SHS ` General Physics 2 Quarter 4 – Week 1 Module 1 – Magnetic Induction and Faraday’s Law i General Physics 2 Grade 12 Quarter 4: Week 1 - Module 1 – Magnetic Induction and Faraday’s Law First Edition, 2021 Copyright © 2021 La Union Schools Division Region I All rights reserved. No part of this module may be reproduced in any form without written permission from the copyright owners. Development Team of the Module Author: DARRYL G. BERSALONA, SST-I Editor: SDO La Union, Learning Resource Quality Assurance Team Illustrator: Ernesto F. Ramos Jr., P II Management Team: Atty. Donato D. Balderas, Jr. Schools Division Superintendent Vivian Luz S. Pagatpatan, Ph.D Assistant Schools Division Superintendent German E. Flora, Ph.D, CID Chief Virgilio C. Boado, Ph.D, EPS in Charge of LRMS Rominel S. Sobremonte, Ed.D, EPS in Charge of Science Michael Jason D. Morales, PDO II Claire P. Toluyen, Librarian II ii Target Almost every modern device or machine, from a computer to a washing machine to a power drill, has electric circuits at its heart. We learned from the previous lessons that an electromotive force (emf) is required for a current to flow in a circuit; and we almost always took the source of emf to be a battery. In the preceding discussion, you have learned about Biot-Savart Law and Ampere’s Law. In this section, we would be going to deal with magnetic induction and Faraday’s Law. After going through this module, you are expected to: 1. Identify the factors that affect the magnitude of the induced emf and the magnitude 2. 3. and direction of the induced current (Faraday’s Law) (STEM_GP12EMIVa-1); Compare and contrast electrostatic electric field and non-electrostatic/induced electric field (STEM_GP12EMIVa-3); Calculate the induced emf in a closed loop due to a time-varying magnetic flux using Faraday’s Law (STEM_GP12EMIVa-4). Before going on, check how much you know about this topic. Answer the pretest on the next page in a separate sheet of paper. 2 Jumpstart For you to understand the lesson well, do the following activities. Have fun and good luck! Direction: Write the letter of the term or phrase that best completes the statement or answers the question. 1. Electromagnetic induction is change in ___________________. A. surface area B. magnetic flux C. magnetic poles D. electric field 2. What would happen if I move a bar magnet in and out of a coil of copper wire? A. Electric current would disappear B. It would produce a gravitational field C. Electric current will flow through the wire D. The magnet would explode 3. ___________ law says that the Induced current is proportional to the change of magnetic flux. A. Lenz's B. Ampere's C. Biot-Savart’s D. Faraday's 4. Where is the strongest attraction force of the magnet? A. at the poles B. above the magnet C. in the middle D. below the magnet 5. What type of current is produced by a battery? A. parallel current B. direct current C. alternating current D. potential current 6. What creates a magnetic field? A. charged particles that do not move C. moving electric charges B. gravity D. an isolated magnetic pole 7. Voltage can be induced in a wire by _______________. A. moving the wire near a magnet B. moving a magnet near the wire C. changing the current in a nearby wire D. all of these 8. A magnet can move in a coil of wire to produce electricity in which system? A. Generator B. Magnet C. Motor D. Transformer 9. Magnetic Field lines around a bar magnet ____________. A. are perpendicular the magnet B. cross back and forth over each other C. spread out from north pole and curve to south D. are perfectly straight 10. How do Maglev trains go up to 311 MPH? A. a train is pulled by a big magnet at the end of the tracks B. magnetized coils repel magnets on the train which moves it C. electric motors push the train and cause it to levitate D. a generator creates electricity which fuels the train 3 Discover In the previous lesson we almost always took the source of emf to be a battery. But for the vast majority of electric devices that are used in industry and in the home (including any device that you plug into a wall socket), the source of emf is not a battery but an electric generating station. Such a station produces electric energy by converting other forms of energy: gravitational potential energy at a hydroelectric plant, chemical energy in a coal- or oil-fired plant, nuclear energy at a nuclear plant. But how is this energy conversion done? The answer is a phenomenon known as electromagnetic induction: If the magnetic flux through a circuit changes, an emf and a current are induced in the circuit. In a power-generating station, magnets move relative to coils of wire to produce a changing magnetic flux in the coils and hence an emf. Other key components of electric power systems, such as transformers, also depend on magnetically induced emfs. The central principle of electromagnetic induction, and the keystone of this chapter, is Faraday’s law. This law relates induced emf to changing magnetic flux in any loop, including a closed circuit. Induction Experiments During the 1830s, several pioneering experiments with magnetically induced emf were carried out in England by Michael Faraday and in the United States by Joseph Henry (1797–1878), later the first director of the Smithsonian Institution. Figure 1 shows several examples. In Fig.1a, a coil of wire is connected to a galvanometer. When the nearby magnet is stationary, the meter shows no current. This isn’t surprising; there is no source of emf in the circuit. But when we move the magnet either toward or away from the coil, the meter shows current in the circuit, but only while the magnet is moving (Fig.1b). If we keep the magnet stationary and move the coil, we again detect a current during the motion. We call this an induced current, and the corresponding emf required to cause this current is called an induced emf. In Fig.1c we replace the magnet with a second coil connected to a battery. When the second coil is stationary, there is no current in the first coil. However, when we move the second coil toward or away from the first or move the first toward or away from the second, there is current in the first coil, but again only while one coil is moving relative to the other. Finally, using the two-coil setup in Fig.1d, we keep both coils stationary and vary the current in the second coil, either by opening and closing the switch or by changing the resistance of the second coil with the switch closed (perhaps by changing the second coil’s temperature). We find that as we open or close the switch, there is a momentary current pulse in the first circuit. When we vary the resistance (and thus the current) in the second coil, there is an induced current in the first circuit, but only while the current in the second circuit is changing. 4 Figure 1. Demonstrating the phenomenon of induced current. Photo credit: University Physics with Modern Physics 13th Edition The common element in all these experiments is changing magnetic flux ๐ฝ๐ฉ through the coil connected to the galvanometer. In each case the flux changes either because the magnetic field changes with time or because the coil is moving through a nonuniform magnetic field. Faraday’s law of induction, the subject of the next section, states that in all of these situations the induced emf is proportional to the rate of change of magnetic flux Φ๐ต through the coil. The direction of the induced emf depends on whether the flux is increasing or decreasing. If the flux is constant, there is no induced emf. Induced emfs are not mere laboratory curiosities but have a tremendous number of practical applications. If you are reading these words indoors, you are making use of induced emfs right now! At the power plant that supplies your neighborhood, an electric generator produces an emf by varying the magnetic flux through coils of wire. This emf supplies the voltage between the terminals of the wall sockets in your home, and this voltage supplies the power to your reading lamp. Indeed, any appliance that you plug into a wall socket makes use of induced emfs. Magnetic Flux The magnetic flux (often denoted Φ or ΦB) through a surface is the component of the magnetic field passing through that surface. The magnetic flux through some surface is proportional to the number of field lines passing through that surface. The magnetic flux passing through a surface of vector area A is โ • ๐ด = ๐ต ๐ด ๐๐๐ ๐ Φ๐ต = ๐ต where B is the magnitude of the magnetic field (having the unit of Tesla, T), A is the area of the surface, and θ is the angle between the magnetic field lines and the normal (perpendicular) to A. From the definition of magnetic flux, we see that its SI unit is the tesla– square meter (๐ โ ๐2 ), which is called the weber (abbreviated Wb): 1 ๐ค๐๐๐๐ = 1 ๐๐ = 1 ๐ โ ๐2 . 5 Faraday’s Law The common element in all induction effects is changing magnetic flux through a circuit. Before stating the simple physical law that summarizes all of the kinds of experiments described in Induction Experiments part, let’s first review the concept of magnetic flux Φ๐ต . For an infinitesimal area element ๐๐ด in a magnetic โ (Fig. 2), the magnetic flux ๐Φ๐ต through the field ๐ต area is โ • ๐๐ด = ๐ต⊥ ๐๐ด = ๐ต ๐๐ด ๐๐๐ ๐ ๐Φ๐ต = ๐ต ๐ธ๐๐ข๐๐ก๐๐๐ 1 Figure 2. Calculating the magnetic flux through an area element. Photo credit: University Physics with Modern Physics (13th Edition) โ perpendicular to the surface of the area element where ๐ต⊥ is the component of ๐ต โ and ๐ is the angle between ๐ต and ๐๐ด. Figure 3. Calculating the flux of a uniform magnetic field through a flat area. Photo credit: University Physics with Modern Physics (13th Edition) The total magnetic flux Φ๐ต through a finite area is the integral of this expression over the area: โ • ๐ด = ๐ต๐ด ๐๐๐ ๐ Φ๐ต = ๐ต ๐ธ๐. 2 Faraday’s law of induction states: The induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. In symbols, Faraday’s law is ๐ ๐ฝ๐ฉ (๐น๐๐๐๐๐๐ฆ ′ ๐ ๐ฟ๐๐ค ๐๐ ๐ผ๐๐๐ข๐๐ก๐๐๐) ๐บ=− ๐ ๐ 6 ๐ธ๐. 3 ΔΦ (We can use this formula from other books and references: ๐ = − ๐ต) Δ๐ก As you will see the formula, the induced emf tends to oppose the flux change, so Faraday’s law is formally written as seen above with the minus sign indicating that opposition. We often neglect the minus sign, seeking only the magnitude of the induced emf. (The minus sign is there to remind us in which direction the induced emf acts. Experiments shows that a current produced by an induced emf moves in a direction so that its magnetic field opposes the original change in flux. This is known as Lenz’s Law.) If we change the magnetic flux through a coil of N turns, an induced emf appears in every turn and the total emf induced in the coil is the sum of these individual induced emfs. If the coil is tightly wound (closely packed), so that the same magnetic flux passes through all the turns, the total emf induced in the coil is ๐ = −๐ ๐Φ๐ต ๐๐ก (๐๐๐๐ ๐๐ ๐ ๐ก๐ข๐๐๐ ) ๐ธ๐. 4 Here are the general means by which we can change the magnetic flux through a coil: 1. Change the magnitude B of the magnetic field within the coil. 2. Change either the total area of the coil or the portion of that area that lies within the magnetic field (for example, by expanding the coil or sliding it into or out of the field). โ and the plane 3. Change the angle between the direction of the magnetic field ๐ต โ is first of the coil (for example, by rotating the coil so that field ๐ต perpendicular to the plane of the coil and then is along that plane). Example 1: Emf and current induced in a loop The magnetic field between the poles of the electromagnet in Fig. 4 is uniform at any time, but its magnitude is increasing at the rate of 0.020 T/s. The area of the conducting loop in the field is 120 cm2, and the total circuit resistance, including the meter, is 5.0 Ω. Figure 4. A stationary conducting loop in an increasing magnetic field. Photo credit: University Physics with Modern Physics (13th Edition) 7 (a) Find the induced emf and the induced current in the circuit. (b) If the loop is replaced by one made of an insulator, what effect does this have on the induced emf and induced current? Solution IDENTIFY and SET UP: The magnetic flux Φ๐ต through the loop changes as the magnetic field changes. Hence there will be an induced emf ๐ and an induced current I in the loop. We calculate Φ๐ต using Eq. โ (Φ๐ต = ๐ต • ๐ด = ๐ต๐ด ๐๐๐ ๐), then find ๐ using Faraday’s law. Finally, we calculate I using ๐ = ๐ผ๐ where R is the total resistance of the circuit that includes the loop. EXECUTE: (a) The area ๐ด vector for the loop is perpendicular to the plane of โ are parallel, and because the loop; we take ๐ด to be vertically upward. Then ๐ด and ๐ต โ is uniform the magnetic flux through the loop is Φ๐ต = ๐ต โ • ๐ด = ๐ต๐ด ๐๐๐ ๐ = ๐ต๐ด. The ๐ต 2 area ๐ด = 0.012๐ is constant, so the rate of change of magnetic flux is ๐Φ๐ต ๐(๐ต๐ด) ๐๐ต ๐ = = ๐ด = (0.020 ) (0.012๐2 ) ๐๐ก ๐๐ก ๐๐ก ๐ = 2.4 ๐ฅ 10−4 ๐ = 0.24 ๐๐ This, apart from a sign that we haven’t discussed yet, is the induced emf ๐. The corresponding induced current is ๐ 2.4 ๐ฅ 10−4 ๐ ๐ผ= = = 4.8 ๐ฅ 10−5 ๐ด = 0.048 ๐๐ด ๐ 5.0 Ω (b) By changing to an insulating loop, we’ve made the resistance of the loop ๐Φ very high. Faraday’s law, Eq. (๐ = − ๐ต ), does not involve the resistance of the ๐๐ก circuit in any way, so the induced emf does not change. But the current will be ๐ smaller, as given by the equation ๐ผ = . If the loop is made of a perfect insulator ๐ with infinite resistance, the induced current is zero. This situation is analogous to an isolated battery whose terminals aren’t connected to anything: An emf is present, but no current flows. EVALUATE: Let’s verify unit consistency in this calculation. One way to do โโ implies that the units this is to note that the magnetic-force relationship ๐น = ๐๐ฃ ๐ฅ ๐ฉ โโ are the units of force divided by the units of (charge times velocity): 1๐ = of ๐ฉ (1๐)/1๐ถ โ ๐/๐ ). The units of magnetic flux are then (1๐)(1๐2 ) = 1๐ โ ๐ โ ๐/๐ถ, and the ๐ ๐ฝ ๐Φ๐ต rate of change of magnetic flux is 1๐ โ = 1 = 1๐. Thus the unit of is the volt, ๐Φ − ๐ต ). ๐๐ก ๐ถ ๐ถ ๐๐ก as required by Eq. (๐ = Also recall that the unit of magnetic flux is the weber 2 (Wb): 1 ๐ โ ๐ = 1๐๐, so 1 ๐ = 1๐๐/๐ . Direction of Induced emf We can find the direction of an induced emf or current by using Eq. (๐ = − together with some simple sign rules. Here’s the procedure: 1. Define a positive direction for the vector area ๐ด. 8 ๐Φ๐ต ๐๐ก ) โ , determine the sign of the From the directions of ๐ด and the magnetic field ๐ต ๐Φ๐ต magnetic flux Φ๐ต and its rate of change . Figure 5 shows several examples. ๐๐ก 3. Determine the sign of the induced emf or current. If the flux is increasing, so ๐Φ๐ต is positive, then the induced emf or current is negative; if the flux is 2. ๐๐ก ๐Φ decreasing, ๐ต is negative and the induced emf or current is positive. ๐๐ก 4. Finally, determine the direction of the induced emf or current using your right hand. Curl the fingers of your right hand around the vector ๐ด, with your right thumb in the direction of ๐ด. If the induced emf or current in the circuit is positive, it is in the same direction as your curled fingers; if the induced emf or current is negative, it is in the opposite direction. In Example 1, in which ๐ด is upward, a positive ๐ would be directed counterโ are upward in clockwise around the loop, as seen from the example. Both ๐ด and ๐ต ๐Φ this example, so Φ๐ต is positive; the magnitude B is increasing, so ๐ต is positive. ๐๐ก Hence by Eq. (3), in Example 1 is negative. Its actual direction is thus clockwise around the loop, as seen from above. If the loop in Fig. 4 is a conductor, an induced current results from this emf; this current is also clockwise, as Fig. 4 shows. This induced current produces an additional magnetic field through the loop, and the right-hand rule shows that this field is opposite in direction to the increasing field produced by the electromagnet. This is an example of a general rule called Lenz’s law, which says that any induction effect tends to oppose the change that caused it; in this case the change is the increase in the flux of the electromagnet’s field through the loop. Figure 5. The magnetic flux is becoming (a) more positive, (b) less positive, (c) more negative, and (d) less negative. Therefore Φ๐ต is increasing in (a) and (d) and decreasing in (b) and (c). In (a) and (d) the emfs are negative (they are opposite to the direction of the curled fingers of your right hand when your right thumb points along ๐ด). In (b) and (c) the emfs are positive (in the same direction as the curled fingers). Photo credit: University Physics with Modern Physics (13th Edition) Lenz’s Law The minus sign in Faraday’s law of induction is very important. The minus means that the EMF creates a current I and magnetic field B that oppose the change in flux ΔΦ this is known as Lenz’ law. The direction (given by the minus 9 sign) of the EMF is so important that it is called Lenz’ law after the Russian Heinrich Lenz (1804–1865), who, like Faraday and Henry, independently investigated aspects of induction. Faraday was aware of the direction, but Lenz stated it, so he is credited for its discovery. Figure 6. Magnet subjected to motion into a coil Photo credit: Lumen Learning Lenz’s Law: (a) When this bar magnet is thrust into the coil, the strength of the magnetic field increases in the coil. The current induced in the coil creates another field, in the opposite direction of the bar magnets to oppose the increase. This is one aspect of Lenz’s law – induction opposes any change in flux. (b) and (c) are two other situations. Verify for yourself that the direction of the induced B coil shown indeed opposes the change in flux and that the current direction shown is consistent with the right-hand rule. Energy Conservation Lenz’ law is a manifestation of the conservation of energy. The induced EMF produces a current that opposes the change in flux, because a change in flux means a change in energy. Energy can enter or leave, but not instantaneously. Lenz’ law is a consequence. As the change begins, the law says induction opposes and, thus, slows the change. In fact, if the induced EMF were in the same direction as the change in flux, there would be a positive feedback that would give us free energy from no apparent source—conservation of energy would be violated. With so much things to consider in this lesson, here are the important things to consider, or in other words, Electromagnetic Induction is the process of using magnetic fields to produce voltage, and in a closed circuit, a current. So how much voltage (emf) can be induced into the coil using just magnetism? Well, this is determined by the following 3 different factors. 1) Increasing the number of turns of wire in the coil – By increasing the amount of individual conductors cutting through the magnetic field, the amount of induced emf produced will be the sum of all the individual loops of the coil, so if there are 20 turns in the coil there will be 20 times more induced emf than in one piece of wire. 2) Increasing the speed of the relative motion between the coil and the magnet – If the same coil of wire passed through the same magnetic field but its 10 speed or velocity is increased, the wire will cut the lines of flux at a faster rate so more induced emf would be produced. 3) Increasing the strength of the magnetic field – If the same coil of wire is moved at the same speed through a stronger magnetic field, there will be more emf produced because there are more lines of force to cut. Applications of Faraday’s Law Following are the fields where Faraday’s law finds applications: 1. Electrical equipment like transformers works on the basis of Faraday’s law. 2. Induction cooker works on the basis of mutual induction which is the principle of Faraday’s law. 3. By inducing an electromotive force into an electromagnetic flowmeter, the velocity of the fluids is recorded. 4. Electric guitar and electric violin are the musical instruments that find an application of Faraday’s law. 5. Maxwell’s equation is based on the converse of Faraday’s laws which states that change in the magnetic field brings a change in the electric field. Example 2: Magnitude and direction of an induced emf A 500-loop circular wire coil with radius 4.00 cm is placed between the poles of a large electromagnet. The magnetic field is uniform and makes an angle of 60° with the plane of the coil; it decreases at 0.200 T/s. What are the magnitude and direction of the induced emf? Figure 7. Circular wire coil subjected to a magnetic field Photo credit: University Physics with Modern Physics (13th Edition) SOLUTION IDENTIFY and SET UP: Our target variable is the emf induced by a varying magnetic flux through the coil. The flux varies because the magnetic field decreases in amplitude. We choose the area vector ๐ด to be in the direction shown in Fig. 7. With this choice, the geometry is similar to that of Fig. 5b. That figure will help us determine the direction of the induced emf. EXECUTE: The magnetic field is uniform over the loop, so we can calculate the flux using Eq. (Φ๐ต = ๐ต๐ด cos ๐) where ๐ = 30°. In this expression, the only ๐Φ ๐๐ต quantity that changes with time is the magnitude B of the field, so ๐ต = ( ) ๐ด ๐๐๐ ๐. ๐๐ก 11 ๐๐ก *CAUTION: Remember how ๐ is defined You may have been tempted to say that ๐ = 60° in this problem. If so, โ , not the angle between ๐ต โ and the remember that ๐ is the angle between ๐ด and ๐ต plane of the loop. From the faraday’s law equation, the induced emf in the coil of N=500 turns is ๐Φ๐ต ๐B ๐ = −๐ =๐ ๐ด ๐๐๐ ๐ ๐๐ก ๐๐ก ๐ = 500(−0.200 )๐(0.0400 ๐)2 (cos 30°) = ๐. ๐๐๐ ๐ฝ ๐ The positive answer means that when you point your right thumb in the direction of the area vector ๐ด (30° below the magnetic field in Fig. 7), the positive direction for ๐ is in the direction of the curled fingers of your right hand. If you viewed the coil from the left in Fig.7 and looked in the direction of ๐ด the emf would be clockwise. EVALUATE: If the ends of the wire are connected, the direction of current in the coil is in the same direction as the emf—that is, clockwise as seen from the left side of the coil. A clockwise current increases the magnetic flux through the coil, and therefore tends to oppose the decrease in total flux. This is an example of Lenz’s law. Induced Electric Field So, we seen that a changing magnetic flux result in an induced emf. Faraday’s law is ๐Φ๐ต ๐=− ๐๐ก Faraday’s law states that induced emf is the negative rate of change of magnetic flux. If magnetic flux changes over time, then there has to be an induced emf. And if a closed conducting path is available, then charges can move and produce a current. We sometimes refer to this current as an induced current because of how it results from an induced emf. The question now is: what is making the charges move? On electrostatics, we learned that a charge placed in an electric field is pushed or pulled by that field. So, if the charge is free to move along a wire, then we have a current. Then, we learned that a moving charge placed in a magnetic field experience a force exerted by the magnetic field itself. Which of these two forces, the one exerted by the electric field or the one by the magnetic field, is responsible for the induced current? Consider a region in space where there is a time-varying magnetic field (see Fig. 8). The magnetic field is going into the paper and a conducting loop is placed perpendicular to the field lines. Let us suppose that this magnetic field is increasing. There will be an induced magnetic field that will oppose this increase. So, the induced magnetic field will be pointing out of the paper. Using the RH grip rule, the induced current is counterclockwise. 12 Figure 8. This conducting loop is placed in a changing magnetic field in space Photo credit: General Physics 2 (Santisteban-Cook 2018) We know there is an induced emf because Faraday’s law tells us so. But where does the emf come from? What makes the charges move? It cannot be the magnetic field because the magnetic field only pushes on charges that are moving already. If it is not magnetic field, it has to be the electric field. This gives us a new way of thinking about Faraday’s law. The changing magnetic field creates an electric field on the loop, in the same direction as the current. Imagine that you are walking around the loop. At every step you take (no matter how small0, you will see an electric field directed tangent to the loop, and this electric field exerts an electric force on the charges all along the loop (see Fig. 9). The induced electric field is different from the electric field we learned before. All the electric fields that are discussed before Faraday’s law are electrostatic fields. This induced electric field is a non-electrostatic field. The equation ๐น ๐ธ= ๐ is still true, but there are some differences. Electrostatic fields start at positive charges and end at negative charges. They have a beginning and an end. Induced electric fields do not start or end at a charge; they just go around in loops. They are also not fixed in time; these fields are always changing. They appear when a changing magnetic flux is present and disappear when there is no change. Another difference is in how we get the potential. In an electrostatic field, we can get the potential using the dot product ๐ธ โ ๐๐, and just like work, if you start and end at the same point, the result is zero. If you travel around a closed loop in an electrostatic field, the potential difference is zero. This is because an electrostatic field is a conservative field, just like the Earth’s gravitational field. If you go around the loop in either Fig. 8 or Fig. 9, you are doing work all around the loop, and when you reach the point where you started on the loop, the work done is not zero. If you go around the loop and calculate all the ๐ธ โ ๐๐′๐ and then get Figure 9. There is an induced electric field all along the loop as a result of the changing magnetic flux Photo credit: General Physics 2 (Santisteban-Cook 2018) 13 the sum over the entire loop, you do not get zero. You get result. This is the emf. ๐Φ๐ต ๐๐ก which is an important So, this is what Faraday’s law now look like: the left side is the emf, but more importantly, it is the summation of all the ๐ธ โ ๐๐′๐ around a closed loop: ๐Φ๐ต โฎ ๐ธโ โ โโโ ๐๐ = − ๐๐ก And because this equation is not equal to zero, we know that an induced electric field is nonconservative, unlike an electrostatic field. One more thing needs to be pointed out about the induced electric field: the conducting loop does not even have to be in the given space for the field to exist. The only thing it requires to exist is varying magnetic field. The electric field will exist with or without free electrons moving around in a loop because the field is a property of the space, not a property of the charges. Let’s take a look at a more detailed explanation. The fact that emfs are induced in circuits implies that work is being done on the conduction electrons in the wires. What can possibly be the source of this work? We know that it’s neither a battery nor a magnetic field, for a battery does not have to be present in a circuit where current is induced, and magnetic fields never do work on moving charges. The answer is that the source of the work is an electric field ๐ธโ that is induced in the wires. The work done by ๐ธโ in moving a unit charge completely around a circuit is the induced emf ε; that is, ๐ = โฎ ๐ธโ โ โโโ ๐๐, where โฎ represents the line integral around the circuit. Faraday’s law can be written in terms of the induced electric field as ๐Φ๐ต โฎ ๐ธโ โ โโโ ๐๐ = − ๐๐ก There is an important distinction between the electric field induced by a changing magnetic field and the electrostatic field produced by a fixed charge distribution. Specifically, the induced electric field is nonconservative because it does net work in moving a charge over a closed path, whereas the electrostatic field is conservative and does no net work over a closed path. Hence, electric potential can be associated with the electrostatic field, but not with the induced field. The following equations represent the distinction between the two types of electric field: โโโ ≠ 0 โฎ ๐ธโ โ ๐๐ โฎ ๐ธโ โ โโโ ๐๐ = 0 ๐ผ๐๐๐ข๐๐๐ ๐ธ๐๐๐๐ก๐๐๐ ๐น๐๐๐๐ ๐ธ๐๐๐๐ก๐๐๐ ๐ก๐๐ก๐๐ ๐ธ๐๐๐๐ก๐๐๐ ๐น๐๐๐๐๐ Our results can be summarized by combining these equations: ๐Φ๐ โโโ = − ๐ = โฎ ๐ธโ โ ๐๐ ๐๐ก 14 Explore Direction: Read and analyze the following problems . Answer them properly. Problem 1. Calculate the flux A square loop of wire 10.0 cm on a side is in a 1.25 T magnetic field B. What are the maximum and minimum values of flux that can pass through the loop? Approach The flux is given by Eq. 2 (Φ๐ต = ๐ต๐ด cos ๐). It is a maximum for ๐ = 0°, which occurs when โ . The minimum value occurs when ๐ = 90° and the plane of the loop is perpendicular to ๐ต โ. the plane of the loop is aligned with ๐ต Problem 2. Change in flux and induced emf A coil of wire is situated in a 0.5 T uniform magnetic field. The area of the coil is 2.0 m2. (a) What is the magnetic flux if the angle between the magnetic field and the normal to the surface of the coil is 60°? (b) After 5 s, the magnetic field is now parallel to the normal to the surface. What is the induced emf? Approach (a) The angle between the magnetic field and the normal to the surface of the coil is 60° (b) The angle between the magnetic field and the normal to the surface of the coil is 0°, because the magnetic field is now parallel to the normal surface (see Fig. 3). Deepen Direction: Read and analyze the following problems . Answer them properly. Activity: Calculating EMF: How great is the induced EMF? Problem 1 A uniform magnetic field is directed at an angle of 30° with the plane of a circular coil of radius 2 cm and 2000 turns. If the magnetic field changes at a rate of 4T per second, calculate the induced emf. 15 Approach We are given the angle 30°, but note that this is the angle of B with respect to the plane of the coil. Thus, the angle with respect to the area vector is 60°. We also know the radius of the coil. Thus, we can calculate its area: ๐ด = ๐๐ 2 = ๐(0.02๐)2 = 1.26๐ฅ10−3 ๐2 The emf is induced because the flux is changing. In this case, the reason for the change in flux is the increasing magnetic field (we know this because the rate of ๐๐ต ๐ change given is positive) at = 4 . We can write this into the law of induction. ๐๐ก ๐ ๐Φ๐ต ๐ ๐ = −๐ = − ๐(๐ต๐ด๐๐๐ 60°) ๐๐ก ๐๐ก The factor A cos 60° is not part of the change so we can take it out of the parentheses. ๐๐ต ๐ = −๐๐ด๐๐๐ 60° ๐๐ก At this point, we are ready to substitute the given. Problem 2 Calculate the magnitude of the induced emf when the magnet is thrust into the coil, given the following information: the single loop coil has a radius of 6.00 cm and the average value of B cos θ (this is given, since the bar magnet’s field is complex) increases from 0.0500 T to 0.250 T in 0.100s. Strategy To find the magnitude of emf, we use Faraday’s law of induction as stated by ΔΦ๐ต ๐ = −๐ Δ๐ก but without minus sign indicates direction: ΔΦ๐ต ๐=๐ Δ๐ก We are given that N=1 and โt=0.100s, but we must determine the change in flux ΔΦ before we can find emf. Since the area of the loop is fixed, we see that ΔΦ๐ต = Δ(๐ต๐ด๐๐๐ ๐) = ๐ด Δ(๐ต cos ๐). Now Δ(๐ต cos ๐) = 0.200 ๐, since it was given that ๐ต cos ๐ changes from 0.500 to 0.250 T. The area of the loop is ๐ด = ๐๐ 2 = ๐(0.060๐)2 = 1.13๐ฅ10−2 ๐2 . Thus, ΔΦ = (1.13๐ฅ10−2 ๐2 )(0.200 ๐) 16 Gauge Directions: Read carefully each item. Write only the letter of the best answer before the number. 1. A vector quantity which defined as the dot product of the magnetic field and the area vector. A. Electric Field B. Magnetic Flux C. Induction D. Induced EMF 2. From the definition of magnetic flux, which of the following is the SI unit for magnetic flux? A. V B. T C. Wb D. J 3. If the magnetic flux through an area bounded by a closed conducting loop changes with time, a current and an emf are produced in the loop, what do you call this process? A. Induction B. Intensity C. Current D. EMF 4. What device was used in conducting various experiments such as with magnetically induced emf by Faraday? A. Voltmeter B. Ohmmeter C. Ammeter D. Galvanometer 5. “Moving the magnet toward or away from the coil.” “Moving a current carrying coil toward or away from the coil.” Based from these actions, what do they have in common? A. All these actions do induce a current in a coil B. All these actions were supported by stationary motion C. All these actions are in a closed circuit D. All these actions used magnets to move the galvanometer to another place. 6. Faraday’s law of induction, the induced emf is proportional to the ___________ of magnetic flux through the coil. A. Type of metal coil B. Color of the coil C. Quantity of magnets D. Rate of change 7. Which of the following statements can describe the process of the law of induction? A. Increase in the number of coils decreases the magnetic flux B. Increase in the number of turns in the coil increases the induced emf C. Decrease in magnetic field decreases the induced current D. Decrease in the speed of relative motion between coil and magnet will result in increased flux 8. The minus sign in Faraday’s law of induction is very important. The minus means that the EMF creates a current I and magnetic field B that oppose the change in flux ΔΦ this is known as ________________. A. Ampere’s Law B. Lenz’ law C. Magnetic flux density D. Induced electric field 17 9. What would be the implication if your curled fingers have the same direction with the induced current or emf in the circuit? A. Positive I and EMF B. No implication was given C. Negative I and EMF D. Both A and B 10. Based on the following statements, which is incorrect about the sign of the induced emf or current? ๐Φ๐ต I. If the flux is increasing, so is positive, then the induced emf or ๐๐ก current is negative ๐Φ๐ต II. If the flux is decreasing, is negative and the induced emf or ๐๐ก current is positive ๐Φ๐ต III. If the flux is increasing, is negative and the induced emf or ๐๐ก current is negatives A. I B. II C. III D. I and II 11. Which of the following is a nonconservative field where it does net work in moving a charge over a closed path? A. Magnetic field B. Electrostatic electric field C. Induced emf D. Induced Electric field 12. The fact that emfs are induced in circuits implies that work is being done on the conduction electrons in the wires. What can possibly be the source of this work? A. Electric flux B. Faraday’s law C. Magnetic flux D. Lenz’s Law For numbers 13-14. A circular coil 50 cm in diameter is rotating in a magnetic field directed upward with a magnitude of 65 mT. Calculate the magnetic flux through the coil at the positions shown below. 13. (a) A. 10.0 Tโm2 B. 0.10 Wb 14. (b) A. 5.00 Wb B. 50.0 Tโm2 C. 1.00 Tโm2 D. -0.01 Wb C. 0.05 Wb D. 0.50 Tโm2 15. A magnetic field B= 0.6T is directed upward through a circular loop of diameter 7 cm and 500 turns. The loop is initially horizontal, so it is perpendicular to the magnetic field. It rotates through a horizontal axis so that the plane of the loop is at 74° with the horizontal axis within 1 second. What is the magnitude of the induced emf? A. -15.3 V B. 17.6 V C. 16.4 V D. -18.18 V 18 Jumpstart 1. B 2. C 3. D 4. A 5. D 6. C 7. D 8. A 9. C 10. B 19 Gauge 1. B 2. C 3. A 4. D 5. A 6. D 7. B 8. B 9. A 10. C 11. 12. 13. 14. 15. D A B C D Explore Problem 1. Calculate the flux โ = 1.25๐ Given: r=10.0 cm, ๐ต Unknown: Φ๐๐๐ฅ =? ๐๐๐ Φ๐๐๐ =? Formula: Φ๐ต ๐๐๐ฅ = ๐ต๐ด cos ๐ ๐๐๐ Φ๐ต ๐๐๐ = ๐ต๐ด cos ๐ Solution: From Eq. 2, the maximum value is Φ๐ต = ๐ต๐ด cos ๐ = (1.25 ๐)(0.100 ๐)(0.100๐) cos 0° = 0.0125 ๐๐. The minimum value is 0 Wb when ๐ = 90° and cos 90° = 0. Answer: Φ๐ต ๐๐๐ฅ = 0.0125 ๐๐ and Φ๐ต ๐๐๐ = 0 ๐๐ Problem 2. Change in flux and induced emf โ = 0.5๐, ๐ = 0°, A=2.0m2 Given: ๐ต Unknown: (a) Φ =? ๐๐๐ ๐ =? ΔΦ Formula: (a) Φ๐ต = ๐ต๐ด cos ๐ ๐๐๐ ๐ = −๐ Δ๐ก๐ต Solution: (a) Φ๐ต = ๐ต๐ด cos ๐ = (0.5๐)(2๐2 ) cos 60° = 0.5 ๐๐ (b) After 5 s, the flux is Φ๐ต = ๐ต๐ด cos ๐ = (0.5๐)(2๐2 ) cos 0° = 1 ๐๐ Solving for the induced emf (1๐๐−0.5๐๐) ΔΦ Answer: ๐ = −๐ Δ๐ก๐ต = − = −0.1 ๐ 5๐ Deepen Activity: Calculating EMF: How great is the induced EMF? Problem 1 Given: θ=30°, r=2cm or 0.02m, N=2000 turns, 4๐/๐ Unknown: ๐ =? ๐Φ Formula: ๐ด = ๐๐ 2 , Φ๐ต = ๐ต๐ด๐๐๐ ๐, ๐ = −๐ ๐ต ๐๐ก Solution: 4๐ Answer: ๐ = −(2000)(1.26๐ฅ10−3 ๐2 )๐๐๐ 60 ( ๐ ) = −5.03 ๐ Problem 2 Given: r=6.00 cm, Δ๐ต๐ = 0.0500 ๐, Δ๐ต๐ = 0.250 ๐, t=0.100 s Unknown: ๐ =? ΔΦ Formula: ๐ = ๐ Δ๐ก๐ต Solution: Entering the determined values into the expression for emf gives (0.200๐)(1.13๐ฅ10−2 ๐2 ) ΔΦ Δ๐ต๐ด ๐ต ๐=๐ =๐ = (1) = 0.0226 ๐ ๐๐ 22.6 ๐๐ Δ๐ก Δ๐ก 0.100๐ Answer: 0.0226 ๐ ๐๐ 22.6 ๐๐ Answer Key References BOOKS Cook, C. S. (2018). Breaking Through GENERAL PHYSICS 2 For Senior High School. C & E Publishing. Retrieved February-March-April 2021 Giancoli, D. C. (2005). PHYSICS: Principles with Applications (6 ed.). New Jersey, USA: Pearson Education, Inc. Retrieved February-March-April 2021 Silverio, A. A. (2006). Exploring Life Through Science. Quezon City, Metro Manila, Philippines: Phoenix Publishing House, Inc.,. Retrieved February-March-April 2021 Walker, J. (2014). Fundamentals of Physics (Halliday and Resnick) (10th ed.). United States of America: John Wiley and Sons, Inc. Retrieved February-March-April 2021 Young , H. D., Freedman , R. A., & Ford, L. A. (2012). University Physics with Modern Physics (13 ed.). San Francisco, California, United States of America: Pearson Education, Inc. (as Addison-Wesley). Retrieved February-March-April 2021 WEBSITES BYJU'S. (2021). Faraday’s Laws of Electromagnetic Induction. Retrieved April 2021, from BYJUS.COM: https://byjus.com/physics/faradayslaw/?fbclid=IwAR3NPPk2bRFzqB6cBrnbu7wTsgIGRlkDTKAnX4WdUlhFbHsVWRNfJMLFoc Electronics Tutorial. (2021). Electromagnetic Induction. Retrieved April 2021, from Electronics Tutorial: https://www.electronicstutorials.ws/electromagnetism/electromagnetic-induction.html LibreTexts, P., & OpenStax. (2020, November 6). Induced Electric Currents. Retrieved April 2021, from Physics Libretexts: https://phys.libretexts.org/Bookshelves/University_Physics/Book%3A_University _Physics_(OpenStax)/Book%3A_University_Physics_II__Thermodynamics_Electricity_and_Magnetism_(OpenStax)/13%3A_Electromagneti c_Induction/13.05%3A_Induced_Electric_Fields Lumen Learning. (2021). Faraday’s Law of Induction: Lenz’s Law. Retrieved April 2021, from Lumen Learning - Boundless Physics: https://courses.lumenlearning.com/physics/chapter/23-2-faradays-law-ofinduction-lenzs-law/ Lumen Learning. (2021). Magnetic Flux, Induction, and Faraday’s Law. Retrieved April 2021, from Lumen Learning - Boundless Physics: https://courses.lumenlearning.com/boundless-physics/chapter/magnetic-fluxinduction-and-faradays-law/ 20