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Lecture 4/13/2017
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Review: Induced EMF
Motional EMF
Induced E-Fields
Eddy Currents
Shrinking Wire Loop
If a coil is shrinking in a
magnetic field pointing
a) clockwise
into the page, in what
b) counterclockwise
direction is the induced
c) no induced current
current?
If a coil is rotated as shown,
in a magnetic field pointing
to the left, in what direction
is the induced current?
a) clockwise
b) counterclockwise
c) no induced current
A wire loop is being pulled away
from a current-carrying wire.
a) clockwise
What is the direction of the
b) counterclockwise
induced current in the loop?
c) no induced current
I
A conducting rod slides on a
conducting track in a
constant B field directed into
the page. What is the
a) clockwise
b) counterclockwise
c) no induced current
direction of the induced
current?
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
x x x x x x x x x x x
v
EMF Induced in a Moving Conductor
EMF Induced in a Moving Conductor
A conducting metal sheet moves with initial velocity 𝑣റ as shown. Over
time,
a) The bar accelerates due to an external magnetic force.
b) The sheet will slow down due to an external magnetic force.
c) The sheet will move at a constant speed because there is no external
magnetic force.
d) The sheet will move at a constant speed, there is a magnetic force but
it does not do work on the system.
EMF Induced in a Moving Conductor
The induced current is in a direction that tends to slow the
moving bar – it will take an external force to keep it moving.
Induced EMF Induced in a Moving Conductor
Lets derive an equation for the induced emf in terms of the relevant quantities.
ℇ=−
𝑑Φ𝐵
𝑑𝑡
Φ𝐵 = 𝐵𝑑𝐴
→
𝑑𝐴 = ℓ𝑑𝑥 = 𝑙𝑣𝑑𝑡
ℇ=−
→
𝐵𝑑𝐴
𝑑𝑡
ℇ=−
𝐵ℓ𝑣𝑑𝑡
𝑑𝑡
Then, the induced emf, ℇ, has a magnitude:
ℇ = −𝐵ℓ𝑣
Is this equation always valid?
This equation is valid as long as B, ℓl, and v are mutually perpendicular (if
not, it is true for their perpendicular components).
A copper bar of length ℓ = 42.0 𝑐𝑚 rotates with angular speed 𝜔 = 2.50 𝑟𝑎𝑑/𝑠
about a pivot at one end. A uniform magnetic field B=1.500 T is directed
perpendicularly to the plane of rotation. Find the motional emf induced between
the ends of the bar.
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1
ℇ = 𝐵𝜔ℓ2
2
Electric Generators
A generator transforms mechanical energy into electrical
energy. This is an ac generator:
Electric Generators
If the loop is rotating with constant angular
velocity ω, the induced emf is sinusoidal:
For a coil of N loops,
Back EMF and Counter Torque
An electric motor turns because there is a
torque on it due to the current.
We would expect the motor to accelerate
unless there is some sort of drag torque.
That drag torque exists, and is due to the
induced emf, called a back emf.
The armature windings of a dc motor have a resistance of 5.0 Ω. The motor is
connected to a 110-V line, and when the motor reaches full speed against its
normal load, the back emf is 98 V. Calculate
(a) the current into the motor when it is just starting up, and (b) the current
when the motor reaches full speed.
98 V
110
V
Back EMF and Counter Torque
A similar effect occurs in a generator – if it is
connected to a circuit, current will flow in it,
and will produce a counter torque. This means
the external applied torque must increase to
keep the generator turning.
Eddy Currents
Induced currents can flow in bulk material as
well as through wires. These are called eddy
currents.
Eddy Currents
Induced currents can flow in bulk material as
well as through wires. These are called eddy
currents.
A) 𝑣𝑖𝑛 > 𝑣𝑜𝑢𝑡
𝑣𝑖𝑛
B) 𝑣𝑖𝑛 < 𝑣𝑜𝑢𝑡
C) 𝑣𝑖𝑛 = 𝑣𝑜𝑢𝑡
D) Impossible to tell with the given information
𝑣𝑜𝑢𝑡