1 electromagnetic induction taking it to the maxwell

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ElectromagneticElectromagneticInductionInduction

Taking It To The MaxwellTaking It To The Maxwell

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Michael Faraday

(1791 – 1867)

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IntroductionIntroductionWe’ve discussed two ways in which We’ve discussed two ways in which

electricity and magnetism are electricity and magnetism are related:related:(1) an electric current produces a (1) an electric current produces a magnetic field.magnetic field.(2) a magnetic field exerts a force on (2) a magnetic field exerts a force on an current carrying wire or moving an current carrying wire or moving electric charge.electric charge.

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IntroductionIntroduction

These discoveries were made in 1820 These discoveries were made in 1820 – 1821.– 1821.

Scientists wondered: Scientists wondered: If electric If electric currents produce a magnetic currents produce a magnetic field, does a magnetic field field, does a magnetic field produce electric currents?produce electric currents?

Joseph Henry (1797 – 1878) and Joseph Henry (1797 – 1878) and Michael Faraday (1791 – 1867) Michael Faraday (1791 – 1867) independently found, ten years later, independently found, ten years later, that this is so.that this is so.

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Hey Mikey!Hey Mikey!

Faraday found that Faraday found that changing a magnetic field produces a current.

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

To move charges requires a force and To move charges requires a force and this force isthis force is

Called the Called the ELECTROMAGNETIC ELECTROMAGNETIC FORCE,FORCE,EMF for short. for short.

You know it as You know it as VOLTSVOLTS of potential. of potential.

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Mikey found that a steady current in X produced no current in Y. Only when the current in X was starting or stopping (i.e., changing) was a current produced in Y.

Faraday suspected that a magnetic field would induce a current, just like a current produces a magnet field.

Induced Current

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The Big IdeaThe Big Idea

To induce a current in a wire ….To induce a current in a wire ….The MAGNETIC FIELD MUST BE

CHANGING WITH RESPECT TO TIME

There are a number of differential equations to describe this, but why complicate things?

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Since there is no change in the magnetic flux, no current is induced.

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I. Distance between coil and magnet decreases.

So the magnetic field (therefore the flux) through the coil increases.

II. To oppose this upward increase in the magnetic filed (flux), the field produced by the induced current points downward.

III. Current is induced.

The Status QuoThe Status Quo

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III. Current is induced in the opposite direction as the previous case.

I. Distance between magnet and coil increases.

So the magnetic field (and therefore the flux) decreases.

II. To oppose the decrease in the upward magnetic field (flux), the induced current produces an upward magnetic field, trying to maintain the “status quo.”

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The Big Idea!The Big Idea!

The induced current moves such that The induced current moves such that its magnetic field tends to oppose its magnetic field tends to oppose and resist the bar magnet’s moving and resist the bar magnet’s moving field.field.

Essentially it wants to Essentially it wants to DAMPENDAMPEN the the other field.other field.

Nature wants the two fields to be in Nature wants the two fields to be in harmony.harmony.

This is This is LENZ’S LAWLENZ’S LAW and it explains and it explains why the magnet falls slowly through why the magnet falls slowly through the copper pipe!the copper pipe!

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Lenz’s LawLenz’s Law

An induced emf always gives rise to a current whose magnetic field opposes the original change in flux.

An induced emf is always in a An induced emf is always in a direction that opposes the direction that opposes the original change in flux that original change in flux that caused it.caused it.

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The Flux CapacitorThe Flux Capacitor

The measure of how the field changes The measure of how the field changes has to do with an amount, or area, of has to do with an amount, or area, of coil or loop of wire exposed to the coil or loop of wire exposed to the field.field.

It doesn’t matter if the field is in It doesn’t matter if the field is in motion, changing in intensity, or if motion, changing in intensity, or if the coil is moving.the coil is moving.

So long as – So long as – FROM THE POINT OF FROM THE POINT OF VIEW OF THE WIRE - the B field VIEW OF THE WIRE - the B field appears to be changing with respect appears to be changing with respect to time.to time.

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The Flux CapacitorThe Flux Capacitor

Magnetic flux is a measure of field Magnetic flux is a measure of field strength B over an area measured in strength B over an area measured in mm22..

Think of it this way. The absolute Think of it this way. The absolute amount of rain fall is 2” per hour amount of rain fall is 2” per hour over the entire state.over the entire state.

We only care about a small part, We only care about a small part, namely a square foot.namely a square foot.

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The Flux CapacitorThe Flux Capacitor

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Phi, Fi, Pho, Phum?Phi, Fi, Pho, Phum?

The unit of flux is the Henry, named The unit of flux is the Henry, named for Joseph Henry. The symbol is the for Joseph Henry. The symbol is the capitol Greek letter Phi….capitol Greek letter Phi….

Φ = B Φ = B XX A A

Φ = B A cosθΦ = B A cosθ

Henry = Tesla • mHenry = Tesla • m22

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Case 1Case 1

Motion

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A current can be induced by changing the area of the field exposed to the coil. Here the area depends on moving the coil into the field.

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A current can be induced by changing the area of the coil exposed to the field by collapsing the coil. Here, the area is changed by shrinking the ring.

Area through the coil decreases

Therefore

Case 2Case 2

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A current can be induced by changing the area of the coil exposed to the field. Here, the area is changed by rotating with respect to the field.

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This side is coming toward you

Case 3 – A GeneratorCase 3 – A Generator

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How to Induce an EMFHow to Induce an EMF

An emf can be induced whenever there is a change in flux.

Since B = BA cos an emf can be induced in three ways:

by a changing magnetic field Bby changing the area of the loop in the

fieldby changing the loop’s orientation

with respect to the field.

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Fleming’s LHR RevisitedFleming’s LHR Revisited

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I

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Area has increased

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dx =

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F

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Umm….SorryUmm….Sorry

But we simply have to have a But we simply have to have a differential equation.differential equation.

Emf = Emf = change in Fluxchange in Flux = B Field x = B Field x Change in AreaChange in Area

change in time change in time change in timechange in time

Emf = B Field x length x velocity x Emf = B Field x length x velocity x change in timechange in time

change in timechange in time

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EMF Induced in a EMF Induced in a Moving ConductorMoving Conductor

= = = = Blv dB B dA Blv dtdt dt dt

This equation is valid as long as B, l, and v are mutually perpendicular.

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Rotating clockwise

F = qv x B is the force on the charges in the wire that produces a current.

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Faraday’s Law of Faraday’s Law of InductionInduction

= = NNdB

Faraday found, experimentally, that the magnitude of the induced emf is proportional to the:

rate of change of magnetic flux.

dt

N = Number of loops of wire

Lenz’s law

Induced emf

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Example 29-5Example 29-5

An ac generator.An ac generator.

The armature of a 60-Hz ac The armature of a 60-Hz ac generator rotates in a 0.15-Tgenerator rotates in a 0.15-T magnetic field. If the area of the coil magnetic field. If the area of the coil is 2.0 x 10is 2.0 x 10-2 -2 mm22, how many loops must , how many loops must the coil contain if the peak output is the coil contain if the peak output is

to be to be 00 = 170 V? = 170 V?

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DC GeneratorDC Generator

A A dc generatordc generator is much like an ac is much like an ac generator or generator or alternatoralternator, except the , except the slip rings are replaced by split-ring slip rings are replaced by split-ring commutators, just as in a dc motor.commutators, just as in a dc motor.

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29-6 29-6 Transformers and the Transformers and the Transmission of PowerTransmission of Power

A A transformer transformer is a device for is a device for increasing or decreasing an ac increasing or decreasing an ac voltage.voltage.

It consists of two coils of wire known It consists of two coils of wire known as the as the primaryprimary and and secondarysecondary coils. coils.

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29-6 29-6 Transformers and the Transformers and the Transmission of PowerTransmission of Power

A A transformertransformer is a device for is a device for increasing or decreasing an ac increasing or decreasing an ac voltage.voltage.

A transformer may be a A transformer may be a step-up step-up transformertransformer (increasing voltage) or (increasing voltage) or a a step-down transformer step-down transformer (decreasing voltage).(decreasing voltage).

A transformer consists of two coils of A transformer consists of two coils of wire known as wire known as primary primary (voltage (voltage input) and input) and secondarysecondary (voltage (voltage output) coils.output) coils.

==

VS VP

NS NP

Transformer equation:

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Example 29-8Example 29-8

Portable radio transformer.Portable radio transformer.

A transformer for home use of a A transformer for home use of a portable radio reduces 120-V ac to portable radio reduces 120-V ac to 9.0-V ac. The secondary contains 30 9.0-V ac. The secondary contains 30 turns and the radio draws 400 mA. turns and the radio draws 400 mA. Calculate (Calculate (aa) the number of turns in ) the number of turns in the primary; (the primary; (bb) the current in the ) the current in the primary; and (primary; and (cc) the power ) the power transformed. transformed.

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Example 29-9Example 29-9

Transmission lines.Transmission lines.

An average of 120 kW of electric An average of 120 kW of electric power is sent to a small town from a power is sent to a small town from a power plant 10 km away. The power plant 10 km away. The transmission lines have a total transmission lines have a total resistance of 0.40 resistance of 0.40 . Calculate the . Calculate the power loss if the power is power loss if the power is transmitted at (transmitted at (aa) 240 V and () 240 V and (bb) ) 24,000 V.24,000 V.

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29-7 29-7 Changing Magnetic Changing Magnetic Flux Produces an Electric Flux Produces an Electric

FieldField

A A changing magnetic fluxchanging magnetic flux induces an emf and a current in induces an emf and a current in a conducting loop. Therefore, it a conducting loop. Therefore, it produces an electric fieldproduces an electric field..

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Changing Magnetic Flux Changing Magnetic Flux Produces an Electric FieldProduces an Electric Field

Changing magnetic flux induces Changing magnetic flux induces current-emf-electric field.current-emf-electric field.

An electric field is An electric field is always always generated generated by a changing magnetic flux, by a changing magnetic flux, even even in free spacein free space where no charges are where no charges are presentpresent

Flux-induced field has properties Flux-induced field has properties different from electrostatic electric different from electrostatic electric field produced by stationary field produced by stationary charges.charges.

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Faraday’s Law – General Faraday’s Law – General FormForm

E E dldl == dB dt

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Forces Due to Changing Forces Due to Changing B are NonconservativeB are Nonconservative

An electric field is induced by a An electric field is induced by a changing magnetic field, even in the changing magnetic field, even in the absence of a conductor.absence of a conductor.

The induced electric field The induced electric field E E thatthat appears in the in the previous appears in the in the previous equation, is a nonconservative, time-equation, is a nonconservative, time-varying field produced by a changing varying field produced by a changing magnetic field.magnetic field.

Electric field lines produced by Electric field lines produced by static charges stop and end on static charges stop and end on charges.charges.

Electric field lines produced by a Electric field lines produced by a changing magnetic field are changing magnetic field are continuous; they form closed loops.continuous; they form closed loops.

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Electrostatic E Field

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Forces Due to Changing Forces Due to Changing B are NonconservativeB are Nonconservative

The fact that the integral of The fact that the integral of E E ddll around a closed path is zero follows around a closed path is zero follows from the fact that electro static force from the fact that electro static force is a conservative force, and so a is a conservative force, and so a potential energy function could be potential energy function could be defined. defined.

The above tells us that the work The above tells us that the work done per unit charge around any done per unit charge around any closed path is zero.closed path is zero.

That is the work done between any That is the work done between any two points is independent of the two points is independent of the path, which is a property of a path, which is a property of a conservative force.conservative force.

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Forces Due to Changing Forces Due to Changing B are NonconservativeB are Nonconservative

But the generalized form of But the generalized form of Faraday’s law tells us that the Faraday’s law tells us that the integral around a closed path is not integral around a closed path is not zero.zero.

Thus we are unable to define a Thus we are unable to define a potential energy function.potential energy function.

Thus we conclude that when the Thus we conclude that when the electric force is produced by a electric force is produced by a changing magnetic field the forces changing magnetic field the forces are not conservative.are not conservative.

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Example 29-10Example 29-10

E E produced by changing B.produced by changing B.

A magnetic filed A magnetic filed B B between the pole between the pole faces of an electromagnetic is nearly faces of an electromagnetic is nearly uniform at any instance over a uniform at any instance over a circular area of radius circular area of radius rr00. The . The current in the windings of the current in the windings of the electromagnet is increasing in time electromagnet is increasing in time so that so that BB changes in time at a changes in time at a constant rate constant rate ddBB//dtdt at each point. at each point. Beyond the circular region (Beyond the circular region (rr > > rr00), ), we assume we assume BB = 0 at all times. = 0 at all times. Determine the electric field Determine the electric field EE at any at any point P a distance point P a distance rr from the center from the center of the circular area.of the circular area.

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Changing Magnetic Flux Changing Magnetic Flux Produces an Electric FieldProduces an Electric Field

Electrons in the moving conductor Electrons in the moving conductor must feel a force since there is a must feel a force since there is a current.current.

A force implies that there is an A force implies that there is an electric field in the conductor.electric field in the conductor.

Therefore, we conclude that a Therefore, we conclude that a changing magnetic flux produces an changing magnetic flux produces an electric field.electric field.

E = = = vBF qvBq q

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IFB

Fext = ?

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IFB

Fext = ?