Physics1C_Chapter29

Physics1C_Chapter29 - 4/6/10 CHAPTER 29 ELECTROMAGNETIC...

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Unformatted text preview: 4/6/10 CHAPTER 29 ELECTROMAGNETIC INDUCTION Vahé Peroomian LET’S BEGIN WITH AN EXPERIMENT!   What happens when a magnet is moved in and out of a coil? MORE EXPERIMENTAL DATA         When B = 0, the galvanometer shows no current. When electromagnet is turned on, increasing B, there is a momentary current. Any changes in the orientation of the coil, or its position, etc., will cause momentary jumps in the needle, but only as long as the motion is taking place. What is changing in all these experiments? 1 4/6/10 MAGNETIC FLUX   The magnetic flux dΦB through an infinitesimal-area element dA in a magnetic field B d Φ B = B ⋅ dA = B⊥ dA = BdA cos φ   Choice of direction of dA is up to us! FARADAY’S LAW OF INDUCTION   The induced emf in a closed loop equals the negative of the time rate of change of the magnetic flux through the loop. ε=− dΦ B dt Where did that minus sign come from??? EXAMPLE 1 A coil consists of 200 turns of wire. Each turn is a square of side d = 18 cm, and a uniform magnetic field directed perpendicular to the plane of the coil is turned on. If the field changes linearly from 0 to 0.50 T in 0.80 s, what is the magnitude of the induced emf in the coil while the field is changing? 2 4/6/10 DIRECTION OF INDUCED EMF . | Faraday’s Law of Induction Choose a direction for A From the directions of A c ont. and B , find the sign of Φ and dΦ/dt. ION 3. Find the sign of the tualize From the description in the problem, imagine magnetic field lines passing through the coil. Because the induced current: t ic field is changing in magnitude, an emf is induced in the coil. If flux is increasing, rize We will evaluate the emf using Faraday’s law from this section, so we categorize this example as a substitution dΦ/dt is positive, . and induced emf or Bf 2 Bi DFB D 1 BA 2 DB 0e0 5 te Equation 31.2 for the situation described here, negative. N current is 5N 5 NA 5 Nd 2 Dt Dt Dt Dt t hat the magnetic field changes linearlyUse right hand rule with 4. to find direction of induced emf. 0 e 0 5 1 200 2 1 0.18 m 2 2 1 0.50 T 2 0 2 5 4.0 V ute numerical values: 1. 2. 0.80 s IF? W hat if you were asked to find the magnitude of the induced current in the coil while the field is changing? u answer that question? 0 e0 r I f the ends of the coil are not connected to a circuit, the answer to this question is easy: the current is zero! es move within the wire of the coil, but they cannot move into or out of the ends of the coil.) For a steady current , t he ends of the coil must be connected to an external circuit. Let’s assume the coil is connected to a circuit and al resistance of the coil and the circuit is 2.0 V. Then, the magnitude of the induced current in the coil is 4.0 V 5 I5 EXAMPLE 2 5 2.0 A R 2.0 V A loop of wire enclosing an area A is placed in a region where the magnetic field is perpendicular to the plane of the loop. The magnitude of B varies in a m pl e . An Exponentially Decaying Magnetic Field Bmaxe–at, time according to the expression B = where a is some constant. That is, at t = 0, the field of wire enclosing an area A is placed in a B is Bmax, where the magnetic field is perpendicular and for t > 0, the field S plane of the loop. The magnitudedof B var- exponentially (see ecreases 2at ime according to the expression B 5igure)., Find the induced emf Bmax f B maxe a is some constant. That is, at t 5 0, the field iexponen- as a function of time. n the loop , and for t . 0, the field decreases Fig. 31.6). Find the induced emf in the loop ction of time. ION Figure . (Example 31.2) Exponential decrease in the magnitude of the magnetic f ield with time. The induced emf and induced current vary w ith time in the same way. t tualize The physical situation is similar to Example 31.1 except for two things: there is only one loop, and the field varies exponentially with time rather nearly. d FB d d 2at 5 2 1 ABmax e 2at 2 5 2ABmax e 5 aABmax e 2at dt dt dt rize We will evaluate the emf using Faraday’s law from this section, so we categorize this example as a substitution . te Equation 31.1 for the situation ed here: e52 EXAMPLE 29.4 pression indicates that the induced emf decays exponentially in time. The maximum emf occurs at t 5 0, where aAB max. The plot of e versus t is similar to the B -versus-t curve shown in Figure 31.6. The figure below shows a simple version of an alternator, a device that generates an emf. A rectangular loop is made to rotate with constant angular speed ω about the axis shown. The magnetic field B is uniform and constant. At time t = 0, φ = 0. Determine the induced emf. 897 897 10/6/09 8:46:17 AM 10/6/09 8:46:17 AM 3 4/6/10 EXAMPLE 29.5 Instead of the alternating current (ac) emf produced by the generator in Example 29.4, we can use a similar scheme to create a direct current (dc) generator that produces an emf that always has the same sign. Consider a motor with a square coil 10.0 cm on a side, with 500 turns of wire. If the magnetic field has magnitude 0.200 T, at what motion speed is the average back emf of the motor equal to 112 V? EXAMPLE 29.6 The figure shows a U-shaped conductor in a uniform magnetic field B perpendicular to the plane of the figure, directed into the page. We lay a metal rod with length L across the two arms of the conductor, forming a circuit, and move the rod to the right with constant velocity v. This induces an emf and a current, which is why the device is called a slidewire generator. Find the magnitude and direction of the reducing induced emf. EXAMPLE 29.7 In the slidewire generator, energy is dissipated in the circuit owing to its resistance. Let the resistance of the circuit at a given point in the slidewire’s motion be R. Show that the rate at which work must be done to move the rod through the magnetic field. 4 4/6/10 PROBLEM 29.10 A rectangle measuring 30.0 cm by 40.0 cm is located inside a spatially uniform magnetic field of 1.25 T, with the field perpendicular to the plane of the coil. The coil is pulled out at a steady rate of 2.00 cm/s traveling perpendicular to the field lines. The region of the field ends abruptly. Find the emf induced in this coil when it is (a) All inside the field (b) Partly inside the field (c) All outside the field. LENZ’S LAW The direction of any magnetic induction effect is such as to oppose the cause of the effect (from our textbook).   The induced current in a loop is in the direction that creates a magnetic field that opposes the change in magnetic flux through the area enclosed by the loop (from Serway and Jewett).   LENZ’S LAW AND THE SLIDEWIRE . | Lenz’s Law Figure . (a) Lenz’s law can be used to determine the direction of t he induced current. (b) When the bar moves to the left, the induced current must be clockwise. Why? As the conducting bar slides to the right, the magnetic flux due to the external magnetic field into the page through the area enclosed by the loop increases in time. S S Bin Bin By Lenz’s law, the induced current must be counterclockwise to produce a counteracting magnetic field directed out of the page. a I R S v R I S v b the area enclosed by the circuit increases with time because the area increases. Lenz’s law states that the induced current must be directed so that the magnetic f ield it produces opposes the change in the external magnetic flux. Because the magnetic flux due to an external field directed into the page is increasing, the induced current—if it is to oppose this change—must produce a field directed out of the page. Hence, the induced current must be directed counterclockwise when the bar moves to the right. (Use the right-hand rule to verify this direction.) If the bar is moving to the left as in Figure 31.11b, the external magnetic flux through the area enclosed by the loop decreases with time. Because the field is directed into the page, the direction of the induced current must be clockwise if it is to produce a f ield that also is directed into the page. In either case, the induced current attempts to maintain the original flux through the area enclosed by the current loop. Let’s examine this situation using energy considerations. Suppose the bar is g iven a slight push to the right. In the preceding analysis, we found that this motion 5 4/6/10 LENZ’S LAW AND A CURRENT LOOP MOTIONAL ELECTROMOTIVE FORCE   Motional emf for a closed conducting loop is given by ε= ( v × B ) ⋅ dl ∫ EXAMPLE 3 CHAPTER | Faraday’s Law The conducting bar illustrated in the figure moves on two frictionless, parallel rails in the presence of a uniform magnetic field directed into the page. The bar has mass m, pl e . Magnetic Force Acting on a Sliding Bar initial velocity v to and its length is l. The bar is given an i the right and is released at t = 0. ting bar illustrated in Figure 31.9 moves on two frictionless, (a) Using Newton’s into find in the presence of a uniform magnetic field directed laws,the the velocity given an as a function ar has mass m , and its length is ,. The bar is of the bar initial t he right and is released at t 5 0. of time. S R (b) Show that the same result is FB ewton’s laws, find the velocity of the bar as a function of found by using an energy approach. S Bin S vi I ze A s the bar slides to the right in Figure 31.9, a counterclockt is established in the circuit consisting of the bar, the rails, tor. The upward current in the bar results in a magnetic force t he bar as shown in the figure. Therefore, the bar must slow r mathematical solution should demonstrate that. x Figure . (Example 31.3) A conducting bar of length , on two fixed conducting rails S is given an initial velocity v i to the right. The text already categorizes this problem as one that uses s. We model the bar as a particle under a net force. m Equation 29.10, the magnetic force is FB 5 2I,B , where the negative sign indicates that the force is to the g netic force is the only horizontal force acting on the bar. n’s second law to the bar in the horizontal 5 B ,v/R from Equation 31.6: dv 5 2I ,B Fx 5 ma 5 m dt m dv 52 B 2 ,2 v 6 4/6/10 EXAMPLE 4 A conducting bar of length l rotates with a constant angular speed ω about a pivot at one end. A uniform magnetic field B is directed perpendicular to the plane of rotation as shown in the figure. Find the motional emf induced between the ends of the bar. PROBLEM 29.49 A loop is being pulled to the right at constant speed v away from an infinitely long straight wire with current I. (a) Calculate the magnitude of the net emf induced in the loop. Do this two ways: (i) by using Faraday’s law of induction and (ii) by looking at the emf induced in each segment of the loop due to its motion. (b) Find the direction of the current induced in the loop. Do this two ways: (i) using Lenz’s law, and (ii) using the magnetic force on the charges in the loop. (c) Check your answer to part (a) in the following special cases to see whether it is physically reasonable: (i) the loop is stationary, (ii) The loop is very thin, so a ~ 0, and (iii) The loop gets very far from the wire. 7 ...
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This note was uploaded on 06/03/2011 for the course PHYSICS 1c taught by Professor Peroomian during the Spring '10 term at UCLA.

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