Induced EMF

1830s Michael FaradayJoseph Henry

There can be EMF produced in a number of ways:

• A time varying magnetic field• An area whose size is varying• A time varying angle between and • Any combination of the above

Br

Sdr

Faraday’s Law of Induction

The induced EMF in a closed loop equals the negative of the time rate of change of magnetic flux through the loop

dtdEMF BΦ

−=

∫Φ

−=⋅dt

drdE Brr

Caution: induction EMF should be not confused with electrostatics:the effect is dynamical

Caution: electric field is not potential anymore: electric field acquired circulation!

cosBd B d A B dA BdA φ⊥Φ = ⋅ = =urr

Ampere’s law

∫contour

0B dl Iμ=ur r

orientation of the contour !

Caution: Sign of the currents enclosed by the contour are determined by the orientation of the contour.

For this orientation of the contour (anticlockwise), currents I1 and I3 are positive while I3 is negative.

∫Φ

−=⋅dt

drdE Brr

For this orientation of the area element the orientation of the contour (direction of the integral) is anticlockwise,

Sign of Induction (orientation of the area is UP)

induced current has clock-wise direction

Orientation of the oval -- as if it is lying on the floor

The same as in (a) onthe previous page, but for opposite orientation;induced current has clock-wise direction

Sign of Induction (orientation of the area is DOWN);direction of the current does not depend on the choice of orientation !

Orientation of the oval -- as if it is on the ceiling

IREMFdt

d B ==Φ

−

Lenz’s principle (law): H.F.E. Lenz (1804-1865)the direction of induction effect is such as to opposethe cause the effectCaution: the induced current opposes the change in flux through the circuit not the flux itself

induced current has anti-clock-wise direction induced current has

clock-wise direction

induced current changed its direction after the magnet passed through the circle

Lenz’s principle (law): H.F.E. Lenz (1804-1865)the direction of induction effect is such as to opposethe cause the effectCaution: the induced current opposes the change in flux through the circuit not the flux itself

IREMFdt

d B ==Φ

−

The force on a charge q moving with a velocityFr vr

The magnitude of the force θsinqvBF =

sec)//(][ meterCoulombNewtonsB ⋅=

Check: Units of the magnetic field and EMF

2

2

1 ( ) 1 /1 1

// 1

1 /[ ]

1

/ 1

1 1 1 /B

T tesla N C m s N A mflux

T m W

T m N m s C J s

b V Wb s

C V s⋅ = ⋅

= ⋅ = ⋅

Φ =

⋅

⋅

=

⋅ = ⋅

=

=

EMFdt

d B =Φ

−

dtdNEMF BΦ

−=

Induction in a coil with N turns

Example 29.4 Generator (a simple alternator)Caution: don’t confuse generator with a motor. Generator of electricity is rotated by an external source (water, wind, gasoline…)

0

0

( ) cos( ) sin

B

B

t ttEMF t

dt

ω

ω ω

Φ = ΦΦ

= − = Φ

Example 29.5 DC Generator

0

0

( ) cos| ( ) / | | sin |

B

B

t tEMF d t dt t

ωω ω

Φ = Φ= − Φ = Φ

0

2

( ) cos| sin |

| sin | 2 /

2 4112 28

4 500 (0.2 )(0.1 ) sec

B t N tEMF NBA t

tEMF EMFfNBA NBA

V revfT m

ωω ω

ω ππω

Φ = Φ=

=

= =

= =⋅ ⋅

Example 29.5 … motor’s back EMFThe motor’s back EMF is the emf induced by the changing magnetic flux in rotating coil of the motor

| sin |

input back

EMF NBA tV EMF Ir

ω ω== +

Example 29.5 … motor’s back EMF

2input back

input work dissipation

P I EMF I r

P P P

= ⋅ +

= +

Comment: power distribution is similar to the examples of Chapter 26, but instead of a lamp – now it is a motor

A series DC motor: What happens when a motor suddenly stops?Did it stop because it was burnt, or it was burnt because it had been

stopped by a jam?Example 27.11, see also Example 29. 5 to understand what is going on here, and why thecurrent has so different values?

2

2

120 4 2 112120 4 480 (4 ) 2 32

/ 120 / 2 60 (60 ) 2 7200

input heat

ab heat

V E A E VP A W P A W

stalledI V r V A P A W

= + × Ω =

= × = = × Ω =

= = Ω = = × Ω =

2

Motor:

input back

input input back

input work dissipation

V EMF Ir

P V I I EMF I r

P P P

= +

= = ⋅ +

= +

Generat| |

|

:

|

or

output

applied

output dissip

EMF V Ir

P EMF I

P P

= +

=

= +

Generator versus Motor: In what direction the current flows??

0

0

( ) cos| ( ) / | | sin |

B

B

t tEMF d t dt t

ωω ω

Φ = Φ= − Φ = Φ

Example 29.6 Slidewire generatorThe increase in the magnetic flux caused by the increase of the area induces the EMF and current.

Caution: actually, there are TWO forces. One (not shown) is applied on the right, and it causes sliding with velocity v. The other force is due to the induced current. It acts in the direction opposite to the direction of sliding (Lenz’s law).

Example 29.6 Slidewire generatorpower distribution in the presence of an output power

2

/| | ( ) /

| | | |

( ) /

( ) /

B

output output

a

applied

appliepplied

applied output applied output dissip

output output dissip ut

d

o put

EMF d dt BLvEMF V Ir BLv V r I

I L B

P v vLBI EMF I

P vLB BLv V r P P P

P V I P BLv V r

F

F

= − Φ = −= + − =

= − ×

= = =

= − = +

= = −

!!!

ur urr

x x xx

x x x x

x x x

x x xx

x x

x x

x x xx

x x x x

x x x

x x x x

B

x

x

x

FMotor

Caution: here, only the applied force that causes sliding is shown. The other force that is due to the induced current is not shown. It acts in the direction opposite to the applied force.

Example 29.6 Slidewire generator; motional EMF

Attention: to generate EMF there is no need in a material frame, like in slidewire. It may be virtual. Then, it is called “motional EMF.” This EMF is not a mystery. It originates from a “probe force” acting on a probe charge moving together with a rod.

Rail-gun; the “motor-counterpart” of the slidewire generator

2

| | | |

( ) ( )

battery back

back

input battery back

back

input mechanical dissipation

V EMF Ir

EMF BL v

P V I I EMF I r

I EMF I BLv v IBL vFP P P

= +

=

= = ⋅ +

⋅ = = == +

Faraday disc dynamo: two ways to get the answer

2

0(2) | ( ) / | / 2

R

BdEMF d t dt B r dr BRdtϕ ω= − Φ = ⋅ =∫

2

0 0 0(1) ( ) ( ) / 2

R R REMF E r dr v r B dr B r dr BRω ω= ⋅ = ⋅ = ⋅ =∫ ∫ ∫

θθ

sinsin

vBEqvBF

==

Lenz’s law: the current directed down toward the sliding contact bcreates a force directed to the left, thus,opposing disc’s rotation

Induced electric fields caused by the varying magnetic flux.

How can it be that the magnetic field induces electric fields outside its location? Caution: Notice that it is a magnetic field changing in time.

∫Φ

−=⋅dt

drdE Brr

Eddy currents

Attention: current in the transmitting coil should be pulsing.

Symmetry: if there are induced electric fields caused by varying magnetic field,why not to look for magnetic fields caused by varying electric fields?

∫Φ

−=⋅dt

drdE Brr

∫ 0( )Ec

dBd r Idt

μ ε Φ= +

ur r

Symmetry (Clerk Maxwell, 1831-1879): if there are induced electric fields caused by varying magnetic field,why not to look for magnetic fields caused by varying electric fields?

∫Φ

−=⋅dt

drdE Brr

∫ 0( )Ec

dBd r Idt

μ ε Φ= +

ur r

displacement displ/ /EI d dt j dE dtε ε= Φ =

The flux term is called “displacement current”

Comparison of the displacement current Idisplacementwith the transport current Ic for charging capacitor

∫Φ

−=⋅dt

drdE Brr

∫ 0( )Ec

dBd r Idt

μ ε Φ= +

ur r

displacement

displacement displ

( / )( )/ /

/ /

E

c E c

E

q CV S d Ed SEI dq dt d dt I I

I d dt j dE dt

ε ε εε

ε ε

= = = = Φ= = Φ =

= Φ =

Maxwell’s equations in the empty space: induced electric fields caused by varying magnetic field,magnetic fields caused by varying electric fields

∫Φ

−=⋅dt

drdE Brr

∫ 0 0EdBd r

dtμ ε Φ

=ur r

Two coupled equations are typical for waves. Theoretical discovery of the electromagnetic waves was made by Maxwell.

∫Φ

−=⋅dt

drdE Brr

∫ 0 0EdBd r

dtμ ε Φ

=ur r

Two coupled equations are typical for waves. Theoretical discovery of the electromagnetic waves was made by Maxwell.

Maxwell’s equations in the empty space: induced electric fields caused by varying magnetic field,magnetic fields caused by varying electric fields

Additional Material:superconductivity

Meissner effect and magnetic levitation

Eddy currents do not let a magnetic field to penetrate inside a superconductor (the Meissner effect); The current in the superconductor in its turn creates a backward magnetic field; This backward magnetic field may cause the levitation of a body, which was the origin of the magnetic field which generated the Eddy current.(Manifestation of the Lenz law at work!)