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Short Version : 27. Electromagnetic Induction. 27.1. Induced Currents. 4 results from Faraday / Henry (1831). v = 0, I = 0. v > 0, I > 0. 1. Current induced in coil by moving magnet bar. v >> 0, I >> 0. v < 0, I < 0.

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27 1 induced currents
27.1. Induced Currents

4 results from Faraday / Henry (1831)

v = 0, I = 0

v > 0, I > 0

1. Current induced in coil by moving magnet bar.

v >> 0, I >> 0

v < 0, I < 0

2. Moving the coil instead of the magnet gives the same result.


3. An induced current also results when a current-carrying circuit replaces the magnet.

4. A current is also induced when the current in an adjacent circuit changes.

 changing B induces currents (electromagnetic induction)

27 2 faraday s law
27.2. Faraday’s Law
  • Magnetic flux
  • Flux & Induced EMF
magnetic flux
Magnetic flux

Magnetic flux:


For a uniform B on a flat surface:

Move magnet right

 more lines thru loop

example 27 1 solenoid
Example 27.1. Solenoid

A solenoid of circular cross section has radius R,

consists of n turns per unit length, and carries current I.

Find the magnetic flux through each turn of the solenoid.



B out of plane

example 27 2 nonuniform field
Example 27.2. Nonuniform Field

A long, straight wire carries current I.

A rectangular wire loop of dimensions l by w lies in a plane containing the wire, with its closest edge a distance a from the wire, and its dimension l parallel to the wire.

Find the magnetic flux through the loop.

Area element for integration

flux induced emf
Flux & Induced EMF

Faraday’s law of induction:

The induced emf in a circuit is proportional to the rate of change of magnetic flux through any surface bounded by that circuit.

C is CCW about S.

Note: dB/dt can be due to

  • changing B caused by
    • relative motion between circuit & magnet,
    • changing current in adjacent circuit,
  • changing area of circuit,
  • changing orientation between B & circuit.



example 27 3 changing b
Example 27.3. Changing B

A wire loop of radius 10 cm has resistance 2.0 .

The plane of the loop is perpendicular to a uniform B that’s increasing at 0.10 T/s.

Find the magnitude of the induced current in the loop.





example 27 4 changing area
Example 27.4. Changing Area

Two parallel conducting rails a distance l apart are connected at one end by a resistance R.

A conducting bar completes the circuit, joining the two rails electrically but free to slide along.

The whole circuit is perpendicular to a uniform B, as show in figure.

Find the current when the bar is pulled to the right with constant speed v.

Let x = 0 be at the left end of rail.






27 3 induction energy
27.3. Induction & Energy



Direction of emf is to oppose magnet’s motion.

RH rule: thumb // m.

Loop ~ magnet with N to left.

Magnet moving right

Lenz’s law :

Direction of induced emf is such that B created by the induced current opposes the changes in  that created the current.



RH rule: thumb // m.

Loop ~ magnet with S to left.

Magnet moving left

motional emf len s law
Motional EMF & Len’s Law

Motional emf: induced emf due to motion of conductor in B.

Square loop of sides L & resistance R pulled with constant speed v out of uniform B.

Force on e:

downward force

  • upward I (CW)

Force on current carrying wire:


< 0






Work done is used to heat up circuit ( E conservation ).

application electric generators
Application. Electric Generators

World electricity generation ~ 2TW.

Rotating loop changes  & induces emf.

Rotating slip rings.

Sinusoidal AC output

Work required due to Lenz’s law.

Electric load

Stationary brushes

Rotating conducting loop

Hand-cranked generator ~ 100W

example 27 5 designing a generator
Example 27.5. Designing a Generator

An electric generator consists of a 100-turn circular coil 50 cm in diameter.

It’s rotated at f = 60 rev/s to produce standard 60 Hz alternating current.

Find B needed for a peak output voltage of 170 V,

which is the actual peak in standard 120 V household wiring.

Loop rotation


EM induction is basis of magnetic recording ( audio, video, computer disks, …).



Card motion

Modern hard disks:

Giant magnetoresistance.

Magnetic strip

Information stored in magnetization pattern

Swiping a credit card.

Patterns of magnetization on the strip induce currents in the coil.

eddy currents
Eddy Currents

Eddy current: current in solid conductor induced by changing .

Usage: non-frictional brakes for rotating saw blades, train wheels, …

Application: Metal Detectors

Induced Current

Nothing between coils

Current detector


Strong I

Transmitter coil

Receiver coil

Weak I : alarm.

Metal between coils

closed open circuits
Closed & Open Circuits

B of induced I points out of page

Setting n // Bin

 C is CCW &  < 0

Bin  d  /d t < 0

 E > 0

 I is CCW

RH rule gives CCW I





E > 0

27 4 inductance
27.4. Inductance


Mutual Inductance:

Changing current in one circuit induces an emf in the other.

Large inductance:

two coils are wound on same iron core.


Transformers, ignition coil, battery chargers, …


Changing current induces emf in own circuit & opposes further changes.


Inductors  frequency generator / detector …

[ L ] = T  m2 / A = Henry

example 27 6 solenoid
Example 27.6. Solenoid

A long solenoid of cross section area A and length l has n turns per unit length.

Find its self-inductance.

B of solenoid:


+E direction = V  along I.

back emf

Rapid switching of inductive devices can destroy delicate electronic devices.

dI /d t < 0


 (lower voltage)

 (lower voltage)



+ (higher voltage)

+ (higher voltage)

example 27 7 dangerous inductor
Example 27.7. Dangerous Inductor

A 5.0-A current is flowing in a 2.0-H inductor.

The current is then reduced steadily to zero over 1.0 ms.

Find the magnitude & direction of the inductor emf during this time.

+ here ( E > 0 ) helps keep I flowing

inductors in circuits
Inductors in Circuits

Current through inductor can’t change instantaneously.

Switch just closed :

I = 0, dI/dt  0; | EL | = E0

L ~ open circuit.

Long after switch closing:

I 0, dI/dt = 0; EL= 0;

L ~ wire.

Switch open: I = 0



I  V = IR 

But rate is 

EL< 0 ; | EL | 

Inductive time constant = L / R

c.f. capacitive time constant = RC

example 27 8 firing up a electromagnet
Example 27.8. Firing Up a Electromagnet

A large electromagnet used for lifting scrap iron has self-inductance L = 56 H.

It’s connected to a constant 440-V power source;

the total resistance of the circuit is 2.8 .

Find the time it takes for the current to reach 75% of its final value.


Switch at B, battery’s shorted out. I exponentially.

Switch at A, I.

Short times: IL can’t change instantaneously.

Long times: EL = 0 ; inductor  wire.

making the connection
Making the Connection

Verify that the current in just after the switch is reopened has the value indicated

Immediately after the switch is closed:

L ~ open circuit.

Long time after the switch is closed:

Immediately after 2nd switch opening :

L ~ short circuit.

I in L ~ continues.

27 5 magnetic energy
27.5. Magnetic Energy

Any B contains energy.

This eruption of a huge prominence from the sun’s surface releases energy stored in magnetic fields.

magnetic energy in an inductor
Magnetic Energy in an Inductor

RL circuit:

Power from battery

Power dissipated

Power taken by inductor

Energy stored in inductor:

example 27 10 mri disaster
Example 27.10. MRI Disaster
  • Superconducting electromagnets like solenoids in MRI scanners store a lot of magnetic energy.
  • Loss of coolant is dangerous since current quickly decays due to resistance.
  • A particular MRI solenoid carries 2.4 kA and has a 0.53 H inductance.
  • When it loses superconductivity, its resistance goes abruptly to 31 m. Find
  • the stored magnetic energy, and
  • the rate of energy release at the instance the superconductivity is lost.

In practice, Cu / Ag are incorporated into the superconducting wires to reduce R.

magnetic energy density
Magnetic Energy Density

Solenoid with length l & cross-section area A :

(Eg. 27.6)

Magnetic Energy Density :

c.f. electric energy density :

27 6 induced electric fields
27.6. Induced Electric Fields

EMF acts to separate charges:

Battery: chemical reaction

Motional emf: F = v  B.

Stationary loop in changing B : induced E

Faraday’s law

Induced E forms loop.

Static E begins / ends on charge.

example 27 11 solenoid
Example 27.11. Solenoid

A long solenoid has circular cross section of radius R.

The solenoid current is increasing, & as a result so is B in solenoid.

The field strength is given by B = b t, where b is a constant.

Find the induced E outside the solenoid, a distance r from the axis.

Symmetry  E lines are circles.

Loop for Faraday’s law



S out  C ccw

conservative nonconservative electric fields
Conservative & Nonconservative Electric Fields


For stationary charges (electrostatics) :

W against E

E is conservative

Induced fields (electromagnetics) :

E does W

E is non-conservative

got it 27 7
GOT IT? 27.7

The figure shows three resistors in series surrounding an infinitely long solenoid with a changing magnetic field;

the resulting induced electric field drives a current counterclockwise, as shown.

Two identical voltmeters are shown connected to the same points A and B.

What does each read?

Explain any apparent contradiction.

Hint: this is a challenging question!

VA  VB = IR

VA  VB = 2IR


Classical model of diamagnetism (not quite right)

Superconductor is a perfect diamagnet (Meissner effect).

B = 0: net = 0

net  0

B 0

This e slows down.

This e speeds up.