Physics 54 Lecture
March 1, 2012
OUTLINE
Micro-quiz problems (magnetic fields and forces)
Magnetic dipoles and their interaction with magnetic fields
Electromagnetic induction
 Introduction to electromagnetic induction
 Production of emf’s by a time varying magnetic field
 Lenz’s Law
 Motional emf’s
 Eddy currents
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A couple of Micro-quiz
problems
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Micro Quiz 1
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Micro Quiz 1
An electron is traveling horizontally east in the magnetic
field of the earth near the equator near the equator.
The direction of the force on the electron is:
N
N
W
B
E
S
W
v
E
-e
S
The electron is deflected downward: Fe = - e v x B
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Micro Quiz 2
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Micro Quiz 2
Fm = q v x B
Fe = q E
up
B
down
v
q v B = q E balance
E/B = v independent
of q or m
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Micro Quiz 3
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Micro Quiz 3
I into page
B
lectron velocity
E
out of page
2
Fe = - e v x B
he electrons move to the right
T
and accumulate on surface 2
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Magnetic dipoles
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Introduction:
monopoles and dipoles
The
simplest electric object in Nature is a point charge q
or electric monopole. An electron is an example of an
ideal point charge.
The
next simplest electric object
is an electric dipole. Atoms and
molecules can carry permanent or
induced electric dipole moments p.
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Introduction:
monopoles and dipoles
No single magnetic single poles have been observed in
Nature. There are apparently no magnetic monopoles!
The
simplest magnetic objects in Nature are magnetic
dipoles given the symbol µ .
Magnetic
dipoles are created by current loops or the
rotation of charged objects. Elementary particles, such
as electrons, carry a magnetic dipole moment.
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Introduction:
monopoles and dipoles
A simple iron bar magnet is a magnetic dipole.
The source of “current loops” in this case are
from the atomic structure of the atoms in the magnet.
Electric dipoles can be broken apart:
+
break up
p
+
electric monopoles
= single charges
- In contrast magnetic dipoles can not:
N
S
µ
N
S
break up
N
S
here are
T
no isolated magnetic
monopoles
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Definition of a magnetic dipole
A magnetic dipole is created by a closed current loop.
Consider a loop with N turns of wire.
Let n be a unit vector
perpendicular to the wire loop
in a direction determined by
the current and the right hand rule.
Define the magnetic moment µ of this loop to be:
v
v
µ=INAn
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Torque on a magnetic dipole
Recall that for an electric dipole in a uniform electric field:
 The net force is zero
 The torque τ = p x E
 The potential energy is U = - p E
.
For a magnetic dipole in a uniform magnetic field:
 The net force is zero
.
 The torque is τ = µ x B (See derivation in text)
 The potential energy is U = - µ B
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Example 1
(torque on a wire loop)
Find the torque on the current loop.
b
a
current I
magnetic field B
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Electromagnetic Induction
(a new topic: Chapter 34)
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Introduction
We will now extend electromagnetic theory by introducing
a new phenomenon called electromagnetic induction.
This is the observation that a time varying magnetic
field creates an electromotive force (emf) , and
at a more basic level an electric field.
Electromagnetic induction is described by two of the
simplest laws of electromagnetic theory:
 Faraday’s Law and Lenz’s Law
The applications of electromagnetic induction are far
reaching. For example electric motors and generators operate
based upon the the principle of electromagnetic induction.
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Faraday’s Law
and induced emf’s
We will first introduce Faraday’s Law in a very practical way.
We will use it to relate a time varying magnetic flux
to an electromotive force (emf) induced in a wire.
Let ΦB = the magnetic flux through the area bounded
by a single closed wire loop.
urface
S
bounded
by the loop
ΦB
S
Normal defined by curve direction
and the right hand rule
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Faraday’s Law
and induced emf’s
With the definition of magnetic flux given on the previous
page, Faraday’s Law predicts that the emf induced in the wire
loop will be:
= | dΦB/dt |
mf induced
e
in wire loop
 agnetic flux
m
through the loop
As stated above the emf is for one loop of wire.
If there are N overlapping loops in series then the emf is:
= N | dΦB/dt | where ΦB = the magnetic flux through one loop
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The sign of the induced emf
is determined by Lenz’s Law
An easy way to keep track of the sign of the induced emf is
formulated by a rule called Lenz’s Law.
The induced emf produces a current, I, that creates
a magnetic field that opposes the change in ΦB .
See the next example.
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Example 2
magnet is moved into a wire loop as shown.
A
In which direction will the current flow?
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Example 2 (solution using
Lenz’s Law)
B2
B1
B2
B1
urrent
c
counterclockwise
The magnetic field B1 caused by the moving magnet
causes a magnetic flux increase through the loop.
his induces an emf which causes a current I
T
that produces a magnetic field B2 opposing B1.
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Example 3
A wire loop is a circle of radius r = 50 cm.
It has a resistance R = 0.1 Ω .
A magnetic field is applied perpendicular to the loop.
B(t)
= 10s
T
o = 1 Tesla
B
B(t)
) Find the direction of the current
a
b) Find I(t)
c) Find the heat generated in the coil.
Bo
T
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Example 4
A loop of wire with resistance R falls under the
influence of gravity.
Io
A. zero
The current induced
in the wire is:
B. non-zero clockwise
C. non-zero counterclockwise
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Example 5
A magnetic field is constant in space but varies with
time: B(t) = [0.3 + 0.5 t2 ] Tesla (perpendicular to page) .
b
a
A. a b t
he magnitude of the emf
T
induced in the above loop is:
B. a b t2
C. ab(0.3 + t)
D. ab( 0.3 + a b t2)
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Example 6
A bar magnet is dropped through a wire loop as shown
S
N
he current induced
T
in the loop will be:
A. zero
B. clockwise
C. counter clockwise
D. clockwise and
counter clockwise
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Motional emf
A time-varying magnetic field through a stationary
conducting loop generates an emf via Faraday’s Law.
stationary
loop
time varying
magnetic fux ΦB
= | d ΦB/dt |
Faraday’s Law also predicts that a conductor moving
through a constant magnetic field will generate an emf.
constant
magnetic field Bo
L
v
= Bo L v
“motional” emf
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Motional emf (con.)
The form of the motional emf can be derived as follows.
Consider a metal rod sliding on two fixed conducting bars
L
constant
magnetic field Bo
out of page
v
x
Magnetic flux in loop = ΦB = Bo L x
and dΦB/dt = Bo L dx/dt = Bo L v
Using Faraday’s Law the magnitude of the emf = dΦB/dt
and the motional emf = = Bo L v
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Example 7
The magnetic field Bo is out of the page
he polarity across the
T
resistor will be:
A. top + bottom B. top - bottom +
C. no potential difference
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Currents generated
in moving conducting plates
When a metal plate or conductor is pulled through a nonuniform magnetic field, currents in the form of “eddys”,i.e.
eddy currents, are formed.
The time-varying magnetic
flux through loops in the
metal generate emf’s
and therefore currents.
The currents react back
on the magnetic field
causing a retarding force.
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Currents generated
in moving conducting plates
Eddy currents therefore can be used to generate a magnetic
brake in which the force increases as the speed of the
moving conductor increases.
In some cases eddy
currents are undesirable
as they transform
energy into heat.
(for example in an AC transformer).
The eddy currents can be suppressed
by cutting or laminating the
metal plates.
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Example 8
A metal plate falls into a region of magnetic field
as shown below.
he current when the plate
T
enters the field will be:
A. clockwise
B. counter clockwise
C. zero
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Example 8
(solution)
he induced eddy currents
T
act as a break to stop the
motion of the metal plate
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Example 9
(eddy currents)
 Find the direction of the force on the copper ring due to
the induced eddy currents.
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End Lecture:
Next one on March 13:
Will cover the remaining material
in Chapter 34
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Example 8
A constant current Iwire = Io flows through the long straight
wire. The loop of wire moves upward with constant
velocity vo so that r(t) = ro + vo t
Find the direction and magnitude of the current I(t)
in the loop of wire if it has resistance R.
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