Electromagnetic Induction

(Chapters 31-32)

• The laws of emf induction: Faraday’s and Lenz’s laws

• Inductance

• Mutual inductance M

• Self inductance L. Inductors

• Magnetic field energy

• Simple inductive circuits

• RL circuits

• LC circuits

• LRC circuits

• We’ve seen that an electric current produces a magnetic field. Should we presume

that the reverse is valid as well? Can a magnetic field produce an electric current?

• Yes, magnetic fields can produce electric fields through electromagnetic induction

• Most of the electric devices that we use as power supplies are electric generators

based on induced emf : these generators convert different forms of energy first into

mechanical energy and then, by induction, into electric energy

The other side of the coin…

Experimental setup:

•A primary coil connected to a battery

•A secondary coil connected to an ammeter

Observations:

• When the switch is closed, the ammeter reads a current and then returns to zero

• When the switch is opened, the ammeter reads an opposite current and then returns to zero

• When there is a steady current in the primary circuit, the ammeter reads zero

Conclusions:

•An electrical current is produced by a changing magnetic field

• The secondary circuit acts as if a source of emf were connected to it for a short time

• It is customary to say that an induced emf is produced in the secondary circuit by the changing

magnetic field. Let’s look closer at why this happens…

• Electromagnetic induction was first demonstrated

experimentally by Michael Faraday

Gauss’s Law – Electric Flux

Def: The electric flux of a

uniform magnetic field B

crossing a surface A under

an angle θ with respect to

the normal to the surface is

B BA

cosB BA

0B

B B dA

B

Aθ

θ

Ex: The uniform field lines

penetrate an area A perpendicular

and then parallel to the surface,

produce maximum and zero fluxes

B

ˆA An

A

B

• In order to quantify the idea of emf induction, we

must introduce the magnetic flux since the emf is

actually induced by a change in magnetic flux rather

than generically by a change in the magnetic field

• Magnetic flux is defined in a manner similar to that

of electrical flux:

2

SIT m Weber WbB

If the field varies across the surface, one must

integrate the contribution to the flux for every

element of surface:

Exercise 1: Gauss’s law can be

reformulated for magnetic sources.

Try to do it yourself: what is the

magnetic flux through a closed

surface around a magnetic source?

Quiz:

1. Given a solenoid, through which of the shown surfaces is the

magnetic flux larger?

A1 A2 A3 A4

2. A long, straight wire carrying a current I is placed along the axis of a cylindrical surface of

radius r and a length L. What is the total magnetic flux through the cylinder?

A1

A2

A3

A4

I

r

Ia)

b)

c) Zero

0ILL

0

2

IL

r

Electromagnetic Induction – Faraday’s Law and Lenz’s Law

Faraday’s Law: The instantaneous emf induced in a circuit equals

the time rate of change of magnetic flux ΦB through the circuit

• The involvement of magnetic flux in the electromagnetic induction is described

explicitly by

Bd

dt

Comments:

• Since ΦB = BAcosθ the change in the flux, dΦB, can be produced by a change in B,

A or/and θ

• If the circuit contains N loops (such as a coil with N turns)

• The negative sign in Faraday’s Law is included to indicate the polarity of the

induced emf, which is more easily found using:

Lenz’s Law: The direction of any magnetic induction effect is to oppose the cause

of the respective effect

Ex: Causes of EM induction can be: varying magnetic fields, currents, emfs, or forces

determining a change in flux for instance by varying the area exposed to the magnetic field.

Induced effects may be magnetic fields, emfs, currents, forces, electric fields

BN d dt

Magnetic flux through a loop can be varied by moving a bar

magnet in proximity

1. If the bar magnet is moved toward a loop of wire:

• As the magnet moves, the magnetic flux increases with time

• The induced current will produce a magnetic field opposing

the increasing flux, so the current is in the direction shown

• This can be seen as a repelling effect onto the incoming bar by

the loop seen as a magnet

2. If the bar magnet is moved away from a loop of wire:

• As the magnet moves, the magnetic flux decreases with time

• The induced current will produce a magnetic field helping the

decreasing flux, so the current is in the direction shown

• This can be seen as a attraction effect onto the bar by the loop

seen as a magnet

Electromagnetic Induction – Applying Lenz’s Law

Bind

Bind

repels the

incoming North

attracts the

departing North

Exercise 2: Lenz’s Law

Consider a loop of wire in an external magnetic field Bext. By

varying Bext, the magnetic flux through the loop varies an a

current flows through the wire. Let’s use Lenz’t law to find

the direction of the current induced in the loop when

a) Bext decreases with time, and

b) Bext increases with time

Bext

Bext

decreasing

increasing

Exercises:

3. Emf induced by moving field: A bar magnet is

positioned near a coil of wire as shown in the figure.

What is the direction of the induced magnetic field in the

coil and the induced current through the resistor when

the magnet is moved in each of the following directions.

a) to the right

b) to the left

4. Emf induced by switching field on and off: Find the

direction of the current in the resistor in the figure at

each of the following times.

a) at the instant the switch is closed

b) after the switch has been closed for several minutes

c) at the instant the switch is opened

Problem:

1. Emf induced by reversing field: A wire loop of radius r lies so that an external magnetic

field B1 is perpendicular to the loop. The field reverses its direction, and its magnitude

changes to B2 in a time Δt. Find the magnitude of the average induced emf in the loop during

this time in terms of given quantities.

vb

va

abV EL FL q vBL

• We’ve seen that, in certain conditions, motion can determine an

emf – we are interested in this phenomenon since it stays behind

converting mechanical energy into electric energy

• To see how it happens, consider a straight conductor of length

L moving with constant velocity v perpendicular on a uniform

field B: the electric carriers in the conductor experience a

magnetic force qvB along the conductor, as on the figure

• Notice that the electrons tend to move to the lower end of the

conductor, such that a negative charge accumulate at the base

• Consequently, a positive charge forms at the upper end of the conductor, such that,

as a result of this charge separation, an electric field E is produced in the conductor

• Charges build up at the ends of the conductor until the upward magnetic force (on

positive carriers forming a current) is balanced by the downward electric force qE

Motional emf – Across a conductor moving in a magnetic field

• The potential difference between the ends of the

conductor is similar with the potential difference

between the plates of a charged capacitor:

• This motional emf is maintained across the conductor as long as there is motion

Ex: The magnetic field of Earth is

about 5×10–5 T. Therefore, if a

straight 1-m metallic rod is moved

perpendicular on the field with a

speed of 1 m/s, the emf produced

across it ends is about 5×10–5 V

+

mF qvB

eF qE

+

–

v

B

Lq

a

b

• Consider now that the moving bar on the previous

slide has a negligible resistance and it slides on rails

connected in a circuit to a resistor R, as in the figure

• As the bar is pulled to the right with a velocity v by

an applied force, the free charges move along the

length of the bar producing a potential difference and

consequently an induced current through R

• The motional emf induced in the circuit acts like a

battery with an emf

• In general, for any conductor moving with velocity

v in a magnetic field B we have an alternative

expression for Faraday’s law:

vBL RI I vBL R

v B dl vBL

force per unit charge (i.e., field) acted

on the element dl of conductor moving

in the external magnetic field B

Motional emf – Producing current in a circuit

The integral is

around a closed

conductor loop

+

+

R

+

–

Lv

B

I

I

The charge carriers are

pushed upward by the

magnetic force

R

I

I

Motional emf – Explained using Faraday and Lenz’s Laws

•Alternatively, we can look at the same situation but using Faraday and Lenz’s

laws: the changing magnetic flux through the loop and the corresponding induced

emf in the bar result from the change in area of the loop

1. Increasing circuit area:

• The magnetic flux through the loop increases

• By Lenz’s law the induced magnetic field Bind

must oppose the external magnetic field Bext.

• The direction of the current that will create the

induced magnetic field is given by RHR #2.

2. Decreasing circuit area:

• The magnetic flux through the loop decreases

• By Lenz’s law the induced magnetic field Bind

must “help” the external magnetic field Bext.

• The direction of the current that is reversed

compared with the case above.

• Then, by Faraday’s Law, we obtain the same

expression for the induced emf:Bd dA dx

B BL BLvdt dt dt

Rv

Iind

extB

indB

mF

R LvIind

extB

indB

dA

dx

appliedF

m

l

R

B

Problems:

2. Gravity as applied force to induce emf: A metallic rod of mass

m slides vertically downward along two rails separated by a distance

ℓ connected by a resistor R. The system is immersed in a constant

magnetic field B oriented into the page.

a) Calculate the current flowing through the resistor R when the

magnetic force on the rod becomes equal to its weight.

b) Calculate the emf induced across the resistor R.

c) Use Faraday’s Law to compute the speed of the rod when the net

force on it is zero.

3. Faraday disk: A thin conducting disk with radius R laying in xy-plane rotates with

constant angular velocity ω around z-axis in a uniform magnetic field B parallel with z. Find

the induced emf between the center and the rim of the disk.

max sin sinBA t t

B

θ

v

I

r

Applications – Electric Generators

•An alternating Current (ac) generator converts mechanical energy to electrical

energy by rotating loops of wire in magnetic fields

• There is a variety of sources that can supply the energy to rotate the loop, including

falling water, heat by burning coal or nuclear reactions, etc.

Basic operation of the generator: as the loop

rotates, the magnetic flux through its surface A

changes with time, such that an emf is induced

• For constant angular speed ω = dθ/dt,

Bθ = ωt

ω

r

v

v

cos sinBd d dBA BA

dt dt dt

Comments:

• The emf polarity varies sinusoidally (ac signal)

• ε = εmax when loop is parallel to the field

• ε = 0 when the loop is perpendicular to the field

• Consider two coils with N1 and N2 turns.

• The variation of current in the first coil corresponds to a

proportionally varying flux through the second:

• By Faraday’s law

22 2

BdN

dt

• By Faraday’s law, changing a current in a coil induces an emf in an adjacent coil:

this coupling is called mutual inductance

21 1 2 2 BM i N

mutual induction

12 21

diM

dt

• The mutual-inductance depends on the geometry of the two coils and on the

presence of a magnetic material as a core. If the material has linear magnetic

properties, the mutual inductance is a constant.

• The discussion is symmetric in the opposite direction, so we have

2 2 1 121 12

1 2

B BN NM M M

i i

Henry H 1W AM

Inductance – Mutual inductance

Inductance – Self inductance

• L is a proportionality constant called the inductance of the coil:

• Notice that nothing prevents a changing flux to produce an emf in the very coil that

produces the actual flux: this phenomenon is called self-inductance: discovered in

the 19th century by Joseph Henry

Ex: Consider a current carrying loop of wire

• If the current increases in a loop, the magnetic flux through the loop surface due to this

current also increases: hence, an emf is induced that opposes the change in magnetic flux

• This opposing emf results in a slowed down increase of the current through the loop

• Alternatively, if the current decreases, the self-inductance will slow down the rate of decrease

diL

dt

negative sign indicates

that a changing current

induces an emf in

opposition to that change

• The self-induced emf is proportional to the

rate of change of the current through the coil:

BL NI

SIHenry (H)L

Def: If a circuit with N loops carrying a current I produces a magnetic flux ΦB

through each loop surface, the self-inductance is given by

εback<0 εback>0

• L characterizes solenoids as elements of circuit called inductors

• Since the flux is proportional to the current, the inductance of a solenoid does not

depend on the current flowing through the coil: it is a characteristic of the device,

depending on geometric factors and the magnetic properties of the interior of the coil

• A device with self-inductance (such as a coil) is called an

inductor: a circuit element with a certain inductance – an

additional circuit element besides capacitors and resistors

Inductance – Inductors

Li a b

Symbol:

Potential difference:

ab a b back

diV V V L

dt

• drop if di/dt > 0

• raise if di/dt < 0

• 0 if i = const.

• Inductance can be interpreted as a measure of opposition to the rate of change in the

current: it determines a potential difference or a back emf across the terminals

Ex: Self inductance of a straight solenoid: A straight solenoid

with n turns per length, and volume V has inductance given by:

0Bn IBA

L N N NI I

I

2 2

0 0 A n A L n V

iA

Energy Stored in a Magnetic Field – Summary of circuit elements

• The work done by a battery to produce an

increasing current against the back emf of an

inductor can be thought of as energy stored

in the magnetic field inside the inductor

• Contrast with the energy dissipated across a current carrying resistor:

• Or the energy stored in the electric field of a charged capacitor:

212LU IL

212RP RI

212CU CV

2

0L n A

R A

0C A d

A

A

d

A

L Ldi

Vdt

RV IR

CVC

Q

0

I

L abU Pdt iV dt L idi

• The magnetic energy density stored in a straight solenoid inductor is given by221

02 1

2

V

V V

LnLIU

u

2

V

I

22

0

0 0

1

2 2

BnI

this is, in general,

the magnetic energy

density in vacuum

• Inside a magnetic material – such that an iron core inside a solenoid – μ0 is to be

replaced with μ : magnetic permeability in the respective material

• An inductor can be combined in series with a resistor into

a dc-RL circuit to obtain a specific behavior

• Recall that the resistance R is a measure of opposition to

the current while the inductance L measures the opposition

to the rate of change of the current. Let’s see what’s

happening in an RL circuit:

• As the current begins to increase, the inductor produces a negative back emf εL < 0

that opposes the increasing current, so the current doesn’t change from 0 to its

maximum instantaneously

• When the current reaches its maximum, the rate of change and the back emf εL = 0

S1:

S2:

εL<0

εL>01. Close S1 and open S2 : the RL series circuit is completed

across a battery ε

2. Open S1 and close S2: the RL series circuit is completed with battery removed

• Since there is no battery, the current starts to decay, such that the inductor produces

a positive back emf εL > 0 to “help” the current.

• If the current becomes zero, the rate of change and the back emf are εL = 0

Inductive circuits – LR-circuit: principles

R

ε

i

i

L

S1

S2

• Kirchhoff rule applies in both cases

(set ε = 0 when current is decaying):0 0L

diL

dtiRiR

1. The current in the RL circuit in series with a battery

increases exponentially to Imax = ε/R: max 1 ti I e

0

ti I e

Inductive circuits – LR-circuit: characteristics

• The time constant, τ = L/R, for an RL circuit is the

time required for the current in the circuit to reach

63.2% of its final value

• A circuit with a large time constant will take a longer

time to reach its maximum current

0 1 Rt LdiiR L i e

dt R

00 Rt LdiiR L i I e

dt

2. The current in the RL circuit without a battery

decays exponentially from its initial value I0:

• If the current reached the maximum value before the

battery was disconnected, it is given by I0 = Imax = ε/R

• An inductor connected across a charged capacitor form an electric oscillator with

oscillating current and charge called a dc-LC circuit. Functionality:

1. As the capacitor discharges, current increases from 0 to a maximum value and the

potential difference across both elements decreases gradually to 0: the electric energy

is stored in the form of magnetic energy

2. When the current reaches its maximum, the capacitor starts to recharge with an

inverse polarity than initially until the current is again zero and the process restarts in

the reverse direction

cycle

Inductive circuits – LC-circuit: principles

electric

energy

magnetic

energy

• The SHO solutions are

Charge:

Current:

0 0 L C

di qv v L

dt C

1 LC

cosq Q t

maximum charge initial phase angle given

by the charge at t = 0

sindq

i Q tdt

L

div L

dt

C

qv

C

Inductive circuits – RL-circuit: characteristics

• Kirchhoff rule can be applied to find the equation describing the

oscillations of charge and current:

22

20

d qq

dt

Angular frequency ω2 = 1/LC

T

t

2T

cosq Q t

2cosq Q t

T

t

q

Q

2T

T/2 3T/2

– Q

φ = 0

φ = π/2

T

t

iimax

2T

di/dt

–imax

i i

L

C+q –q

• We see that the charge on the capacitor satisfies

an equation similar with that of a Simple

Harmonic Oscillator with angular frequency

max

di

dt

max

di

dt

Problem:

5. LC oscillator: A power supply with emf ε is used to fully charge up a capacitor C. Then the

capacitor is connected to an inductor L.

a) What is the frequency and period of the LC circuit?

b) Find the maximum charge, the maximum current and the maximum rate of change of

current in the circuit.

c) Write out the time dependency of the charge, current and rate of change of current

considering t = 0 the first time when the capacitor holds only half of its maximum charge.

d) Sketch the q vs t graph.

0 x

k

222

2

2

0

SHO

SHO

F kxd x

xd xdtF ma m

dt

2 k

m

Exercise 5: LC oscillations compared with a mechanical

analog: The periodic motion of a Simple Harmonic Oscillator,

containing a mass m connected to a light spring of force

constant k oscillating on a frictionless horizontal surface, is

given by Newton’s 2nd Law as following:

m

Find the oscillating quantities analogue to electric quantities oscillating in the LC circuit:

LC: charge q current i change in current di/dt

Spring: