1 W09D2: Faraday’s Law Today’s Reading Assignment Course Notes: Sections 10.1-10.4Announcements Math Review Week Ten Tuesday from 9-11 pm in 26-152 PS 7 due Week Ten Tuesday at 9 pm in boxes outside 32-082 or 26-152 Next Reading Assignment: W09D3 Course Notes: Sections 9.10.2; 10.7, 10.9-10 23 Outline Faraday’s Law Applications of Faraday’s Law Problem Solving Experiment 3: Faraday’s Law4 Faraday’s Law of Induction dΦBdt∝ IinducedChanging magnetic flux induces a current5 Demonstration: Induction and Simulation of Induction http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%203&show=0 http://public.mitx.mit.edu/gwt-teal/FaradaysLaw2.html6 Demo: Electromagnetic Induction http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%203&show=07 Demonstration: Magnet Falling Through Plastic Tube and Aluminum Tube http://tsgphysics.mit.edu/front/?page=demo.php&letnum=H%2016&show=08 Magnetic Flux Thru Wire Loop ΦB=B ⋅ dAS∫∫Analogous to Electric Flux (Gauss’ Law) (1) Uniform (2) Non-Uniform ΦB= B⊥A = B Ac o sθ=B ⋅A B B9 Electromotive Force (emf) Imagine a charge q moving around a closed path under the action of a force . Then the work done per unit charge is called the electromotive force (emf). Has the same dimensions as electric potential so SI unit is the volt. If the closed path is a circuit with resistance R then the electromotive force will cause a current to flow according to ε= IR F ε=Fq⋅ dsclosd path∫10 Motional emf Pull a loop with speed v Through uniform magnetic field (shaded area). At one instant in time, electromotive force is ε=Fmagq⋅ dsloop∫=qv ×Bq⋅ dsleft leg∫= vBw11 Changing Magnetic Flux and Motional emf Magnetic flux through loop is decreasing −ΔΦmagΔt=BwΔsΔt=BwvΔtΔt= vBw=ε12 Concept Question: Loop in Uniform Field While a rectangular wire loop is pulled upward though a uniform magnetic field B field penetrating its bottom half, as shown, there is 12 1. a current in the loop. 2. no current in the loop. 3. I do not understand the concepts of current and magnetic field. 4. I understand the concepts of current and magnetic field but am not sure of the answer.13 Concept Q.: Loop in Uniform Field While a rectangular wire loop is pulled sideways though a uniform magnetic field B field penetrating its bottom half, as shown, there is 13 1. a current in the loop. 2. no current in the loop. 3. I do not understand the concepts of current and magnetic field. 4. I understand the concepts of current and magnetic field but am not sure of the answer.14 Ways to Induce emf Quantities which can vary with time: • Magnitude of B • Area A enclosed by the loop • Angle between B and normal vector to loop ε= −ddtBAcosθ( )15 Problem: Changing Area Conducting rod pulled along two conducting rails in a uniform magnetic field B at constant velocity v a) Calculate the magnitude of the derivative of the magnetic flux with respect to time. b) Calculate the magnitude of EMF due to the magnetic force. Does your result agree with your result from part a)? c) What is the magnitude of induced current?16 Generalization: Faraday’s Law Loop is at rest, and magnet is moving resulting in a changing magnetic flux through loop and an induced emf and current in the loop ε= −dΦBdt17 Induced Electric Field and Electromotive Force (emf) ε=E ⋅ dsclosd path∫An induced electric field appears in the current loop resulting in an electric force on the charges in the conductor producing an induced current ε= IR18 Faraday’s Law of Induction If C is a stationary closed curve and S is a surface spanning C then The changing magnetic flux through S induces a non-electrostatic electric field whose line integral around C is non-zero E ⋅ dsC∫= −ddtB ⋅ dAS∫∫ ε= −dΦBdt19 Sign Conventions: Right Hand Rule By the right hand rule (RHR) clockwise integration direction for line integral of electric field (emf) requires that unit normal points into plane of figure for magnetic flux surface integral Magnetic flux positive into page, negative out of page : E ⋅ dsclosed path∫= −ddtB ⋅ dAopen surface∫∫20 Sign Conventions: Right Hand Rule By the right hand rule (RHR) counterclockwise integration direction for line integral of electric field (emf) requires that unit normal points out of plane of figure for magnetic flux surface integral Magnetic flux positive out of page, negative into page E ⋅ dsclosed path∫= −ddtB ⋅ dAopen surface∫∫21 Problem: Calculating Electric Field Consider a uniform magnetic field which points into the page and is confined to a circular region with radius R. Suppose the magnitude increases with time, i.e. dB/dt > 0. Find the magnitude and direction of the electric field in the regions (i) r < R, and (ii) r > R. (iii) Plot the magnitude of the electric field as a function r. E ⋅ dsC∫= −ddtB ⋅ dAS∫∫22 Minus Sign? Lenz’s Law Induced EMF is in direction that opposes the change in flux that caused it ε= −dΦBdt23 For a Closed Conducting Path Induced current is source of induced magnetic flux that opposes the change in magnetic flux that caused it IindR = −dΦBdt24 Concept Question: Loop The magnetic field through a wire loop is pointed upwards and increasing with time. The induced current in the coil is 1. Clockwise as seen from the top 2. Counterclockwise25 Direction of Induced Current ε= −dΦBdt> 0⇒ IindcounterclockwiseFor a counterclockwise integration direction for the line integral hence a upwards unit normal for the surface integral ! ε= −dΦBdt< 0⇒ Iindis clockwise!26 Concept Question: Moving Loop A circuit in the form of a rectangular piece of wire is pulled away from a long wire carrying current I in the direction shown in the sketch. The induced current in the rectangular circuit is 1. Clockwise 2. Counterclockwise 3. Neither, the current is zero27 Lenz’s Law Induced current, torque, or force is always directed to oppose the change that caused it28 Induced Current Loop Acts Like Magnet Force between magnet and induced current loop is like force between magnets29 Concept Question: Faraday’s Law: Loop A coil moves up from underneath a magnet with its north pole pointing
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