Electromagnetic induction So far we have found that kQ = E r - PDF document

Electromagnetic induction So far we have found that kQ = ˆ E r v Electric charges create E-fields 2 r µ × ˆ q v r v = B v 0 Moving electric charges create B-fields π 2 4 r But this is not the whole story! Electromagnetic induction Electric Fields can also be created by a changing Magnetic Field. This is described by “Faraday’s Law”. Magnetic Fields can also be created by a changing Electric Field. This will help complete the set of equations called Maxwell’s equations of electricity and magnetism. Electricity and Magnetism can be unified in a single theory! The electricity and magnetic forces are manifestations of a single “electromagnetic interaction” 2

Electromagnetic induction Magnetic Flux

Magnetic Flux The flux through an element of area is: To calculate the flux through the surface, we need to integrate: B For the simple case of a flat surface, we obtain: φ Magnetic flux

Clicker Question At what angle of orientation of the surface is the flux through the surface half of the maximum possible flux? A) φ = 0 degrees B) φ = 30 degrees B C) φ = 45 degrees φ D) φ = 60 degrees E) None of the above 7 Faraday’s Law of Induction A changing (time-dependent) magnetic field “induces” (generates) an EMF ε . The magnitude of the EMF equals minus the rate of change of the flux. Φ d ε = − B dt Two new quantities introduced. 1. Magnetic Flux = Φ B 2. Electro Motive Force (EMF) = ε 8

Important observations: • Recall: EMF is not a force, but a source of voltage capable of generating power. • The EMF is called “motional” EMF, to distinguish from “chemical” EMF (batteries) • Recall: The magnetic flux through a closed surface is zero! (Gauss’ Law for B-fields) • The surface for Faraday’s Law is an open surface! Clicker Question We have a square imaginary loop with side of length a. There is a Magnetic Field that is uniform throughout the region as shown. How does the Magnetic Flux magnitude | Φ B | change if we halve the Magnetic Field strength and double the sides of the square loop? A) Flux goes to ½ original value B) Flux goes to ¼ original value B X C) Flux goes to 2 times original value D) Flux does not change E) None of the above 10

Changing the magnetic flux 1. Change the strength of the Magnetic Field 2. Change the area of the loop 3. Change the orientation of the loop (A vector) and the B-field 11 Clicker Question A loop of wire is moving rapidly through a uniform magnetic field as shown. Is a non-zero EMF induced in the loop? A) No, there in no EMF B) Yes, there is an EMF 12

Clicker Question A loop of wire is spinning rapidly about a stationary axis in uniform magnetic field as shown. True or False: There is a non-zero EMF induced in the loop. A) False, zero EMF B) True, there is an induced EMF F = φ BA cos and the angle φ is changing. 13 Direction of the EMF around a loop

Lenz’s Law The induced EMF tends to induce a current in the direction that opposes the changes in magnetic flux Φ d ε = − B dt The (-) negative sign in Faraday’s Law is more of a reminder. Lenz’s Law is just another way to express Faraday’s Law. Moreover, it can be deduced from it. 15 Lenz’s Law If we move the magnet closer to the loop, what is the direction of the induced EMF and thus the direction of the current in the loop? Φ = ⋅ B v d A v 1. The Magnetic B ∫ Flux is increasing. surf Φ d > B 0 2. Therefore dt 3. Induced current must create B-field that fights the change! 16

Lenz’s Law Φ d > B 0 dt i Induced current must create B-field that B fights the change! An induced B-field down through the loop will create a slight decrease in Φ B . Thus, we have determined the direction of the induced current and EMF. 17 Lenz’s Law “The Magnetic Flux Φ B Up Through the Loop is Increasing . To Fight this Change, an Induced B-field is made that Decreases the Flux Up Through the Loop. This is accomplished by the induced current.” 18

Lenz’s law (summary) Clicker Question A bar magnet is positioned below a loop of wire. The magnet is pulled down, away from the loop. As viewed from above, is the induced current in the loop clockwise, counterclockwise, or zero? eyeball B i A: clockwise B: counter-clockwise C: zero Answer: Counterclockwise. As the magnet is pulled away, the flux is decreasing . To fight the decrease, the induced B-field should add to the original B-field. 20

Clicker Question A loop of wire is sitting in a uniform, constant magnet field as shown. Suddenly, the loop is bent into a smaller area loop. During the bending of the loop, the induced current in the loop is ... A: zero B: clockwise C: counterclockwise B(in) B(in) Answer: Clockwise. The flux into the page is decreasing as the loop area decreases. To fight the decrease, we want the induced B to add to the original B. By the right hand rule (version II) , a clockwise induced current will make an induced B into the page, adding to the original B. 21 Clicker Question Is there an induced current in this circuit? If so, what is its direction? A. Yes, clockwise B. Yes, counterclockwise C. No * There is zero Magnetic Flux at all times through the loop. 22

Clicker Question A current-carrying wire is pulled away from a conducting loop in the direction shown. As the wire is moving, is there a clockwise current around the loop, a counterclockwise current or no current? A. There is a clockwise current around the loop. B. There is a counterclockwise current around the loop. C. There is no current around the loop. 23 Area = (0.1m) 2 =0.01 m 2 Example with numbers: dB/dt = +0.1 Tesla/second B (increasing) Φ d d dB | ε = = = | B ( BA ) A dt dt dt m c 0 ε = = 1 | | ( 0 . 01 )( 0 . 1 ) 0 . 001 Volts Suppose there are 1000 loops to the wire coil. V 10 cm Φ Φ d ( N ) d ( 1 ) ε = = = | | B N B 1 Volt dt dt 24

Generators Generators use Faraday’s Law to convert mechanical energy into electrical energy. This is the opposite of motors which convert electrical into mechanical energy. 25 The alternator • Wire loop in a constant external B-field. • Turning a crank makes the loop rotate. • This then induces a current in the wire loop. B B i (induced) X X 26

The alternator • Wire loop in a constant external B-field. • Turning a crank makes the loop rotate. • This then induces a current in the wire loop. 27 Clicker Question Instead of turning a crank and inducing a current, what if you hook up the wire loop to a battery and push current through it. What happens to the loop? B A) Nothing B X i B) It feels a net force X C) It feels a net torque D) The current stops flowing Note that the wire loop is a rectangle that is slightly rotated from the plane of the page. 28

Generator Motor Converts mechanical energy Converts electrical energy into into electrical energy. mechanical energy. Crank or otherwise Battery creates a current in a mechanically turns a wire loop loop. This interacts with an in an external B-field. This external B-field to create a induces an electric current in mechanical torque. the loop. 29 The “sliding bar generator” Sliding metal bar on metal rails Bar is pulled by an external agent to the v right at constant B L velocity. X X X X X X There is a uniform B- field surrounding the x=vt rails and loop system. 30

The “sliding bar generator” This creates a wire loop v whose area is growing with B L X X X X X X time. = = A Lx Lvt x=vt Thus, the Magnetic Flux is increasing with time! d d d Φ = = = ( BA ) B ( Lvt ) BLv B dt dt dt 31 Clicker Question What is the direction of v the induced EMF and B L thus current in the wire X X X X X X loop? A) Clockwise x=vt B) Counter Clockwise ε = BLv Motional EMF induced 32

The “sliding bar generator” There is another way to see why the current in the bar is up. Charges in the bar are moving to the right i with the bar itself. There is a B-field in this region. v Thus, there is an upward force on the charges. = × = F v q v B v qvB (up) X v B Thus, there is an upward current in the bar! 33 The “sliding bar generator” One more consequence…. If there is an upward current in the wire and there i F M F Agent is an external B-field, the we get? = × = F Mag v I L v B v ILB (left) v But, there is also the external agent pulling the rod to the right. X If the velocity is constant, what is true about the B forces? They must be equal and opposite! 34

Clicker Question A conducting loop is halfway into a magnetic field. Suppose the magnetic field begins to increase rapidly in strength. What happens to 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 pulled to the left, into the magnetic field. D.The loop is pushed to the right, out of the magnetic field. E. The tension is the wires increases but the loop does not move. Eddy currents If a metal object (something in which electric charges can move) and an external source of Magnetic Field are in relative motion so that there is changing Magnetic Flux through the metal, then Faraday says there is an EMF and thus an Eddy current. If the metal is moving, then the direction of the current is always such as to cause a force which opposes the motion (slows the metal).