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van der Wielen 1 Matthew van der Wielen Chapter 20 (Electromagnetism) Review Area covered by AP Physics B Exam: Electromagnetism (Electromagnetic Induction, including Faraday’s Law and Lenz’s Law) Breakdown of Objectives for Physics B Exam: 1. Students should understand the concept of magnetic flux, so they can calculate the flux of a uniform magnetic field through a loop of arbitrary orientation. 2. Student’s should understand Faraday’s Law and Lenz’s Law, so they can: a) Recognize situations in which changing flux through a loop will cause an induced EMF (or current) in the loop. b) Calculate the magnitude and direction of the induced EMF and current in a loop of wire or a conducting bar when the magnitude of a related quantity such as magnetic field or area of the loop is changing at a constant rate. References used: Physics Textbook (Chapter 20 deals with Electromagnetic Induction) Classroom Notes (Classroom notes can be found directly after/inside notes on magnetism) AP Website (specifically the Physics B section and course description pages) Hyperphysics (Pages for magnetic flux and EMF) van der Wielen 2 Breakdown, Section 1: Flux, in and of itself, is defined as field lines per unit area. Specifically, Magnetic Flux is the number of magnetic field lines that pass through the area of a loop. The illustration to the left is an example of magnetic flux. The blue arrows represent magnetic field lines, while the gray shaded region bounded by the yellow loop is the area of focus. The Flux of this picture is 9, as there are 9 magnetic field lines passing through the area of the loop. To determine flux, one must take into account the a) Strength of the B field, b) The area of the loop, and c) The orientation of the loop in the B field. As all the above properties are proportional to flux, we get the equation: Where: i. (Flux symbol) is the symbol for magnetic flux. ii. β is the magnetic field strength, and indicates the direction of the magnetic field. iii. A is the area of the loop, and indicates the vector of the loop, which is always perpendicular to its surface. iv. θ is the angle between the β vector and the A vector (in degrees). van der Wielen 3 v. The units of flux are generally Tesla’s *meters2 , or Webbers. Though the text on the picture to the left isn’t exactly clear, the images themselves are an excellent representation of the equation given above. In the top most image, the strength of the magnetic field is being manipulated. As the number of magnetic field lines inside the loop increases, so does the flux inside the loop. In the middle of the image, the angle of the loop in the β field is being manipulated. The right image shows the loop with the Area vector parallel to the magnetic field vector, and accurately shows the loop catching the most field lines. However, as the angle is increased, as shown in the left image, the number of field lines caught by the loop decreases. Finally, in the bottom most picture, the area of the loop is being changed, and the images show that as the Area of the loop increases, so does the number of field lines caught. Overview Section 1: In review: Magnetic Flux can be calculated using the equation , and therefore flux changes when either the magnetic field strength, area of the loop, or angle of the loop in relation to the magnetic field changes. Below are some sample problems (followed by their calculations and answers), to test your knowledge of magnetic flux. Sample Problems: van der Wielen 4 1. A magnetic field of strength 0.30 T is directed perpendicular to a plane circular loop of wire of radius 25 cm. Find the magnetic flux through the area enclosed by this loop. 2. A circular loop with a radius of .200 meters is placed in a uniform magnetic field of magnitude 0.850 Tesla’s. The normal to the loop makes an angle of 30 degrees with respect to the direction of the magnetic field. Calculate the magnetic flux through the area enclosed by the loop. Solutions: 1. Answer: 5.9 * 10-2 Webbers i. Convert radius of loop to proper units: 25 cm -> .25 m. ii. Find the area of the loop: A = π * r2. A = π * (.25)2 = approximately .196 m2 iii. Calculate magnetic flux: = β * A. = .196 * .3 = 5.9*10-2 Webbers 2. Answer: 9.2 * 10-2 Webbers i. Find the area of the loop: A = π * r2. A = π * (.2)2 = approximately .126 ii. Calculate flux inside the loop: = β * A * cosθ = (.825)(.126)(cos30) = 9.2*10-2 Webbers Breakdown, Section 2, part 1 The electromotive force (referred to as EMF from here on) is not in fact a real force, but is more simply a voltage induced in a wire due to the movement of a magnet relative to it. An induced EMF creates a current in the wire, called the induced current, as it is created by the induced EMF. Faraday’s Law states that the instantaneous EMF induced in a circuit equals the time rate of change of magnetic field through the circuit. van der Wielen 5 In English, Faraday’s Law gives us the equation: EMF = (-N * delta )/delta t Where: EMF is the induced voltage N is the number of loops is the number of field lines per unit area, and the delta symbol just means change. T is time, in seconds, and the delta symbol means, once again, change. The negative sign is only an indication of Lenz’s Law (Conservation of Energy), which will be discussed later. The induced voltage in the above equation is generally created when a magnet is placed within or near a loop of wire, and then something is done to the system that changes the flux inside the loop, such as changing the area of the loop or more simply just moving the magnet. The AP test requires that students “Recognize situations in which changing flux through a loop will cause an induced EMF (or current) in the loop”. The answer to that is simpler than it might sound. Any change in the flux passing through the plane of the loop will cause an induced voltage (and therefore current) in that loop. Therefore, EMF will change anytime when any of the three components of flux (magnetic field strength, area of the loop, or angle of the loop relative to the magnetic field) are changed. Faraday’s Law of Induction was found by doing a simple experiment involving an Ammeter, a magnet, and a loop of wire (with the loop connected to the Ammeter). When the magnet is placed relatively near van der Wielen 6 (and stationary) the loop, the ammeter reads nothing. However, when the magnet (or loop) is moved toward or away from the loop, the ammeter deflects. From this experiment, Faraday determined that an induced voltage was created by a change in magnetic flux, and so the above equation was born. Overview Section 2, part 1: In review: An induced EMF is created in a loop when the magnetic flux passing through that loop changes, per the equation EMF = (-N * delta )/delta t. Because EMF changes when magnetic flux changes, EMF changes when any of the components of the flux equation (magnetic field strength, area, or angle) changes. Breakdown, Section 2, part 2 However, the above experiment didn’t only give us the one law, but also another, equally important: Lenz’s Law. Though Lenz’s Law is a simple statement of conservation of energy, without it the world would be a totally different place. A formal statement of Lenz’s Law reads as follow: The polarity of the induced EMF is such that it produces a current whose magnetic field opposes the change in magnetic flux through the loop. That is, the induced current tends to maintain the original flux through the coil. Though the above definition may seem daunting, one just needs to remember that Lenz’s Law is simply a statement of the conservation of energy and things become clearer. The Law of the Conservation of Energy states that the total amount of energy in a closed system remains constant, and therefore energy can not be created or destroyed. When the magnetic flux inside the loop is changed, it induces an EMF, or Voltage (and therefore current) inside the wire. In Chapter 19 of the class textbook, we learn that a current carrying wire produces a B field whose direction can be determined using the right hand rule (see chapter 19 notes for more information). Knowing Lenz’s Law and the information on current carrying wires, we van der Wielen 7 can determine the direction of the current induced in the wire. We know that the changing flux creates voltage. If the created current created an even larger change in flux (by increasing the magnetic field lines passing through the loop), the created voltage would increase again, creating another change in flux, resulting in yet another increase in voltage, and so on and so forth. Obviously, this “infinite voltage” scenario violates the Conservation of Energy (or, in this instance, Lenz’s Law). Therefore, we know the current induced in the wire must oppose the change in flux (and equal it in strength), to prevent the “infinite voltage” scenario. Overview, Section 2, part 2: By knowing Faraday’s Law and Lenz’s Law, one can calculate the induced EMF and direction of the current in the wire. EMF can be calculated using the equation EMF = (-N * delta )/delta t, and the direction of the current can be found by observing the conservation of energy of the system. Sample Problems: (assume magnetic field vector and area vector are parallel for below problems) 1. A 20-turn circular coil of wire is placed in a uniform magnetic field of 1.0 T. The radius of the loop is .5 m. The strength of the magnetic field is decreased to .1 T over a period of 1 s. What is the induced EMF in the loop? 2. A single loop of wire is placed in a uniform magnetic field of strength 5.0 T (magnetic field lines travelling from East to West). The loop of wire has a radius of 20 cm. If the strength of the magnetic field is increased to 10.0 T, what is the direction of the current induced in the wire? Solution: Problems 1 1. First, find the change in flux. van der Wielen 8 2. To find the change in flux, find the flux before the change in magnetic field strength, then again at the end, and subtract the two from each other. = β * A * cos θ. Delta = π * (.5)2 - .1 * π * (.5)2 = approximately .71. 3. Finally, plug all numbers into Faraday’s equation to get the correct answer. EMF = -N * delta / delta t –> EMF = -14.14 volts, or 14.14 volts (the negative symbol just gives direction). Solution: Problems 2 1. This problem doesn’t actually need a calculation to solve it. Think back on Lenz’s Law. Because the magnetic field strength is increasing, the current created will oppose the increase in magnetic field strength that created it. If the field lines are moving from East to West, the induced current will flow counterclockwise (which, by right hand rule, creates a magnetic field travelling from West to East, thus cancelling out the change that created the current, and keeping the Law of Conservation of Energy). van der Wielen 9 Works Cited "Collegeboard AP Physics B." 2004 Free Response Questions. Collegeboard, 2004. Web. 28 Mar 2010. <http://www.collegeboard.com/prod_downloads/ap/students/physics/ap04_frq_physics_b.pdf>. "Magnetic Flux." Magnetic Flux. Web. 28 Mar 2010. <http://upload.wikimedia.org/wikipedia/commons/0/06/Magnetic_flux.png>. Mahoney, John. "Flux, Induction, and EMF Waves." physics.ucdavis.edu. Physics Department, UC Davis,28 08 2008. Web. 28 Mar 2010. Nave, Carl R. "Hyperphysics." Hyperphysics. Apple Corporation, 2006. Web. 28 Mar 2010. <http://hyperphysics.phy-astr.gsu.edu/hbase/HFrame.html>. <http://physics.ucdavis.edu/physics7/Physics7/7C200807/lectures/Flux%20Induction%20and%20Electro magnetic%20Waves.pdf> Serway, Raymond, and Jerry Faughn. College Physics. 6th Edition. Volume 1. Canada: Thomson Learning Academic Resource Center, 2003. 620-51. Print. van der Wielen 10 Quiz: Multiple Choice 1. A uniform magnetic field of strength 2.0 Tesla’s passes through a loop of radius 50 cm. The Area vector is parallel to the magnetic field vector. What is the flux within the area of the loop? a) 1.57 Webbers b) 15707.96 Webbers c) 1.48 Webbers d) .5 Webbers 2. A loop of radius 2 m is placed within a uniform magnetic field of strength 1 * 10^-2 T. The loop is then tilted so it’s area vector lies at a 30 degree angle with the magnetic field. What is the flux within the area of the loop? a) 12.57 Webbers b) 3.14 Webbers c) .11 Webbers d) 42 Webbers 3. A loop of radius 5 cm is placed within a uniform magnetic field of strength 3.0 T. If the Area vector lies perpendicular to the magnetic field vector, what is the flux within the area of the loop? a) .024 Webbers b) 0 Webbers c) 200 Webbers van der Wielen 11 d) None of the above 4. A 15 turn loop of radius 100 cm is placed within a uniform magnetic field of strength 5.0 Tesla’s. If the area of the loop is decreased to 50 cm in 10 seconds, what is the EMF produced in the loop? (Assume parallel vectors) a) 17.67 Volts b) 3.53 Volts c) -17.67 Volts d) More than one of the above choices is correct. 5. Based on the information given in question 4, what would the direction of the induced current be if the magnetic field lines travelled from North to South? a) Clockwise b) Counterclockwise c) There is no induced current Free Response Questions: 1. A square loop of wire of side .2 m is positioned in a uniform magnetic field B of .03 T. The magnetic field is directed into the page, perpendicular to the plane of the loop. van der Wielen 12 a) Calculate the magnetic flux through the loop. b) If the field strength is increased uniformly to .2 T in .5 seconds, what is the EMF induced in the loop? c) Is the direction of the current induced in the loop clockwise or counterclockwise? Justify your answer. d) Describe a method you could use to induce a current in the loop if the magnetic field were kept constant. 2. Explain, in detail, why the current induced by the EMF must create a B field that opposes the change in the B field that created it. van der Wielen 13 Answers: 1. a 2. c 3. b 4. d 5. a FR 1. (This problem was taken from 2004 FR Questions a) Flux = 1.2 * 10^-3 Webbers b) EMF = .014 Volts c) Counterclockwise: Current will be produced that opposes the increase in flux into the page. By RHR, the current will be counter clockwise. d) Change the area of the loop, change the orientation of the loop, remove the loop from the field. 2. Any correct explanation of Lenz’s Law or Conservation of Energy will do.
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