore, there is no induced current Even though the magnet is in the loop The magnet is moved away from the loop, and the ammeter deflects in the opposite direction 2
Induced Current Experiment, Summary EMF Produced by a Changing Magnetic Field, Summary The ammeter deflects when the magnet is moving toward or away from the loop The ammeter also deflects when the loop is moved toward or away from the magnet Therefore, the loop detects that the magnet is moving relative to it Faraday s Experiment Set Up of Induction Statements A primary coil is connected to a switch and a battery The wire is wrapped around an iron ring A secondary coil is also wrapped around the iron ring There is no battery present in the secondary coil and the secondary coil is not directly connected to the primary coil The emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit Remember F B is the magnetic flux through the circuit and is found by If the circuit consists of N loops, all of the same area, and if F B is the flux through one loop, an emf is induced in every loop, and Faraday s law becomes 3
Example Ways of Inducing an emf Assume a loop enclosing an area A lies in a uniform magnetic field; determine the magnetic flux and the induced emf. Find the magnetic flux through the loop and the induced emf. We can induce an emf in any one of four ways: 1. 2. 3. 4. Applications of GFCI Applications of Electric Guitar Pickup A GFCI (ground fault circuit interrupter) protects users of electrical appliances against electric shock The pickup coil of an electric guitar uses Faraday s law 4
Example: Inducing EMF in a coil Create a coil that consists of 200 turns of wire on a square of side d = 18 cm. A uniform magnetic field directed perpendicular to the plane of the coil is applied. If the field changes linearly from 0 T to 0.5 T in 0.8 seconds, what is the magnitude of the induced emf in the coil while the field is changing? Motional EMF Motional emf Sliding Conducting Bar A motional emf is the emf induced in a conductor moving through a constant magnetic field A conducting bar moving through a uniform field (left) and the equivalent circuit diagram (right). Assume the bar has zero resistance, and that the stationary part of the circuit has a resistance R. Find the induced current I in the bar. 5
Sliding Conducting Bar, Energy Considerations The applied force does work on the conducting bar This moves the charges through a magnetic field and establishes a current Example #2: Another Sliding Conducting Bar in a B Field The conducting bar in the figure slides on two frictionless, parallel rails in the presence of a uniform magnetic field directed into the page. The bar has a mass m, a length, and an initial velocity v to the right, starting at t = 0. Find the velocity of the bar as a function of time. The change in energy of the system during some time interval must be equal to the transfer of energy into the system by work The power input is equal to the rate at which energy is delivered to the resistor Example #3: Motional EMF in a Rotating Bar A conducting bar of length rotates with a constant angular speed w about a pivot point at one end. A uniform magnetic field B is directed perpendicular to the plane of rotation. Find the motional EMF induced between the ends of the rod. Lenz s Law 6
Lenz s Law Lenz Law, Example Faraday s law indicates that the induced emf and the change in flux have opposite algebraic signs; this has a physical interpretation that has come to be known as Lenz s law The conducting bar slides on the two fixed conducting rails. How does the magnetic flux change with the changing area? What direction does the induced current flow? How does the current change as the bar moves in different directions? Induced Current Directions Example A magnet is placed near a metal loop. Find the direction of the induced current in the loop when the magnet is pushed toward the loop (a and b). Find the direction of the induced current in the loop when the magnet is pulled away from the loop (c and d). Induced EMF and Electric Fields 7
Example: Electric Field induced by a Changing Magnetic Field in a Solenoid A long solenoid of radius R has n turns of wire per unit length and carries a time-varying current that varies sinusoidally as in which I max is the maximum current and w is the angular frequency of the alternating current source. Determine the magnitude of the induced electric field at a distance r > R from its long central axis. Then compute the magnitude of the induced electric field inside the solenoid, a distance r from the axis. Generators and Motors Generators Generators The generation of the vast majority of the energy transferred to homes and businesses by electrical transmission is based on Faraday s law We will now consider generators In the simplest form, the AC generator consists of a loop of wire rotated by some means in a magnetic field Electric generators take in energy by work and transfer it out by electrical transmission Use the active figure to adjust the speed of rotation and observe the effect on the emf generated 8
Rotating Loop Example: Induced emf in an AC Generator Assume a loop with N turns, all of the same area rotating in a magnetic field The flux through the loop at any time t is: The induced emf in the loop is This is sinusoidal, with The coil in an AC generator consists of 8 turns of wire, each of area A = 0.0900 m 2 and a total resistance of 12.0 W. The coil rotates in a 0.500-T magnetic field at a constant frequency of 60.0 Hz. What s the maximum induced emf? What s the maximum induced current when the output terminals are connected to a lowresistance conductor? DC Generators DC Generators, cont. The DC (direct current) generator has essentially the same components as the AC generator The main difference is that the contacts to the rotating loop are made using a split ring called a commutator Use the active figure to vary the speed of rotation and observe the effect on the emf generated In this configuration, the output voltage always has the same polarity and pulsates with time To obtain a steady DC current, commercial generators use many coils and commutators distributed so the pulses are out of phase 9
Example: Circular saw Saw Bonus You have a new circular saw that has a total resistance of 10 W and runs on normal household voltage (120 V). When the motor is running, the back emf is 70V. Find the current in the coil at the instant you pull the trigger to fire the saw. Find the current when the motor has reached maximum speed. You are happily sawing along until the blade gets jammed in the piece of wood you are cutting and the motor (along with the blade) can no longer turn. By what percentage does the power input to the motor increase when it is jammed? Eddy Currents Eddy Currents Circulating currents called eddy currents are induced in bulk pieces of metal moving through a magnetic field The eddy currents are in opposite directions as the plate enters or leaves the field Eddy currents are often undesirable because they represent a transformation of mechanical energy into internal energy 10
Eddy Currents, Example Eddy Currents, Final The magnetic field is directed into the page. The induced eddy current is counterclockwise as the plate enters the field. It is opposite when the plate leaves the field. The induced eddy currents produce a magnetic retarding force and the swinging plate eventually comes to rest. To reduce energy loses by the eddy currents, the conducting parts can Be built up in thin layers separated by a nonconducting material Have slots cut in the conducting plate Both prevent large current loops and increase the efficiency of the device Interesting Problem #41 Interesting Problems to Solve Suppose you drop an aluminum rod of length 1.0 m out of a window at a place where the horizontal magnetic field of the Earth is 2.0 x 10 5 T. The rod is oriented horizontally (at right angles to the magnetic field). What is the induced emf between the ends of the rod when its instantaneous downward velocity reaches 12 m/s? 11
Interesting Problem #42 Interesting Problem #43 To measure the velocity of the flow of blood in the mesenteric artery in the abdomen of a dog, a researcher places the animal in a magnetic field of 3.0 x 10 2 T, inserts small electrodes through the wall of the artery from each side, and measures the emf with a voltmeter. The inner diameter of the artery is 0.30 cm and the measured emf is 1.8 x 10 6 V. What is the velocity of blood flow? A rectangular coil of 150 loops forming a closed circuit measures 0.20 m x 0.10 m. The resistance of the coil is 5.0 W. The coil is placed between the poles of an electromagnet, face on to a magnetic field B. Suppose that when we switch the electromagnet off, the strength of the magnetic field decreases at the rate of 20 teslas per second. What is the induced emf in the coil? What is the direction of the induced current? Interesting Problem #44 An electromagnetic generator consist of a rectangular coil of N loops of wire that rotates about an axis perpendicular to a constant magnetic field B. Sliding contacts connect the coil to an external circuit. What emf does the coil deliver to the external circuit? The coil has an area A and rotates with an angular frequency w. Interesting Problem #45 The circular pole faces of the electromagnet shown in the figure have a radius R = 0.10 m. The magnetic field is suddenly shut off, decreasing (briefly) at a rate of 2.0 x 10 4 T/s. What is the induced electric field for r R? For r R? What is the numerical value of this field at r = R? Hint: Assume that there is only a magnetic field between the pole faces. 12
Key Review Points Faraday s law of induction relates the induced emf in a loop is directly proportional to the time rate of change of the magnetic flux through the loop: Key Review Points where is the magnetic flux through the loop. When a conducting bar of length moves at a velocity v through a magnetic field B, in which B is perpendicular to the bar and to v, the motional emf induced in the bar is: A general form of Faraday s law of induction is: in which E is the non-conservative electric field produced by the changing magnetic field Key Review Points Lenz s law states that the induced current and induced emf in a conductor are in such a direction as to set up a magnetic field that opposes the change that produced them. For a flat surface in a uniform field, the magnetic flux is: For a coil of N turns, Faraday s law of induction says that 13