Transcript m, R

Chapter 27
Electromagnetic Induction
and Faraday’s Law
Induced EMF (1)
Electric current
magnetic field
Faraday’s experiments (1820-1821)
Changing magnetic field can produce a current
2
Induced EMF (2)
Phenomenon of electromagnetic induction
An induced EMF (electromotive force in P566) is
produced by a changing magnetic field.
3
Faraday’s law of induction
The EMF induced in a circuit is equal to the
changing rate of magnetic flux through the circuit.
dB
 
dt
Faraday’s law of induction
1) EMF is produced even if no current can flow
(as when the circuit is not complete)
2) N loops coil:    N d  B
dt
3) Induction current in circuit:
I  /R
4
Lenz’s law
An induced EMF is always in a direction that opposes
the original change in flux that caused it.
This is known as Lenz’s law
1) Conservation of energy
2) Three ways to create EMF
 B   BdS cos 
5
Rotating coil
Example1: A circular coil is rotating in uniform
magnetic field. Determine the EMF.

Solution: The magnetic flux:
 B  BS cos    R 2 B cos(t  0 )
R
dB
  
  R 2 B sin(t   0 )
dt
If the magnetic field is changing:
B
B  B0 e  t
   R 2 B0 e  t sin(t  0 )  R 2  B0 e  t cos(t  0 )
6
Induced charges
Thinking:A small coil moves far away from the
position in figure. How to determine the total
induced charge going through the coil?
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EMF induced in moving conductor
Conductor moves in B → motional EMF
++
Caused by Lorentz force:
F  qv  B  qEk
-
F
Ek : non-electrostatic field
Motional EMF:
All ⊥ case:
v
--




   Ek  dl   (v  B)  dl
   Bvdl
Direction?
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Derived by Faraday’ law
The result can also be derived by Faraday’ law
dl moves with velocity v
dS
It sweeps out an area dS
dS  (vdt )  dl
Magnetic flux:
v
dl
d B  B  dS  B  (v  dl )dt
EMF:    d  B   B  (v  dl )  (v  B)  dl


dt
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Motion in uniform field
Example2: Determine the EMF induced in the
conductor in a uniform magnetic field.
(a)
(b)
v
v
B
l
lab
EMF:   Blv
Force: F  BIl  B2l 2v / R
Power: P  Fv  I R
2
→ straight wire:
  (v  B)  lab
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Rotates in uniform field
Example3:A conductor rod rotates about axis o.
Determine the induced EMF. (B, L, ω)

Solution: For an infinitesimal:
d   B   r  dr
dr v
r 
o
Total EMF in the conductor:
1
1 2
B
   B rdr  B L2
S

L

0
2
2
dB
d 1 2 1
Faraday’s law:   
   BL  B L2
dt 2
2
dt
L
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Rotates in nonuniform field
Question:A conductor rod rotates about axis o.
Determine the induced EMF when the two wires
are perpendicular to each other.
I
0 IL

(1  ln 2)
2

o
L
L
B
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Motion on rails
Example4: Conducting rod rests on frictionless
parallel rails with an EMF source. Determine the
speed of rod if the source puts out (a) constant I;
(b) constant EMF. (c) What is the terminal speed?
Solution: (a)
F  ma  BIl
BIl
 v  at 
t
m
B

F
l
m, R
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(b) source puts out constant EMF:
I
  Blv
R
t Bl
v
dv BIl (  Blv) Bl
dv



dt  
0 mR
0   Blv
dt
m
mR
 v

Bl
(1  e
B2l 2t

mR
)
B
(c) Terminal speed
v


F
l
Bl
m, R
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Changing B produces E
Conductor stays at rest, magnetic field changes
→ Induced EMF → induced current
Forces on static charges?
Maxwell: It’s caused by electric field
Induced (vortex) electric field: produced by changing
magnetic field, and acts on electric charges.
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Faraday’s law → general form
Generalize the definition of V : Vab 
For induced EMF in a closed circuit:



b
a
E  dl
dB
dt
dB
d
   B  dS
Ei  dl  
dt
dt
where Ei is the induced electric field
B
 Ei  dl   t  dS
General form of
Faraday’s law
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Induced electric field
B
 Ei  dl   t  dS
B
   Ei  
t
1) Ei is produced by changing B, not by charges
2) Even if there is no conductor, Ei still exists
3) Induced electric field is nonconservative
4) Minus sign shows the direction of Ei
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Comparison of fields
Electrostatic / induced electric / magnetic field
E
 dl  0;
E
Qin
s
s
 dS 
0
Field lines:
dB
 Ei  dl   dt ;
 E  dS  0
i
 B  dl
 0 I in
 B  dS  0
vortex
dB
dt
I
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Vortex electric field
Example5:Uniform magnetic field in cylindrical
space changes as dB/dt=C>0. Determine the
induced electric field.
Solution: Analyze the symmetry
dB
 Ei  dl  Ei  2 r   dt  C  S
C  r2
C
Ei
r  R : Ei  
 r
2 r
2
CR 2
C   R2

r  R : Ei  
o
2r
2 r
R
r
R
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EMF in a wire
Example6:Magnetic field in a solenoid changes
as dB/dt=C>0. A straight wire lies tangent to the
solenoid at its center. What is the EMF in wire?
Solution: Induced electric field:
CR 2
r  R : Ei  
2r
r

R
CR 2
  Ei  dl   
cos  dl
2r
L

CR
R C
 
d  
 / 4
2
4
 /4
2
Ei
2
dl
l
2R
R  r cos 
l  R tan 
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Another Solution:
1
Imagine a closed circuit
R
2
2R
1   Ei  dl  0,  2  0
d B
d  R2 B
 R 2C
     
 (
)
dt
dt
4
4
Discussion:
 ab   ab ?
dB
 Ei  dl   dt  0!
 ab   ab
a
b
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Self inductance
Thinking:If a solenoid with current I is cut off
from the battery, will I drop abruptly to 0? How
does I change over time?
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*Applications of induction
Sound systems /
microphones:
Transformers:
Recording tape /
computer memory
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*Vortex current
Vortex electric field → vortex current
Heating effect → induction cooker
Electromagnetic damping
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