Chapter29: Electromagnetic Induction
Motional EMF’s
Conductor Moving through a magnetic field
Magnetic Force on charge carriers
F = qv ×B
Accumulation of charge
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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I
Consider circuit at right with total circuit
resistance of 5 Ω, 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.
dE = v ×B⋅dl
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Faraday disk dynamo = DC generator
dE = v × B ⋅ dl
= ω rBdr
R
E = ∫ ω rBdr
dl=dr
0
R2
=ω B
2
I
ω
B
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Faraday’s Law of Induction
Changing magnetic flux induces and EMF
dΦ B = B ⋅ dA = BdA cos φ
Φ B = ∫ B ⋅ dA
Φ B = B ⋅ A = BA cos φ for a uniform magnetic field
=−
dΦ B 
dΦ B

for N loops 
 = −N
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
dΦ = BdA = Blvdt
dΦ
E=−
= − Blv
dt
watch sign conventions + RHR
Lenz' s Law + induced currents
I
dA=l vdt
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Simple Alternator
θ=ωt
B
Φ B = AB cos ω t
dΦ B
dt
= − AB(−ω sin ω t )
E =−
= ω AB sin ω t
I=
ω
E
ω AB sin ω t
R
(
ω AB sin ω t )2
2
Pelec = I R =
R
2
ω A B sin ω t
M = IA =
R
Pmech = τω =| M × B | ω = (MB sin θ )ω
=
R
=
ω A2 B sin ω t
R
B sin ω t ω = 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 determination of B vs r (lab experiment)
I=I0 sin(wt)
B(r , t ) =
r
µ0 I µ0 I 0
=
sin(ω t )
2π r 2π r
Φ ≈ B (r , t ) Acoil
Φ = B(r,t)
Ecoil = −ω NA
Ecoil ∝
µ0 I 0
cos(ω t )
2π r
1
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
dΦ 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 = 2π rE
dΦ B
dt
1 dΦ B
E=
2π r dt
=
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Qenc
E
∫ ⋅ dA =
εo
Maxwell’s Equations
4πρ
∇⋅ E =
B
∫ ⋅ dA =0
dΦ E
µ
ε
B
⋅
dl
=
(
I
+
)
o
c
o
∫
dt
dΦ M
E
⋅
dl
=
−
∫
dt
εo
∇⋅ B = 0
dE
∇ × B = µ o ( Jc + ε 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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