Chapter 13 Electromagnetic Induction

In this chapter we will study the following topics:

-Faraday’s law of induction -Lenz’s rule -Electric field induced by a changing magnetic field -Eddy Currents -Electric Generators

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In a series of experiments Michael Faraday in England and Joseph Henry in the US were able to generate electric currents without the use of batteries Below we describe some of the

Faraday's experiments

se experiments thathelped formulate whats is known as "Faraday's lawof induction"

The circuit shown in the figure consists of a wire loop connected to a sensitiveammeter (known as a "galvanometer"). If we approach the loop with a permanent magnet we see a current being registered by the galvanometer. The results can be summarized as follows:

A current appears only if there is relative motion between the magnet and the loop Faster motion results in a larger current If we

1.2.3. reverse the direction of motion or the polarity of the magnet, the currentreverses sign and flows in the opposite direction. The current generated is known as " "; the emf that appears induced current is known as " "; the whole effect is called " "induced emf induction 2

Faraday'sLawof InductionIn the figure we show a second type of experimentin which current is induced in loop 2 when the switch S in loop 1 is either closed or opened. Whenthe 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 begenerated either by a permanent magnet or by anelectric current in a coil.

loop 1 loop 2

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.

Faraday's law is not an explanation of induction but merely a description of of what induction is. It is one of the four "Maxwell's equations of electromagnetism"all of which are statements of experimental results. We have already encounteredGauss' law for the electric field, and Ampere's law. 3

B!

n̂

dA

Magnetic Flux ΦB =!B i!A

The magnetic flux through a surface that bordersa loop is determined as follows: 1. We divide the surface that has the loop as its borderinto area elements of area dA.

For each element we calculate the magnetic flux through it: cos

ˆHere is the angle between the normal and the magnetic field vectorsat the position of the element.

We integrate a

Bd BdA

n B

Φ =2.

3.

!

2: T m known as the Weber (symbol

ll the terms. cos

We can express Faraday's law of induction in the folowin W

g b)

form:

B BdA B dAΦ = =

⋅

⋅∫ ∫SI magnetic flux unit

!!

B

The magnitude of the emf induced in a conductive loop is equal to rateat which the magnetic flux Ö through the loop changes with time

E

B B dAΦ = ⋅∫!!

BddtΦ= −E

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d!Aθ

Changing of ΦB through a loop

ΦB =!B i!A = BAcosθ

ε = −dΦB

dt= BA d cosθ(t)

dtChange in Angle

" #$ %$+ B A(t)

dtChange inArea

&cosθ + B(t)

dtChange inB− field

&Acosθ

1. Change the angle θ between !B and

!A.

2. Change either the total area of the coil orthe portion of the area within the magnetic field.

3. Change the magnitude of B within the loop.

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B!

dA

d!Aθ

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B!

n̂

loop

1. An Example of a rotating wire frame

ΦB = NABcosθ = NabBcos ωt( )where θ =ωt

E = −dΦB

dt=ωNabBsin ωt( )

E =ωNabBsin ωt( ) ω = 2π f d!A

θ

x x x x x x x x x x x x x x x x

x x x x x x x x x x x x x x x x x x x x

t=0 Δt after

2. An Example of a change in B(t)

ΦB = NABcosθ = AB(t) here N = 1, θ = 0o

Let B(t) = 4T + (2Ts

)t let A = 1m2

E = −dΦB

dt=− A

dBdt

= −(1m2 )(2Ts

) = −2V

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3. An Example of a change in A(t)

ΦB = NABcosθ = A(t)B here N = 1, θ = 0o

The loop reduces from r = 1m to r = 0.5m in1s

A(t) = πr(t)2 let B = 1T

E = −dΦB

dt=− dA

dtB = −2πr(t)(

drdt

) ⋅B = −2π (1T )ΔrΔt

ΔrΔt

= (0.5−1.0)m1s

= −0.5m / s

E = 2π (1T )(0.5m / s) = 3.14V

Note the induced CW current adds B − field linesbackin to the smaller loop.

x x x x x x x x x x x x x x x x

t=0 Δt after x x x x x x x x x x x x x x x x

BddtΦ= −E

Reverse EMFThe negative sign in the equation that expresses Faraday's law indicates that the genrated EMF opposes the change in flux. This is known as a reverse or back EMF.

Lenz's RuleThe direction of the current induced in a conductor by a changing magnetic field is such that the magnetic field created by the induced current opposes the initial changing magnetic field.

In the figure we show a bar magnet approaching a loop. The flux through the loopis increasing. The induced current in the loop flows in the direction to cancel the building flux. The loop is equivalent to a magnet whose north pole faces the corresponding north pole of the bar magnet approaching the loop. The loop repelsthe approaching magnet and thus opposes the change in ΦB.

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Lenz's Rule : Opposition to flux changeAs the magnet approaches the conductive hoopa countercurrent develpes which opposes the increase in magnetic flux Φ. The countercurrent creates its own magnetic ficld which opposes that of the bar magnet. The hoop deflects away from the bar magnet.

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!

✖

!

✖

Induction, energy transferand PowerBy Lenz's rule, the induced current always opposesthe external agent that produced the induced current.Thus the external agent must always do work on theloop-magnetic field system. This work appears asthermal energy that gets dissipated on the resistance R of the loop wire. Consider the loop of width L shown in the figure. Part of the loop is located in a region where a uniform magnetic field B exists. The loop is beingpulled outside the magnetic field region with constantspeed υ. The magnetic flux through the loop ΦB = BA = BLx The flux decreases with time

E =dΦB

dt= BL

dxdt

= BLυ i =E

R= BLυ

R

Pth = i2R = (BLυ

R)2 R Pth =

B2L2υ 2

R

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DragForceExerted

P =!F ⋅!υ = B2L2υ 2

R

Fx =B2L2υx

RFdrag = Fhand

← → a = 0!

Eddy currentsNow replace the wire loop in the previous example with a solid conducting plate and move the plate out of the magnetic field as shown in the figure.The motion between the plate and

!B induces a current in the conductor and we

encounter an opposing dragforce. With the plate the free electrons do not follow one path as in the case of the loop. Instead the electrons swirl around the plate. These currents are known as "eddy currents". As in the case of the wire loop the net result is that mechanical energy that moves the plate is transformed into thermal energy that heats up the plate.The drag force is used in magnetic braking, rocket sleds,metal detection, frictionless rail, etc.

https://www.youtube.com/watch?v=MglUIiBy2lQ

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12

Eddy current

Cu tube

Nd Magnet

Caseof Drag Force on Magnet inFreefallIf I drop a strong magnet in to a Cu pipe a circulating counter eddy current will develop in the region near the magnetforming a countering drag force. This forceis independent of the magnet's orientation, always reacting to the changing flux lines from either pole of the magnet.

ma = mg - Fdrag = 0dυdt

= 0

υ =υ0 = cons tan t

The magnet will slowly float down at constant velocity. υ0

https://www.youtube.com/watch?v=nV4yA8OYwmE https://www.youtube.com/watch?v=otu-KV3iH_I

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Generationof ElectricityWe can convert mechanical energy in to electricity by Farday's Law ofin Induction.A common electrical general consists of a coil of N turns being rotated by mechanical

means in a magnetic field . Φ=NBAcos(ω t) ε = − d Φdt

= NBAω sin(ω t)

Let N = 200, B = 0.80 T, A=0.01m2,T = 100.0 ms. with ω=2π /T

ε=(200)(0.8T)(0.01m2 )(2π

(0.1s))sin(20π t) = ε0 sin(20π t) ε0 =101V = amplitude

ε(t) = ε0 sin(ωt)

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Pulsed DCWe can split the generators "armiture", the brushes that pick up the current fromthe coil, to invert the negative pulses, producing "pulsed DC". ε(t) = ε0 | sin(ωt) |

Time AveragedVoltage

Given f (t) then favg =1T

f (t)dt0

T

∫ε(t) = ε0 | sin(ωt) |

εavg =1T

ε0 | sin(2πT

t) |0

T

∫ dt

=2ε0

Tsin(

2πT

t)0

T /2

∫ dt=−2ε0

T(

T2π

) cos(2πT

t) |0T /2

εavg = −(επ

)[cos(π )− cos(0)]=2ε0

πεavg =

2πε0

STOP HERE

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