Chapt 8 Phys210

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    Chapt er 8: FaradayChapt er 8: Faraday s Laws LawSo far, our studies in electricity and magnetism have focused onSo far, our studies in electricity and magnetism have focused on the electric fieldsthe electric fields

    produced by stationary charges and the magnetic fields producedproduced by stationary charges and the magnetic fields produced by moving charges.by moving charges.

    This chapter explores the effects produced by magnetic fields thThis chapter explores the effects produced by magnetic fields that vary in time.at vary in time.

    8.1. Faraday8.1. Faradays Law of I nduct ions Law of I nduct ion

    Yes, it is possible.Yes, it is possible.

    In Chapter 29, we discussed two way in which electricity and maIn Chapter 29, we discussed two way in which electricity and magnetism are related:gnetism are related:

    1.1.An electric current produces a magnetic field;An electric current produces a magnetic field;

    2.2.A magnetic field exerts a force on an electric current or movinA magnetic field exerts a force on an electric current or moving electric charge.g electric charge.

    I f elect ric cur rents produce a magnet ic f ield, is it possibleI f elect ric current s produce a magnetic field, is it possible

    that a magnet ic field can produce an elect ric current?that a magnet ic field can produce an elect ric current?

    Michael Faraday and Joseph Henry have found this first on the 1Michael Faraday and Joseph Henry have found this first on the 199

    thth

    century.century.

    I nduced emfI nduced emf

    A coil of wire, X, is connected to a battery.A coil of wire, X, is connected to a battery.

    The current that flows through XThe current that flows through X

    produce a magnetic field that is intensifiedproduce a magnetic field that is intensified

    by the iron core.by the iron core.

    Would a steady current in X produce aWould a steady current in X produce agreat enough magnetic field to a producegreat enough magnetic field to a produce

    a current in a second coil Y?a current in a second coil Y?Sketch of Faraday's Experiment toSketch of Faraday's Experiment to

    Induce EMFInduce EMF

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    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    The answer is :The answer is : No!No! AAsteadysteady current incurrent in

    X producesX produces nono current in Y.current in Y.

    Only when the current X is starting orOnly when the current X is starting or

    stopping a current is produced in Y.stopping a current is produced in Y.

    Therefore, aTherefore, a

    changingchanging

    magnetic field can produce an electric current!magnetic field can produce an electric current!

    Such a current is called anSuch a current is called an induced currentinduced current ..

    An induced emf is produced by a changing magnet ic f ield.An induced emf is produced by a changing magnet ic f ield.

    (a) A current is induced when a(a) A current is induced when a

    magnet is moved toward a coil.magnet is moved toward a coil.

    The next experiment onThe next experiment on eletromagneticeletromagnetic

    inductioninduction , as this phenomena is called, shows, as this phenomena is called, shows

    that if a magnet is quickly moved into a ciol ofthat if a magnet is quickly moved into a ciol of

    wire, current is induced in the wire.wire, current is induced in the wire.

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    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    (b) The induced current is(b) The induced current is

    opposite when the magnet isopposite when the magnet ismoved away from the coil. Notemoved away from the coil. Note

    that the galvanometer zero is atthat the galvanometer zero is at

    the center of the scale and thethe center of the scale and the

    needle deflects left or right,needle deflects left or right,

    depending on the direction of thedepending on the direction of thecurrent.current.

    (c) No current is induced if the(c) No current is induced if themagnet does not move relativemagnet does not move relative

    to the coil. It is the relativeto the coil. It is the relative

    motion that counts: the magnetmotion that counts: the magnet

    can be held steady and the coilcan be held steady and the coil

    moved, which also induces anmoved, which also induces anemf.emf.

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    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    Faraday found first of all that theFaraday found first of all that the more rapidlymore rapidly the magnetic field changes, thethe magnetic field changes, thegreatergreater the induced emf.the induced emf.

    The emf is proportional to the rate of the change ofThe emf is proportional to the rate of the change of

    thethe magnetic flux,magnetic flux, BB, passing through the circuit or, passing through the circuit or

    loop of area A.loop of area A. Magnetic flux for a uniform magnetic field is defined as:Magnetic flux for a uniform magnetic field is defined as:

    A.BcosBAABB

    ===

    Here is the component of the magnetic field perpendiculaHere is the component of the magnetic field perpendicular to the face ofr to the face of

    the loop, andthe loop, and is the angle between and the vector (representing the areais the angle between and the vector (representing the area))

    whose direction is perpendicular to the face of the loop (as showhose direction is perpendicular to the face of the loop (as shown in the figure).wn in the figure).

    B B

    B

    A

    If the area is of some other shape, or the magneticIf the area is of some other shape, or the magneticfield is not uniform, the magnetic flux can be written:field is not uniform, the magnetic flux can be written:

    = Ad.BB

    The fluxThe flux BB can be thought of being proportional tocan be thought of being proportional tothe total number of lines passing through the loopthe total number of lines passing through the loop..

    Unit:Unit: weberweber ((WbWb)= tesal. Meter)= tesal. Meter22 (T.m(T.m22).).

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    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    55

    Faraday Law of I nduct ionFaraday Law of I nduct ion

    The emf induced in a circuit is equal to the rate of change ofThe emf induced in a circuit is equal to the rate of change of magnetic fulxmagnetic fulx

    through the circuit:through the circuit:

    )1.31(dt

    dN B

    =

    The minus sign is placed there to remind us in which directionThe minus sign is placed there to remind us in which direction the induced emf acts.the induced emf acts.

    LenzLenzs Laws Law

    An induced emf gives rise t o cur rent w hose magnet ic f ield opposeAn induced emf gives rise t o cur rent w hose magnet ic f ield opposes t hes t he

    original change in f lux .original change in f lux .

    An induced emf is always in t he direct ion t hat opposes t he origiAn induced emf is always in t he direct ion t hat opposes t he original changenal change

    in f lux t hat causes it .in f lux t hat causes it .

    It is important to note that an emf isIt is important to note that an emf is

    induced whenever there is a change in flux.induced whenever there is a change in flux.

    Then an emf can be induced in three ways:Then an emf can be induced in three ways:

    (a)(a) By changing magnetic field BBy changing magnetic field B

    (b)(b) By changing the area A of the loop in the fieldBy changing the area A of the loop in the field

    Example 31.1: Inducing an emf in a Coil (Page 870).Example 31.1: Inducing an emf in a Coil (Page 870).

    (c)(c) By changing the loopBy changing the loops orientations orientation with respect to the filed (see figure).with respect to the filed (see figure).

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    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    8.2 Mot ional emf (emf I nduced in a Moving Conductor )8.2 Mot ional emf (emf I nduced in a Moving Conductor )

    A straight wire of lengthA straight wire of length ll is moving with speedis moving with speedvv perpendicular to a magnetic field of strengthperpendicular to a magnetic field of strength BB

    has an emf induced between its ends equal to:has an emf induced between its ends equal to:

    B lB lvv..

    BlvdtlvdtB

    dtdAB

    dtd B ====

    8.5 Elect r ic Generator8.5 Elect r ic Generator

    The most important practical result of FaradayThe most important practical result of Faradayss

    great discovery was development of thegreat discovery was development of the electricelectricgeneratorgenerator oror dynamodynamo..A generator transforms mechanical energy intoA generator transforms mechanical energy into

    electric energy. This is just the opposite of what aelectric energy. This is just the opposite of what a

    motor does.motor does.

    A simplified diagram of anA simplified diagram of an ac generatorac generator is shownis shown

    in the figure besides.in the figure besides.

    Example 31.3: Magnetic Force Acting on a Sliding Bar (Page 873)Example 31.3: Magnetic Force Acting on a Sliding Bar (Page 873)

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    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    77

    A generator consists of many coils of wire (onlyA generator consists of many coils of wire (only

    one is shown) wound on an armature that can rotateone is shown) wound on an armature that can rotate

    in a magnetic field. The axle is turned by somein a magnetic field. The axle is turned by some

    mechanical means (falling water, car motor belt), andmechanical means (falling water, car motor belt), and

    an emf is induced in the rotating coil. An electrican emf is induced in the rotating coil. An electric

    current is thus thecurrent is thus the outputoutputof a generator.of a generator.

    In the figure the equation F = q.v XIn the figure the equation F = q.v X B tells us that,B tells us that,

    with rotating counterclockwise, the (conventional)with rotating counterclockwise, the (conventional)

    current in the wire labeled A on the armature iscurrent in the wire labeled A on the armature is

    outward; therefore it is outward at brush A, as shown.outward; therefore it is outward at brush A, as shown.(Each brush presses against a continuous slip ring.)(Each brush presses against a continuous slip ring.)

    After oneAfter one--half revolution, wire A will be where wire C is now in the drawihalf revolution, wire A will be where wire C is now in the drawing, andng, and

    the current then at brush A inward. Thus the current produced isthe current then at brush A inward. Thus the current produced is alternating.alternating.

    Let us assume the loop is being made to rotate in a uniform magLet us assume the loop is being made to rotate in a uniform magnetic fieldnetic field BB

    with a constant angular velocitywith a constant angular velocity .. From FaradayFrom Faradays law, the induced emf iss law, the induced emf is::

    [ ]B.A.cosdt

    dAdB

    dt

    d

    dt

    dB ==

    =

    .

    where A is the area of the loop andwhere A is the area of the loop and is the angle betweenis the angle between BB andand AA..

    C

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    SinceSince == dd/dt,/dt, thenthen == 00++ t. We arbitrarilyt. We arbitrarily

    taketake 00 = 0, so:= 0, so:

    ( ) .sincos tBAtdtdBA ==

    If the rotating coil contains NIf the rotating coil contains N loops,loops,

    ttBAN sinsin 0== Thus the output emf is sinusoidal with amplitudeThus the output emf is sinusoidal with amplitude NBANBA ..

    The frequencyThe frequency ff= w/2= w/2 is 60 Hz for general use in the United States and Canada,is 60 Hz for general use in the United States and Canada,

    although 50 Hz is used in many countries.although 50 Hz is used in many countries. In electric power generating plants, the armature is mounted onIn electric power generating plants, the armature is mounted on a heavy axlea heavy axle

    connected to a turbine, which is the modern equivalent of a wateconnected to a turbine, which is the modern equivalent of a waterwheel.rwheel.

    Water pressure at a dam can turn the turbine at a hydroelectricWater pressure at a dam can turn the turbine at a hydroelectric plant. Most ofplant. Most of

    the power generated at present in the United States, however, isthe power generated at present in the United States, however, is done at steamdone at steamplants, where the burning of fossil fuels (coal, oil, natural gaplants, where the burning of fossil fuels (coal, oil, natural gas) boils water tos) boils water to

    produce highproduce high--pressure steam at turns the turbines.pressure steam at turns the turbines.

    Likewise, at nuclear power plants, the nuclear energy based isLikewise, at nuclear power plants, the nuclear energy based is used to produceused to produce

    steam to turn turbines.steam to turn turbines.

    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    Ch F dCh 8 F d LL

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    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    31.4 I nduced emf and Elect ric Fields31.4 I nduced emf and Elect r ic Fields

    We can relate an induced current in a conducting loop to an elecWe can relate an induced current in a conducting loop to an electric field by claimingtric field by claimingthatthat an electric field is created in the conductor as a result of thean electric field is created in the conductor as a result of the changingchanging

    magnetic fluxmagnetic flux..

    The induction of a current in the loop implies the presenceThe induction of a current in the loop implies the presence

    of an induced electric fieldof an induced electric field EE, which must be tangent to the, which must be tangent to theloop because that is the direction in which the charges in theloop because that is the direction in which the charges in the

    wire move in response to the electric force. The work donewire move in response to the electric force. The work done

    by the electric field in moving a test chargeby the electric field in moving a test charge qqonce aroundonce around

    the loop is equal tothe loop is equal to qq..

    A conducting loop of radius r in aA conducting loop of radius r in a

    uniform magnetic field perpendicularuniform magnetic field perpendicular

    to the plane of the loop. If Bto the plane of the loop. If Bchanges in time, an electric field ischanges in time, an electric field is

    induced in a direction tangent to theinduced in a direction tangent to the

    circumference of the loop.circumference of the loop.

    Because the electric force acting on the charge isBecause the electric force acting on the charge is q.q.EE, the, the

    work done by the electric field in moving the charge oncework done by the electric field in moving the charge once

    around the loop isaround the loop is q.E(2q.E(2r)r), where, where 22rr is theis the

    circumference of the loop. These two expressions for thecircumference of the loop. These two expressions for the

    work done must be equal; therefore,work done must be equal; therefore,

    )2/()2( rEThusrqEq ==

    Using this result along with Equation 31.1 and thatUsing this result along with Equation 31.1 and that

    BB == BA = BBA = B rr22

    for a circular loop, the inducedfor a circular loop, the inducedelectric field can be expressed as:electric field can be expressed as:

    dt

    dBr

    dt

    d

    rE B ==

    22

    12

    Ch t 8 F dCh t 8 F d LL

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    Home Work 8:Home Work 8:

    Solve the following problems starting page 887, 7Solve the following problems starting page 887, 7thth edition of the textbook:edition of the textbook: P31.1,P31.1,

    31.7, 31.8, 31.18, 31.22, 31.27, 31.31, and 31.35.31.7, 31.8, 31.18, 31.22, 31.27, 31.31, and 31.35.

    Chapt er 8: FaradayChapt er 8: Faraday s Laws Law

    TheThe emfemf for any closed path can be expressed as the line integral offor any closed path can be expressed as the line integral of over thatover that

    path: In more general cases,path: In more general cases, EEmay not be constant and the path may notmay not be constant and the path may not

    be circle. Hence, Faradaybe circle. Hence, Faradays law of induction,s law of induction, == -- ddBB//dtdt, can be written in the general, can be written in the generalform:form:

    s.E

    d.s.E

    d=

    =

    dt

    dd BsE. General form of FaradayGeneral form of Faradays laws law

    The induced electric fieldThe induced electric field EE in equation above is a nonin equation above is a non--conservative field that isconservative field that is

    generated by a changing magnetic field.generated by a changing magnetic field.

    Example 31.7 Electric Field Induced by a Changing Magnetic FieldExample 31.7 Electric Field Induced by a Changing Magnetic Field in a Solenoid (Pagein a Solenoid (Page

    879)879)