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Physics 2113 Jonathan Dowling Lecture 29: MON 02 NOV Induction and Inductance I Fender Stratocaster Solenoid Pickup Faraday's Experiments In a series of experiments, Michael Faraday in England and Joseph Henry in the U.S. were able to generate electric currents without the use of batteries. The circuit shown in the figure consists of a wire loop connected to a sensitive ammeter (known as a "galvanometer"). If we approach the loop with a permanent magnet we see a current being registered by the galvanometer. 1. A current appears only if there is relative motion between the magnet and the loop. 2. Faster motion results in a larger current. 3. If we reverse the direction of motion or the polarity of the magnet, the current reverses sign and flows in the opposite direction. The current generated is known as "induced current"; the emf that appears is known as "induced emf"; the whole effect is called "induction." Changing B-Field Induces a Current in a Wire Loop Note Current Changes Sign With Direction No Current When Magnet Stops loop 1 loop 2 In the figure we show a second type of experiment in which current is induced in loop 2 when the switch S in loop 1 is either closed or opened. When the current in loop 1 is constant no induced current is observed in loop 2. The conclusion is that the magnetic field in an induction experiment can be generated either by a permanent magnet or by an electric current in a coil. Faraday summarized the results of his experiments in what is known as "Faraday's law of induction." An emf is induced in a loop when the number of magnetic field lines that pass through the loop is changing. Loop Two is Connected To A Light Bulb. Loop One Has a 60 Hz Alternating Current The Current in Loop One Produces a Rapidly Changing Magnetic Field in Loop Two That Induces a Current in Loop Two — Lighting the Bulb! Induction Charges My Electric Toothbrush! Faraday’s Law: What? The Flux! • A time varying magnetic FLUX creates an induced EMF • Definition of magnetic flux is similar to definition of electric flux. Units of Magnetic Flux: = Telsa • m2 = Weber! E EMF dFB =dt B dA • Take note of the MINUS sign!! • The induced EMF acts in such a way that it OPPOSES the change in magnetic flux (“Lenz’s Law”). dF B dB EEMF = =A dt dt EMF is proportional to the slope dB/dt Eb > Ed = Ee > Ea = Ec = 0 Lenz’s Law • The Loop Current Produces a B Field that Opposes the CHANGE in the bar magnet field. • Upper Drawing: B Field from Magnet is INCREASING so Loop Current is Clockwise and Produces an Opposing B Field that Tries to CANCEL the INCREASING Magnet Field • Lower Drawing: B Field from Magnet is DECREASING so Loop Current is Counterclockwise and Tries to BOOST the Decreasing Magnet Field. I a = Ib > I c = 0 ICPP 30.4.1. Consider the situation shown. A triangular, aluminum loop is slowly moving to the right. Eventually, it will enter and pass through the uniform magnetic field region represented by the tails of arrows directed away from you. Initially, there is no current in the loop. When the loop is entering the magnetic field, what will be the direction of any induced current present in the loop? a) clockwise b) counterclockwise c) No current is induced. 30.4.1. Consider the situation shown. A triangular, aluminum loop is slowly moving to the right. Eventually, it will enter and pass through the uniform magnetic field region represented by the tails of arrows directed away from you. Initially, there is no current in the loop. When the loop is entering the magnetic field, what will be the direction of any induced current present in the loop? a) clockwise b) counterclockwise c) No current is induced. ICPP 30.4.2. Consider the situation shown. A triangular, aluminum loop is slowly moving to the right. Eventually, it will enter and pass through the uniform magnetic field region represented by the tails of arrows directed away from you. Initially, there is no current in the loop. When the loop is exiting the magnetic field, what will be the direction of any induced current present in the loop? a) clockwise b) counterclockwise c) No current is induced. 30.4.2. Consider the situation shown. A triangular, aluminum loop is slowly moving to the right. Eventually, it will enter and pass through the uniform magnetic field region represented by the tails of arrows directed away from you. Initially, there is no current in the loop. When the loop is exiting the magnetic field, what will be the direction of any induced current present in the loop? a) clockwise b) counterclockwise c) No current is induced. ICPP 30.4.3. A rigid, circular metal loop begins at rest in a uniform magnetic field directed away from you as shown. The loop is then pulled through the field toward the right, but does not exit the field. What is the direction of any induced current within the loop? a) clockwise b) counterclockwise c) No current is induced. 30.4.3. A rigid, circular metal loop begins at rest in a uniform magnetic field directed away from you as shown. The loop is then pulled through the field toward the right, but does not exit the field. What is the direction of any induced current within the loop? a) clockwise b) counterclockwise c) No current is induced. B Example • A closed loop of wire encloses an area of A = 1 m2 in which in a uniform magnetic field exists at 300 to the PLANE of the loop. The magnetic field is DECREASING at a rate of 300 60° = BAcos(600 ) = BA/2 dB/dt = 1T/s. The resistance of the dFB A dB wire is 10 . E = = dt 2 dt • What is the induced current? Is it …clockwise or …counterclockwise? E A dB i= = R 2R dt (1m2 ) i= (1T/s) = 0.05A 2(10W) ICPP • 3 loops are shown. B II • B = 0 everywhere except in III I the circular region I where B is uniform, pointing out of the page and is increasing at a steady rate. • III encloses no flux so EMF=0 • Rank the 3 loops in order of • I and II enclose same flux so EMF same. increasing induced EMF. • Are Currents in Loops I & II – (a) III < II < I ? Clockwise or Counterclockwise? – (b) III < II = I ? – (c) III = II = I ? ICPP 30.4.4. A coil of wire that forms a complete loop is moving with a constant speed v toward a very long, current carrying wire, only a portion of which is shown. What affect, if any, does the current carrying wire have on the coil of wire? a) Since the magnetic field increases as the coil approaches the wire, a current is induced in the coil. B µ1/ r b) The rectangle will be distorted as it is pulled in the direction of the current in the wire. c) Close to the wire, a magnetic force acts on the loop that accelerates the loop away from the wire. d) Since the magnetic field around the wire is not changing, there is no effect on the coil. e) Since the coil and the wire are not touching, there is no effect. 30.4.4. A coil of wire that forms a complete loop is moving with a constant speed v toward a very long, current carrying wire, only a portion of which is shown. What affect, if any, does the current carrying wire have on the coil of wire? a) Since the magnetic field increases as the coil approaches the wire, a current is induced in the coil. b) The rectangle will be distorted as it is pulled in the direction of the current in the wire. c) Close to the wire, a magnetic force acts on the loop that accelerates the loop away from the wire. d) Since the magnetic field around the wire is not changing, there is no effect on the coil. e) Since the coil and the wire are not touching, there is no effect. Example B= m0i 2pr i • An infinitely long wire carries a constant current i as shown R r=R+x • A square loop of side L is moving x L towards the wire with a constant velocity dR/dt=v L v. • What is the EMF induced in the loop Choose a “strip” of width dx when it is a distance R from the loop? located as shown. Flux thru this “strip” FB = L ò 0 m0iLdx 2p (R + x) m0iLdx dF = BLdx = 2p (R + x) L é m 0iL ù =ê ln( R + x)ú ë 2p û0 m 0iL é R + L ù = ln ê ú 2p R ë û dFB E =dt m0 Li d ì é =ílnê1+ 2p dt î ë L ùü ý ú R ûþ Example dFB E =dt m 0 Li d ì é L ù ü =íln ê1 + ú ý 2p dt î ë R û þ m 0 Li dR é R ù L = 2p dt êë R + L úû R 2 2 é ù m 0i L = vê ú 2p ë ( R + L) R û i R dR/dt=v L x What is the DIRECTION of the induced current? • Magnetic field due to wire points INTO page and gets stronger as you get closer to wire • So, flux into page is INCREASING • Hence, current induced must be counter clockwise to oppose this increase in flux = CCW B Example : The Generator • A square loop of wire of side L is rotated at a uniform frequency f in the presence of a uniform magnetic field B as shown. • Describe the EMF induced in the loop. q = wt f = 2pw L B B q = BL cos(q ) dFB 2 dq E == BL sin(q ) = BL2 (2pf )sin(2pft ) dt dt 2 30.3.2. A circular ring is rotated clockwise at a constant rate for an extended period of time using the apparatus shown. Which of the graphs below correctly shows the magnetic flux through the ring as a function of time? Note: At time t = 0 s, the plane of the ring is perpendicular to the direction of the magnetic field. ICPP 30.3.2. A circular ring is rotated clockwise at a constant rate for an extended period of time using the apparatus shown. Which of the graphs below correctly shows the magnetic flux through the ring as a function of time? Note: At time t = 0 s, the plane of the ring is perpendicular to the direction of the magnetic field.