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TRANSCRIPT
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!)