PHY 1214 General Physics II

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1 PHY 1214 General Physics II Lecture 20 Magnetic Flux and Faraday s Law July 6-7, 2005 Weldon J. Wilson Professor of Physics & Engineering Howell Hall 221H

2 Lecture Schedule (Weeks 4-6) We are here. July 6, 2005 PHY Lecture 20 2

3 Air Flux The amount of air flow through the loop depends on the orientation of the loop with respect to the air-flow direction. Aeff = abcosθ = Acosθ July 6, 2005 PHY Lecture 20 3

4 The number of arrows passing through the loop depends on two factors: (1) The density of arrows, which is proportional to B (2) The effective area A eff = A cos θ of the loop We use these ideas to define the magnetic flux: Magnetic Flux Flux : Φ A B = AB cosθ m eff 2 Flux units : 1 weber = 1 Wb = 1 Tm July 6, 2005 PHY Lecture 20 4

5 Magnetic Flux You are given a loop of wire The wire is in an uniform magnetic field B The loop has an area A The flux is defined as B = B A = B A cos is the angle between B and the normal to the plane July 6, 2005 PHY Lecture 20 5

6 Area Vector Define the area vector A of a loop such that it has the loop area as its magnitude and is perpendicular to the plane of the loop. If a current is present, the area vector points in the direction given by the thumb of the right hand when the fingers curl in the direction of current flow. If the area is part of a closed surface, the area vector points outside the enclosed volume. With this definition: Φ A B = AB θ = A B m eff cos July 6, 2005 PHY Lecture 20 6

7 Magnetic Flux The flux can be visualized with respect to magnetic field lines The value of the magnetic flux is proportional to the total number of lines passing through the loop When the area is perpendicular to the lines, the maximum number of lines pass through the area and the flux is a maximum When the area is parallel to the lines, no lines pass through the area and the flux is 0 July 6, 2005 PHY Lecture 20 7

8 Magnetic Flux (a) When the field is perpendicular to the plane of the loop, = 0 and B = B, max = BA (b) When the field is parallel to the plane of the loop, = 90 and B = 0 The flux can be negative, for example if = 180 SI units of flux are T m² = Wb (Weber) July 6, 2005 PHY Lecture 20 8

9 Electromagnetic Induction When a magnet moves toward a loop of wire, the ammeter shows the presence of a current (a) When the magnet is held stationary, there is no current (b) When the magnet moves away from the loop, the ammeter shows a current in the opposite direction (c) If the loop is moved instead of the magnet, a current is also detected July 6, 2005 PHY Lecture 20 9

10 Lentz s Law (1) Heinrich Friedrich Emil Lenz ( ) In 1834, Heinrich Lenz announced a rule for determining the direction of an induced current, which has come to be known as Lenz s Law. Here is the statement of Lenz s Law: There is an induced current in a closed conducting loop if and only if the magnetic flux through the loop is changing. The direction of the induced current is such that the induced magnetic field always opposes the change in the flux. July 6, 2005 PHY Lecture 20 10

11 Lentz s Law (2) If the field of the bar magnet is already in the loop and the bar magnet is removed, the induced current is in the direction that tries to keep the field constant. If the loop is a superconductor, a persistent standing current is induced in the loop, and the net flux through the loop remains unchanged. Superconducting loop July 6, 2005 PHY Lecture 20 11

12 Six Induced Current Scenarios July 6, 2005 PHY Lecture 20 12

13 Example: Lentz s Law The switch in the circuit shown has been closed for a long time. What happens to the lower loop when the switch is opened? July 6, 2005 PHY Lecture 20 13

14 Example: Lentz s Law The figure shows two solenoids facing each other. When the switch for coil 1 is closed, does the current in coil 2 flow from right to left or from left to right? July 6, 2005 PHY Lecture 20 14

15 Example: A Rotating Loop A loop of wire is initially in the xy plane in a uniform magnetic field in the x direction. It is suddenly rotated 90 0 about the y axis, until it is in the yz plane. In what direction will be the induced current in the loop? Initially there is no flux through the coil. After rotation the coil will be threaded by magnetic flux in the x direction. The induced current in the coil will oppose this change by producing flux in the x direction. Let your thumb point on the x direction, and your fingers will curl clockwise. Therefore, the induced current will be clockwise, as shown in the figure. July 6, 2005 PHY Lecture 20 15

16 Example: A Rotating Loop A loop of wire is initially in the xy plane in a uniform magnetic field in the x direction. It is suddenly rotated 90 0 about the y axis, until it is in the yz plane. In what direction will be the induced current in the loop? Initially there is no flux through the coil. After rotation the coil will be threaded by magnetic flux in the x direction. The induced current in the coil will oppose this change by producing flux in the x direction. Let your thumb point on the x direction, and your fingers will curl clockwise. Therefore, the induced current will be clockwise, as shown in the figure. July 6, 2005 PHY Lecture 20 16

17 Example: A Circular Loop Rotating in a Magnetic Field The figure shows a 10 cm diameter loop rotating in a uniform T magnetic field. What is the magnitude of the flux through the loop when the angle is θ=0 0, 30 0, 60 0, and 90 0? A = π R = π (0.005 m) = m Wb for θ = Wb for θ = 30 Φ m = AB cosθ = Wb for θ = 60 0 Wb for θ = 90 July 6, 2005 PHY Lecture 20 17

18 Faraday s Law and Electromagnetic Induction The instantaneous emf induced in a circuit equals the time rate of change of magnetic flux through the circuit If a circuit contains N tightly wound loops and the flux through each loop changes by during an interval t, the average emf induced is given by Faraday s Law: ε = N Φ t July 6, 2005 PHY Lecture B

19 Faraday s Law and Lenz Law The minus sign is included because of the polarity of the emf. The induced emf in the coil gives rise to a current whose magnetic field OPPOSES ( Lenz s law) the change in magnetic flux that produced it July 6, 2005 PHY Lecture 20 19

20 Example: Electromagnetic Induction in a Circular Loop The magnetic field shown in the figure decreases from 1.0 T to 0.4 T in 1.2 s. A 6.0 cm diameter loop with a resistance of Ω is perpendicular to the field. What is the size and direction of the current induced in the loop? ε = B t ε Φ t m = = 0.6 T 1.2 s AB f t = 0.5 T/s 2 = πr = π I AB B t ε 1.41 mv = = R Ω i = A B t = πr 2 B t 2 ( 0.03 m) ( 0.50 T/s) = 1.41mV = 141mA The current direction is such as to reinforce the diminishing B field. Therefore, the current I will be clockwise. July 6, 2005 PHY Lecture I

21 What does Faraday s Law Tell Us? Faraday s Law tells us that all induced currents are the associated with a changing magnetic flux. There are two fundamentally different ways to change the magnetic flux through a loop: (1) The loop can move, change size, or rotate, creating motional emf; (2) The magnetic field can change in magnitude or direction. We can write: ε = Φ t m = B A t motional emf + A B t new physics The second term says that an emf can be created simply by changing a magnetic field, even if nothing is moving. July 6, 2005 PHY Lecture 20 21

22 Question A very long solenoid with no field outside passes through a conducting loop. The current in the solenoid is increased so that the B field inside the solenoid increases. (B outside = 0). Is a current induced in the loop? YES! Since the flux through the loop changes, an emf is induced in the loop, even though the field that produces the flux does not touch the loop. How can this happen? Faraday would say that when the number of lines of force in the solenoid increases, they must come in from infinity and must cut through the loop on their way in. July 6, 2005 PHY Lecture 20 22

23 Question 1 A conducting loop is half way into a magnetic field. Suppose that the field begins to increase rapidly in strength. Which statement describes the behavior of the loop? (a) The loop is pushed upward, toward the top of the page; (b) The loop is pushed downward, toward the bottom of the page; (c) The loop is pushed to the left, into the magnetic field; (d) The loop is pushed to the right, out of the magnetic field; (e) The tension in the wire increases, but the wire does not move. July 6, 2005 PHY Lecture 20 23

24 Metal Detectors Metal detectors like those used at airports can detect any metal objects, not just magnetic materials like iron. They operate by induced currents. A transmitter coil sends high frequency alternating currents that will induce current flow in conductors in its field. Because of Lenz s Law, the induced current opposes the field from the transmitter, so that net field is reduced. A receiver coil detects the reduction in the magnetic fields from the transmitter and registers the presence of metal. July 6, 2005 PHY Lecture 20 24

25 Applications of Faraday s Law Ground Fault Interrupters The ground fault interrupter (GFI) is a safety device that protects against electrical shock Wire 1 leads from the wall outlet to the appliance Wire 2 leads from the appliance back to the wall outlet The iron ring confines the magnetic field, which is generally 0 If a leakage occurs, the field is no longer 0 and the induced voltage triggers a circuit breaker shutting off the current July 6, 2005 PHY Lecture 20 25

26 Applications of Faraday s Law Electric Guitar A vibrating string induces an emf in a coil A permanent magnet inside the coil magnetizes a portion of the string nearest the coil As the string vibrates at some frequency, its magnetized segment produces a changing flux through the pickup coil The changing flux produces an induced emf that is fed to an amplifier July 6, 2005 PHY Lecture 20 26

27 Applications of Faraday s Law Apnea Monitor The coil of wire attached to the chest carries an alternating current An induced emf produced by the varying field passes through a pick up coil When breathing stops, the pattern of induced voltages stabilizes and external monitors sound an alert July 6, 2005 PHY Lecture 20 27

28 Application Tape Recorder A magnetic tape moves past a recording and playback head The tape is a plastic ribbon coated with iron oxide or chromium oxide x

29 Application Tape Recorder, cont. To record, the sound is converted to an electrical signal which passes to an electromagnet that magnetizes the tape in a particular pattern To playback, the magnetized pattern is converted back into an induced current driving a speaker July 6, 2005 PHY Lecture 20 29

30 End of Lecture 20 Before the next lecture, read sections 22.3 thru July 6, 2005 PHY Lecture 20 30

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