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Physics 101

Spring 2010

Lecture “whatever”

Electromagnetic Induction

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Agenda• Proffesor Benton is ill

– Terrence A. Mulera• HR 102 • X5701• mulera @usfca.edu• Hours: catch me when you can

• Faraday’s Law of Electromagnetic Induction– Examples

• Counter EMFs and Counter Torques• Eddy currents• General form of Faraday’s Law• Inductance

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http://terryspeaks.wiki.usfca.edu/

No lab this week

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sin

B B

B

d

t dt

B A BA

3 ways to generate an induced EMF

1) Change B

2) Change A

3) Change the orientation of B and A

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Example: Pulling a 100 loop coil from a magnetic field in t = 0.100 sec

22

3 2

Initial 0.600 T 5.00 10 m

1.50 10 Tm

At 0

B

B

BA

t

3 20 1.50 10 Tm100

0.100 sec1.50 Wb/sec

=1.50 V

B constt

Lenz’s law says EMF must be such that it acts to maintain the flux.

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If the 100 turn coil has a resistance of 100 1.5 V

15.0 mA100

IR

Lenz’s law current circulates clockwise

Energy dissipated

22 2 -31.50 10 A (100 )(0.100 sec) 2.25 10 JE Pt I Rt

Force exerted to pull coil out of field can be calculated 2 ways

(1) Directly: (100)(0.015 A)(0.600 T) 0.0450 NF NIlB

(2) Energy considerations:

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Energy dissipated E = work W needed to pull coil out of field

3

-2

2.25 10 J0.0450 N

5.00 10 m

WF

l

Example: Double the linear dimensions of coil by stretching, don’t move it. Same time interval

Stretch the coil

2 edge 4 area or 4

3

Bf Bi

B Bf Bi Bi

34 1.50 10 Wb4

0.100 sec 6.00 V

BiB

t t

Note sign difference. Is this consistent with Lenz’s law?

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Example: AC generator

cos cosB B A BA BA t

sinB BA tt

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Example: Betatron

Changing magnetic field

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Donald Kerst

University of Illinois

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Counter EMF:

Electric motor turns and produces mechanical energy when current runs through it.

As motor turns induction produces a counter EMF which acts to oppose the motion.

Counter EMF speed

Motor reaches some equilibrium speed

Add mechanical load, motor is slowed, counter EMF is reduced and current flows.

A stalled electric motor can draw large current and burn out

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Counter torque:

Turning produces an EMF. If generator is not connected to anything we get an EMF but no current flows.

G G Load

Add current drawing load. Current flows in generator coils and magnetic field exerts a torque which opposes the rotation.

The greater the load, the greater the counter torque.

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Eddy currents:

Induced currents in bulk material

Rotating metal wheel

Point c, B = 0 → B in Induced current is counterclockwise to oppose change

Point e, B in → B = 0. Induced current is clockwise.

Magnetic force acts to oppose rotation of the wheel.

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Eddy currents can be useful for braking or oscillation damping but also waste energy throgh I2R heating in the iron cores of motors, generators and transformers.

Thin sheets of Fe insulated from one another

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Transformers revisited:

BS SV N

t

BP PV N

t

Kirchoff loop rule

Assuming no flux is lostS S

P P

V N

V N

Conservation of energy output power = input power

or S PS S P P

P S

I NV I V I

I N

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General form of Faraday’s Law:

Changing magnetic flux induces a current in a wire

An electric field doing the work to move charges in the wire

A changing magnetic field produces an electric field

The emf induced in a circuit is the work done by E to move a unit charge around the circuit

i ii

E l E dl

B Bd

t dt

BdE dl

dt

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0

QE dA

0B dA

Bd

E dxdt

Maxwell’s Equations (integral form)

0 0 0Ed

B dx Idt

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Example: Magnetic field between pole faces of a magnet

22

2

BdE dl

dtB

E r rt

r BE

t

Valid for r < r0, for r > r0

202

BE r r

t

20

2

r BE

r t

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Note that the more general form of Faraday’s Law is valid not only in conductors but in any region of space,

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InductanceMutual Inductance:

2 21 21 121 21

1 2

or N M I

MI N

212 2Faraday's Law

dN

dt

12 21Combining:

dIM

dt

M21 depends only on the geometry and contents of the 2 coils

(like capacitance)

Now reverse the situation: 2 1 121 12 12

2

where dI N

M Mdt I

12 21Symmetry M M M

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Units of M are henries (H)

1 H = 1V sec / A = 1 sec

Self Inductance:

Changing current in a coil → changing flux through coil → opposing EMF.

is called the - of the coil (H)BNL self inductance

I

then dI

Ldt

All circuits have some (self) inductance

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Example:

Apply Lenz’s Law

(a) I is increasing so isInduced EMF acts to retard I

(b) I is decreasing so is (-)

Induced EMF acts to assist I

Example: Inductance of a solenoid, N = 100, l = 5.0 cm and A = 0.30 cm2

0

0

where N

B nI nl

NIABA

l

For an air filled coil

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20

27 -3 2

3

4 10 T m/A 100 3.0 10 m7.5 H

5.0 10 m

B N ANL

I l

Now, add an Fe core with = 4000 0 to the solenoid

L will be 4000 x larger or

L = 30 mH

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Networking rules for inductances:

Series:

1 1Parallel:

ii

i i

L L

L L

The above neglects mutual inductance