Chapter 30[1].Magnetic

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Chapter 30. Induction and Inductance

30.1. What is Physics?30.2. Two Experiments30.3. Faraday's Law of Induction30.4. Lenz's Law

30.5. Induction and Energy Transfers30.6. Induced Electric Fields30.7. Inductors and Inductance30.8. Self-Induction30.10. Energy Stored in a Magnetic Field30.11. Energy Density of a Magnetic Field30.12. Mutual Induction

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What is Physics?

Can a magnetic field produce an electric

field that can drive a current?

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Relative motion between a magnet and a coil

Changing the area of a coil

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Conductor moving in the magnetic field

the number of magnetic field lines that passthrough the loop is changing.

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The current in the coil induced by a changingmagnetic field or changing the area of a coil methodsis called an induced current. A closed circuit isnecessary for the induced current to flow.

The emf produced in the coil which drives theinduced current is called the "induced emf". Theinduced emf exists whether or not the coil is part ofa closed circuit.

The phenomenon of producing an induced emf withthe aid of a magnetic field is called electromagneticinduction.

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What is the cause of induced emf?

The number of magnetic field lines that passthrough the loop is changing.

The faster the number of magnetic field lines thatpass through the loop changes, the greater theinduced emf

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MAGNETIC FLUX

This unit is called a weber(Wb), after the German physicist WilhelmWeber: 1 Wb = 1 T m2

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Example. Magnetic Flux

A rectangular coil of wire is situated in a constantmagnetic field whose magnitude is 0.50 T. The coil hasan area of 2.0 m2. Determine the magnetic flux for

the three orientations,=0, 60.0, and 90.0, shownin Figure.

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Faraday's Law of Induction

The magnitude of the emf induced in aconducting loop is equal to the rate at whichthe magnetic flux through that loop changeswith time.

If we change the magnetic flux through a coil of Nturns, an

induced emf appears in every turn and the total emfinduced in the coil is the sum of these individual inducedemfs.

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Check Your UnderstandingA coil is placed in a magnetic field, and thenormal to the plane of the coil remainsparallel to the field. Which one of the

following options causes the average emfinduced in the coil to be as large as possible?(a) The magnitude of the field is small, and itsrate of change is large. (b) The magnitude of

the field is large, and its rate of change issmall. (c) The magnitude of the field is large,and it does not change.

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Sample

The long solenoid S shown (in cross section) in Fig. 30-3 has 220

turns/cm and carries a current i=1.5 A ; its diameter Dis 3.2cm. At its center we place a 130-turn closely packed coil C ofdiameter d=2.1 cm. The current in the solenoid is reduced tozero at a steady rate in 25 ms. What is the magnitude of theemf that is induced in coil C while the current in the solenoidis changing?

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Lenz's Law

An induced current has a direction such thatthe magnetic field due to the currentopposesthe change in the magnetic flux that induces

the current.

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Example The Emf Produced by aMoving Copper Ring

In Figure there is a constantmagnetic field in arectangular region of space.This field is directed

perpendicularly into the page.Outside this region there isno magnetic field. A copperring slides through theregion, from position 1 to

position 5. For each of thefive positions, determinewhether an induced currentexists in the ring and, if so,find its direction.

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Sample ProblemFigure 30-8 shows a conducting loop consisting of a

half-circle of radius r=0.20m and three straight

sections. The half-circle lies in a uniform magneticfield that is directed out of the page; the fieldmagnitude is given by B=4.0t2+2.0t+3.0, with Binteslas and tin seconds. An ideal battery with emfbet=2.0V is connected to the loop. The resistance ofthe loop is 2.0.

(a) What are the magnitude and direction of the emfinduced around the loop by B field at t=10 s?

b) What is the current in the loop at t=10 s?

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Example

An electromagnet generates a magnetic field which"cuts" through a coil as shown. What is the polarity ofthe emf generated in the coil if the applied field, B(a) points to the right and is increasing? (b) points tothe right and is decreasing? (c) is pointing to the left

and increasing? (d) is pointing to the left anddecreasing?

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An AC Generator

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Induction and Energy TransfersYou pull a closed conducting loop out of a magnetic field at

constant velocity v. While the loop is moving, a clockwise

current iis induced in the loop, and the loop segmentsstill within the magnetic field experience forces F1, F2and F3.

The rate at which you do work is:

1F iL B

The rate at which thermal energy appears in the loop:

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Checkpoint

The figure shows four wire loops, with edge lengths ofeither Lor 2L. All four loops will move through a regionof uniform magnetic field B (directed out of the page)at the same constant velocity. Rank the four loopsaccording to the maximum magnitude of the emf induced

as they move through the field, greatest first.

I d d El i Fi ld

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Induced Electric Fields Let us place a copper ring of radius rin a uniform

external magnetic field. Suppose that we increasethe strength of this field at a steady rate.

If there is a current in the copper ring, an electricfield must be present along the ring because anelectric field is needed to do the work of moving theconduction electrons. It is called as induced electricfield .

As long as the magnetic field is increasingwith time,the electric field represented by the circular fieldlines in Fig. cwill be present. If the magnetic fieldremains constantwith time, there will be no inducedelectric field and thus no electric field lines.

A changing magnetic field produces anelectric field.

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Comparison between Induced electric

fields and static electric fields

Electric fields produced in either way exert forceson charged particles: F=qE

The field lines of induced electric fields form closed

loops. Field lines produced by static charges never doso but must start on positive charges and end onnegative charges.

0E ds

For electric fields that are produced by staticcharges, , therefore, Electric potential hasmeaning; for electric fields that are produced byinduction, , therefore, electric potentialhas no meaning.

0E ds

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A Reformulation of Faraday's Law

Consider a particle of charge q0moving

around the circular path of Fig. b. The workWdone on it in one revolution by theinduced electric field is W=q0, where isthe induced emf

From another point of view, the work is

Faraday's law

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Inductors and Inductance consider a long solenoid (more specifically, a

short length near the middle of a long solenoid)as our basic type of inductor (symbol ) toproduce a desired magnetic field

The inductance of the inductor is

Unit is:

Inductance of a solenoid:

S lf I d ti

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Self-Induction

An induced emf appears in anycoil in which the current is changing.

This process is called self-

induction, and the emf that appearsis called a self-induced emf.

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Checkpoint

The figure shows an emf induced in a coil. Whichof the following can describe the current through thecoil: (a) constant and rightward, (b) constant andleftward, (c) increasing and rightward, (d) decreasing

and right-ward, (e) increasing and leftward, (f)decreasing and leftward?

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Energy Stored in a Magnetic Field

The left side of Eq. represents the rate atwhich the emf device delivers energy to the restof the circuit.

The rightmost term represents the rate atwhich energy appears as thermal energy in theresistor.Energy that is delivered to the circuit but doesnot appear as thermal energy must, by theconservation-of-energy, be stored in the

magnetic field of the inductor.

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Energy Density of a Magnetic FieldConsider a length lnear the middle of a long solenoid of cross-sectional areaAcarrying current i; the volume associated with this length is Al.

The energy stored per unit volume of the field is

2

0L n lA

S mpl P bl m

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Sample ProblemA long coaxial cable consists of two thin-walledconcentric conducting cylinders with radii aand b. The

inner cylinder carries a steady currenti, and the outercylinder provides the return path for that current. The

current sets up a magnetic field between the twocylinders. (a) Calculate the energy stored in themagnetic field for a length of the cable. (b) What is

the stored energy per unit length of the cable ifa=1.2mm, b=3.5mm , and i=2.7A ?

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Mutual InductionThe mutual inductance M21of coil 2 with respect to coil 1 as

Is a magnetic flux through coil2 associated with the current in

coil 1

21

2

d

dt

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Sample Problem

Figure 30-26 shows two circular close-packed coils, the

smaller (radius R2, with N2 turns) being coaxial with thelarger (radius R1 with N1 turns) and in the same plane.

Derive an expression for the mutual inductance M for this

arrangement of these two coils, assuming that R1 >>R2.