VIII. Magnetic Induction Dr. Bill Pezzaglia Updated 2014Mar.

VIII. Magnetic Induction

Dr. Bill Pezzaglia

Updated 2014Mar

VIII. Magnetic Induction

A. Dynamos & Generators

B. Faraday’s Law

C. Inductance

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Michael Faraday (1791 - 1867)

• 1821 Creates first motor

• 1831 Creates “dynamo”, a DC generator

• 1831 Law of induction

• 1846 Discovers Diamagnetism

3

A. Dynamos & Generators

1) Dynamo Rule

2) Motional EMF

3) Mechanical to Electric Power

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1. Dynamo Rule

(a) Faraday (1831)• Move a wire so that it “cuts”

magnetic field lines will generate current

• Recall

• Positive charges in wire will move one direction, negative the other

5

BvqF

1.(a).ii Planetary Dynamos

Moon Io moving orbiting through Jupiters magnetic field generates BIG current

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1.(a).iii Planetary Dynamos

The electric currents generate so much heat the moon has active volcanoes!

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(b) Magnetohydrodynamic Generators (MHD)

1832 Faraday’s Fluid Generator experiment at Waterloo Bridge: attempts to measure current generated from velocity of Thames cutting through earth’s magnetic field (didn’t work too well).

8

BvqF

+

-

(b).ii MHD Generators 9

(c). Faraday’s Disk Dynamo (1831)

This was the first practical DC generator. It gives high current, but low voltage.

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I

2. EMF (Electromotive Force)

(a) Definition of EMF ()• Misnomer: its not really a “force”, it’s a

“voltage” (i.e. ENERGY per charge)

• Chemical EMF: A battery is like a “pump”. When charges pass through, their energy is increased. Change in voltage=“EMF”

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q

Work

2.(b). Motional EMF

• Magnetic (“Dynamo”) Work done by moving a length “L” wire through magnetic field “B” with velocity “v” generates an EMF

• General Result, only the partof velocity, magnetic field andwire path which are mutuallyperpendicular will contribute.

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q

LqvB

q

FL

q

Work )(

LBv

)(

2.(c). AC Generator1892 Tesla (working for Edison) invents AC Generator. Edison hates the idea (he is using DC dynamos), and so Tesla sells it to Westinghouse. Edison goes on a campaign to convince people AC is DANGEROUS while DC is safe.

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http://www.youtube.com/v/i-j-1j2gD28

2.(c).ii AC Generator Details

• Rectangle area: A=ab

• Velocity:

• EMF:

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2

av

sin)2(2

bBa

sinBA

sin)( vBLLBv

t

3. Mechanical to Electric Power

(a) Electric Power Output:

(b) Current created will experience force from magnetic field, which is in opposite direction as “v”

(c) Mechanical Work: by Newton’s 3rd law, we must PUSH the wire through with force F, or power:

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BLIF

IP

ILBVIvILBvFP )()(

B. Faraday’s Law

1) Magnetic Flux

2) Faraday’s Law

3) Lenz’s Law

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1831 Faraday’s Three Experiments

Generated current by:

• Moving a coil in and out of a magnetic field

• Moving magnet in and out of coil

• When current turned on or off in a coil, current is generated in a nearby coil.

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He explained all of these effects with one single law

Michael Faraday

1. Magnetic Flux

(a) Definition:• Magnetic Flux is the Magnetic Field “B”

(aka “Magnetic Intensity Vector”) times area it “flows” through

• Units: Weber=Teslam2

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AB

Greek Letter: “phi” or

(b) Lambert’s Law (Orientation matters!)

Sunlight coming in at a low altitude angle will have its energy spread out over more area.

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Lambert’s Law (1760)Intensity is reduced by cosine of angle of incidence

Flux is the dot product of the electric field vector with the “area vector” (which is “normal” to the surface)

cosBAAB

(c) Magnetic Flux is Conserved

• Because there are no magnetic monopoles, there are no “sources” of magnetic field lines.

• Magnetic Field Lines must be continuous (i.e. continue through magnet)

• Gauss’s law for magnetism: total magnetic flux through a closed surface is ZERO.

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2. Faraday’s Law of Induction (1831)

• Possibly done 1830 by Henry (unpublished)

• The EMF generated in a loop of wire is equal to the change in magnetic flux through the loop (with respect to time)

• You can get a change of flux in 2 ways

21

t

)()( ABAB

(a) Change the Area of Loop

• Consider sliding wire in constant magnetic field

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t

AB

t

vLt

A

BvL)(• This is equivalent to

Dynamo Rule (motional emf)

(b) Or, change the orientation of loop

• Consider AC generator, where we twist the loop

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cosBA

t

BAt

sin

sinBA• This is equivalent to Dynamo

Rule (motional emf)

(c) Change Magnetic Field

• You can NOT explain this one by dynamo rule, as no wires move!

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• However, it makes perfect sense because motion is relative. Whether you think the magnet is moving with the coil still, or the magnet still with the coil moving merely depends upon the motion of the observer!

t

BA

t

3. Lenz’s Law (1834)(a) The law: Direction of induced current is such as to oppose the cause producing it.

• It’s the minus sign in Faraday’s Law

25

t

Heinrich Lenz1804-1864

(b). Eddy Currents

• 1824, François Arago discovers when conductor is exposed to changing magnetic field, small circular “eddy currents” (also known as Foucault currents) are generated.

• 1855, J.B.L. Foucault rotates a copper disc with rim between poles of magnet and discovers that the induced eddy currents in the metal cause:

• Inductive Braking: the force required for the rotation of a copper disc becomes greater with magnet (even though copper is not attracted by a magnet)

• Inductive Heating: the disc becomes heated by the eddy current (i.e. “friction”), because of resistance of metal.

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http://www.magnet.fsu.edu/education/community/slideshows/eddycurrents/index.html

Braking (tubes) http://www.youtube.com/watch?v=sPLawCXvKmg

(c). Magnetic Levitation

• Lenz’s law is usually demonstrated by “Elihu Thomson's jumping ring” (1887? 1897?).

• The induced current in the metal ring creates an electromagnetic that opposes the applied field, lifting the ring.

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Jumping Ring: http://www.youtube.com/watch?v=Pl7KyVIJ1iE

(d). Magnetic Levitation

• Meissner Effect (1933) for superconductors, the eddy currents are so strong that they become perfectly diamagnetic, cancelling the external field, inducing magnetic levitation.

• Application: Magnetically levitated trains!

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Meissner Effect: http://www.youtube.com/watch?v=94-Z2QgHl-s&feature=related Train: http://www.youtube.com/watch?v=GHtAwQXVsuk

JR-Maglev EDS suspension is due to the magnetic fields induced either side of the vehicle by the passage of the vehicle's superconducting magnets. --Wikipedia

C. Inductance

1) Mutual Induction (Transformers)

2) Self Inductance & RL circuits

3) Energy in Inductors

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1. Mutual Induction

(a) 1831 Faraday notes when current turned on or off in a coil, voltage is generated in a nearby coil.

• Define Mutual Inductance “M” (units of Henries=ohmsec)

• If first coil is solenoid, then can show

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t

IM

1212

1

221021 A

NNM

(a).ii Coefficient of Coupling [DETAILS]

• Loosely Coupled: if coils are far apart, not all of magnetic flux from first coil goes through the second (flux leakage). Early transformers were very inefficient because of this.

• Tightly Coupled: either have coils wrapped around each other, or share same iron core so that nearly all flux from one goes through other.

• Reciprocity: If make coils same length with same area, and tightly coupled, then mutual inductance is same both ways (“L” is self inductance to be discussed in next section)

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211221 LLMM

(b) Ruhmkorff Induction Coil• Probably first invented by invented by Nicholas

Callan in 1836.

• 1851 Ruhmkorff shows if secondary coil has many more windings than primary, then a BIG voltage can be generated from a small one.

• DC current in primary creates magnetic field

• Current is periodically “interrupted” by a vibrating switch, causing field to collapse

• BIG voltage is generated in secondary by Faraday’s law

• This was how early Cathode Ray, X-ray and “neon signs” were powered.

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https://www.youtube.com/watch?v=1C4lOAPBu7A

(c) Transformer Equation

• Nikola Tesla pioneered the idea of using AC current, transmitting power at high voltages (low current) and then using transformers to step it back down to low voltage for user.

• 1884 “Closed Flux Transformer” invented: the flux through both coils is the same:

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2

1

1

2

1

2

I

I

N

N

V

V

tNV

tNV

11

22 Divide the equations to get the transformer rule:

Note that power is conserved!

2211 IVIV

2. Self Inductance

(a) Probably done first by Joseph Henry 1830, a year before Faraday.

• Current in coil makes a magnetic field. Change in current changes field, which by Faraday’s law creates a voltage in the coil (by Lenz law a “back emf” which opposes change). Definition of (self) inductance:

• Details: Self Inductance “L” is a function only of geometric (and magnetic permeability) of coil. Generally goes like square of number of turns N. For solenoid of length l and area “A” with iron core of permeability :

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t

ILV

I

BNL

A

NL 2

b. Inductor Combinations

• Rules for networks of inductors is similar to resistors (hence opposite of capacitors)

• In Series: Inductors ADD

• In Parallel: Inductors combine:

35

21

111

LLL

21

21

LL

LLL

21 LLL

or

(c) Inductor Behavior

• In brief, Inductors resist change in current

• The inductor has zero current initially, and when you close the switch, the inductor creates a back voltage to resist current trying to flow through it. Instead the current flows freely through the light bulb.

• Over time, the current starts to flow through the inductor and energy is stored in the magnetic field it creates.

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• When you open the switch, the inductor try to keep the current flowing, so it will attempt to supply any voltage necessary to do so. The bulb will light until the magnetic field collapses (exponentially). However, the BIG voltage surge might burn the bulb out!

Demo of Back EMF: http://www.youtube.com/watch?v=aSmMFog10D0

(c) RL Circuits (incomplete)

• No notes here, see section 26.5 in book

• In brief, RL circuits behave complementary to RC circuits. Inductors try to keep the current flowing, so if there is an abrupt loss of current, inductors will attempt to supply any voltage necessary to keep the current flowing, the current will decay exponentially with time constant:

• Complementary, the RC circuit: capacitors try to keep the voltage the same, so if there is an abrupt loss of voltage, the capacitor will attempt supply any current necessary to keep the voltage up. The voltage will decay exponentially with time.

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R

L

ItI t

exp)( 0

3. Energy Stored in Inductor

• Energy in Inductor:

• You can think of it as “kinetic energy of current” stored in the inductor (whereas a capacitor stores “potential energy” of charge)

• Using the equations for magnetic field and inductance of a solenoid coil, you can show:

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221 LIU

volumeBU 2

2

1

b. Energy Density

• Divide by volume (of a solenoid coil)

• In other words, the energy of an inductor is stored in the magnetic field itself! [Whereas for a capacitor its stored in the electric field]

• This energy creates a “magnetic pressure” that tries to push a coil apart

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2

2

1B

vol

Uu

c. Forces on Cores etc.

• If you insert a core, you increase the inductance by factor Km:

• Hence, increase the energy. Hence inductors will want to push cores out (doorbell or induction gun).

• By inserting core, you change inductance, so more correct formula for voltage (with possible sign errors):

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t

LI

t

ILV

LKL

K

m

m

0

References

•1831 Faraday Law of Induction•Neat video: http://www.youtube.com/watch?v=yahQtxBV5vA&feature=related•http://www.youtube.com/watch?v=VPxdl1zpcC8&feature=related•Magnetic flux and orientation: http://www.youtube.com/watch?v=KXFXUrBWp-c&feature=related

•AC generator http://www.youtube.com/watch?v=gqA3WoOunEA&feature=related•Even better AC generator: http://www.youtube.com/watch?v=i-j-1j2gD28&NR=1•Variable AC generator: http://www.youtube.com/watch?v=mCvXa_VVFh4&feature=related•Simulations: http://www.esjd.pt/recursos_educativos/phet1.0/new/simulations/index1057.html?cat=All_Sims

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Schedules/Reading

In Knight “College Physics” (2nd ed)•Faraday’s Law, Chapter 25.1-25.4•Transformer 26.2•RL circuit 26.5, and lab notes.

•Skip 25.5-25.8 for now (Maxwell’s Equations)

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