Chapter30: Electromagnetic Induction
Motional EMF’s
Conductor Moving through a magnetic field
Magnetic Force on charge carriers
Accumulation of charge
F = qv B
Balanced Electrostatic/Magnetic Forces
qvB=qE
Induced Potential Difference
E = EL = vBL
Generalized
d E = v Bdl
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With conduction rails:
I
F = qv B
demonstrations with galvanometer
moving conductor
moving field!
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Consider circuit at right with total circuit
resistance of 5 W, B = .2 T, speed of
conductor = 10 m/s, length of 10 cm.
Determine:
(a) EMF,
(b) current,
(c) power dissipated,
(d) magnetic force on conducting bar,
(e) mechanical force needed to maintain
motion and
(f) mechanical power necessary to maintain
motion.
I
dE = v Bdl
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Faraday disk dynamo = DC generator
  
dE  v  B  dl
dl=dr
  rBdr
R
E    rBdr
0
2
R
 B
2
I

B
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Faraday’s Law of Induction
Changing magnetic flux induces and EMF
 
d B  B  dA  BdA cos 
 
 B   B  dA
 
 B  B  A  BA cos  for a uniform magnetic field
E 
d B 
d B



N
for
N
loops


dt 
dt

Lenz’s Law: The direction of any magnetic induction effect
(induced current) is such as to oppose the cause producing it.
(opposing change = inertia!)
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B
(increasing)
A
E
B
(decreasing)
E
A
E
B
(increasing)
A
A
E
B
(decreasing)
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Conducting rails: changing flux from changing area
d  BdA  Blvdt
d
E 
  Blv
dt
watch sign convention s + RHR
Lenz' s Law + induced currents
I
dA=l vdt
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Simple Alternator
qt
B

 B  AB cos  t
d B
E 
dt
  AB( sin  t )
  AB sin  t
E  AB sin  t
I 
R
R
2

t

sin
AB


Pelec  I 2 R 
R
 A2 B sin  t
M  IA 
R
Pmech   | M  B |   MB sin q 

 A2 B sin  t
R
B sin  t   Pelec
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Example: A coil of wire containing 500 circular loops with radius 4.00 cm is placed in
a uniform magnetic field. The direction of the field is at an angle of 60o with respect to
the plane of the coil. The field is decreasing at a rate of .200 T/s. What is the
magnitude of the induced emf?
60o
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A “new” lab experiment
I=I0 sin(wt)
r
 = B(r,t)
0 I 0 I0
B(r , t ) 

sin( t )
2 r 2 r
  B(r , t ) Acoil
0 I 0
Ecoil   NA
cos( t )
2 r
1
Ecoil 
r
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Eddy Currents
Induced currents in bulk conductors
magnetic forces on induced currents
energy losses to resistance
ex: Conducting disk rotating through a perpendicular
magnetic field
Ferromagnets ~ Iron (Electrical Conductor)
Constrain Eddy currents with insulating laminations
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Induced Electric Fields
Changing magnetic Flux produces an EMF E
B (increasing)
E
Induced Electric Field
 
E   E  dl
 
d B
E

d
l



dt
Not an Electrostatic field
Not Conservative!
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I’ induced E
I’ induced E
B
I
increasing
I
increasing
 
 E  dl  2 rE
d B

dt
1 d B
E
2 r dt
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  Qenc
 E  dA 
 
 B  dA 0
o
Maxwell’s Equations
  4
 E 
 
d E
 B  dl  o ( I c   o dt )
 
d M
 E  dl   dt
o
 
 B  0

 

dE
  B   o ( Jc   o
)
dt

 
dB
E  
dt
Lorentz Force Law

  
F  q ( E  v  B)
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Superconductivity
zero resistivity below Tc
(with no external magnetic field)
with field above the critical field Bc, no superconductivity
at any temperature
B
Bc
Normal
Superconducting
Tc
T
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More magnetic fun with superconductivity: the Meissner Effect
Within a superconducting material, Binside = 0!
magnetic field lines are expulsed from superconductor
Relative permeability Km = 0
internal field is reduced to 0
=> Superconductor is the perfect diamagnetic material!
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mechanical effects
ferromagnetic or paramagnetic materials attracted to
permanent magnet
superconductor repels permanent magnet!
Type-II superconductors
small filaments of “normal phase” coexist within bulk
superconductor
some magnetic field lines penetrate material
two critical fields field just begins to penetrate material
Bc1 : magnetic field just begins to penetrate material
Bc2 :bulk material ”goes normal”
more practical for electromagnets: higher Tc and Bc2
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