Summary
Lecture 19
The interaction between the electric and magnetic fields is what we refer to as
electromagnetism.
If a magnetic field is changing in some space, it results in appearance of the
electric field in this space.
Magnetic Flux and Electric Field
EMF Induced in Moving Conductor
Generators
Magnetic flux through the area A:
ΦB = B⊥ A = BAcosθ
Units for magnetic flux:
T ⋅ m 2 = weber = Wb
Faraday’s law: the induced EMF in a closed
loop is proportional to the rate of change of
magnetic flux through the loop.
ε = −N
ΔΦB
Δt
Lenz’s law: the current produced by an induced EMF moves in a direction such
that its magnetic field opposes the original change in flux.
Physics 112, Spring 2010, Feb 26, Lecture 19
Physics 112, Spring 2010, Feb 26, Lecture 19
Faraday’s Law
Lenz’s Law
Doubling the radius of a loop of wire produces what kind of change on the
According to Lenz's law, the direction of an induced current in a conductor will
be that which tends to produce which of the following effects?
induced EMF, assuming all other factors remain constant?
1) the induced EMF is 4 times as much
1) enhance the effect which produces it
2) the induced EMF is twice times as much
2) produce a greater heating effect
3) the induced EMF is half as much
3) produce the greatest voltage
4) there is no change in the induced EMF
ε=
2
4) oppose the effect which produces it
ΔΦB BΔA BΔ(π r 2 )
=
=
Δt
Δt
Δt
Physics 112, Spring 2010, Feb 26, Lecture 19
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Physics 112, Spring 2010, Feb 26, Lecture 19
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Lenz’s Law
Changing Magnetic Flux Induces Electric Field
A coil lies flat on a level table top in a region where the magnetic field vector
points straight up. The magnetic field suddenly grows stronger. When
viewed from above, what is the direction of the induced current in this coil as
the field increases?
Faraday’s law: changing magnetic flux induces
an EMF and can induce a current.
The electric field will be induced at any point in space
where there is a changing magnetic filed (even
without any conductor).
Open loop.
1) counterclockwise
2) clockwise
Induce current means that there is a closed loop
or the external resistor in series with the loop.
3) clockwise initially, then counterclockwise before stopping
Closed loop.
4) There is no induced current in this coil.
Electric field in a conductor moving in a magnetic field:
E=
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Physics 112, Spring 2010, Feb 26, Lecture 19
Conducting Rod Moving in Magnetic Field (#1)
r
B
F qvB
=
= vB
q
q
Space without loop.
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Physics 112, Spring 2010, Feb 26, Lecture 19
Conducting Rod Moving in Magnetic Field (#2)
r
v
l
Apply right-hand-rule #3.
A conducting rod is moved to the
right on a U-shaped conductor.
A conducting rod moves to the right with velocity, v, perpendicular to a
magnetic field, B.
Δx = vΔt
An EMF is induced in the rod of magnitude:
ΔA = lΔx = lvΔt
ε=
ε = Blv⊥
ΔΦB BΔA BlvΔt
=
=
= Blv
Δt
Δt
Δt
Physics 112, Spring 2010, Feb 24, Lecture 18
ε=
7
work (force)(distance) (qvB)l
=
=
= Blv
q
q
q
Physics 112, Spring 2010, Feb 24, Lecture 18
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Electromagnetic Blood-Flow Measurement
Airplane Flying in the Earth’s Magnetic Field
Boing 737 is flying at 720 km/h through a
region where the Earth’s magnetic field is
5 x 10-5 T and pointing down.
A
Blood contains charged ions. If the
blood vessel is 2 mm in diameter,
the magnetic field is 0.08 T, the
measured EMF is 0.1 mV, what is
the flow velocity of the blood?
How much potential difference is
created across the 35 m wingspan ?
r
B
Conductor is moving in a magnetic field!
ε = Blv⊥
v
B
1. An induced EMF:
ε = Blv⊥
B is a magnetic field of the Earth.
l is the length of the conductor (wingspan).
v is the speed or the conductor (wingspan).
2. Velocity of the blood:
(720 km / h)(1000 m / h)
720 km / h =
= 200 m / s
3600 s / h
v⊥ =
ε = Blv = (5 ×10−5T )(35 m)(200 m / s) = 0.35V
EMF Induced in Moving Rod
r
A conducting rod a 20 cm length is moving
B
Left end
1) zero
2) 2 V with right end positively charged
3) 2 V with right end negatively charged
4) 4 V with right end positively charged
5) 4 V with right end negatively charged
1.0 ×10−4V
= 0.63 m / s
(0.080T )(2.0 ×10−3 m)
Physics 112, Spring 2010, Feb 26, Lecture 19
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Electric Generators
A generator transforms mechanical energy into electrical energy.
Axle
r
v
v v⊥
The axle is rotated by an external
force, e.g. falling water or steam.
Right end
As it turns at constant speed v, an
alternating EMF, is induced in
each coil along edges a and b
B
ε = Blv⊥
ε = Blv⊥
ε = Blv⊥ = (2T )(0.2 m)(10 m / s) = 4 V
Physics 112, Spring 2010, Feb 26, Lecture 19
Bl
=
9
Physics 112, Spring 2010, Feb 24, Lecture 18
perpendicular to the magnetic field as
shown at speed of 10 m/s. If the magnetic
field is 2 T, what EMF will be induced and
how the right end will be charged?
ε
Measurements of blood
velocity from induced EMF.
v⊥ perpendicular to B
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Physics 112, Spring 2010, Feb 26, Lecture 19
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Generator Equation
Electric Generators
ε = Blv⊥
The EMF induced in the segments ab
and cd, whose velocity components
perpendicular to the magnetic field B
are:
v⊥ = v sin θ
ε = 2Blv sin θ
ε = 2NBlv sin θ
h = ad = bc
ε 0 = NB ωA
ω is angular frequency (radians/s)
f is frequency (Hz = 1/s)
ω = 2πf
vII to B
Electrical Energy
An electric generator can be
used as a motor and vice versa.
θ = ωt
ε = 2 NBlv sin ωt
ε = 2 NBlω (h / 2) sin ωt
v = ωr = ω (h / 2)
ε = NBωA sin ωt
A = lh
ε = NBωlh sin ωt
Physics 112, Spring 2010, Feb 26, Lecture 19
ε = NB ωA sin ωt = ε 0 sin ωt
Mechanical Energy
If the coil is rotating with constant angular velocity ω :
ω =θ /t
v⊥ to B
An AC generator produces an alternating
voltage; the output EMF:
Faraday’s law can be generalized:
A changing magnetic flux induces an electric field.
Note that an electric field is produced regardless of whether
there are any conductors around.
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Physics 112, Spring 2010, Feb 26, Lecture 19
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Generator
A generator has a coil of wire rotating
in a magnetic field.
The rotation rate increases.
What happens to the maximum output
voltage of the generator?
1) it increases
2) it decreases
3) it varies sinusoidally
ε = NBωA sin ωt = ε 0 sin ωt
ε 0 = NBωA
4) it stays the same
Physics 112, Spring 2010, Feb 26, Lecture 19
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