General Physics (PHY 2140)
Lecture 16
¾ Electricity and Magnetism
9Induced voltages and induction
9 Magnetic flux and induced emf
9 Faraday’s law
http://www.physics.wayne.edu/~apetrov/PHY2140/
Chapter 20
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Lightning Review
Last lecture:
1. Magnetism
9 Ampere’s law and applications
2. Induced voltages and induction
9 Magnetic flux
µ0 I
B=
2π r
B = µ0 nI
Φ = BA cos θ
Review Problem: A sphere of radius R is placed near a long, straight wire that
carries a steady current I. The magnetic field generated by the current is B. The
total magnetic flux passing through the sphere is
1. µoI.
2. µoI /(4 πR2).
3. 4 πR2 µoI
4. zero.
5. need more information
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Recall: right hand rule II
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20.1 Induced EMF and magnetic flux
Faraday’s experiment
Picture © Molecular Expressions
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Two circuits are not connected:
no current?
However, closing the switch
we see that the compass’
needle moves and then goes
back to its previous position
Nothing happens when the
current in the primary coil is
steady
But same thing happens when
the switch is opened, except
for the needle going in the
opposite direction…
What is going on?
4
20.2 Faraday’s law of induction
Induced current
I
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v
S
N
5
20.2 Faraday’s law of induction
I
v
B
S
I
N
B
I
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v
A current is set up in the circuit as long as
there is relative motion between the magnet
and the loop.
6
Does there have to be motion?
I
(induced) I
-
+
AC Delco
1 volt
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Does there have to be motion?
I
-
+
AC Delco
1 volt
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Does there have to be motion?
I
(induced)
-
+
AC Delco
1 volt
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NO!!
Does there have to be motion?
-
+
AC Delco
1 volt
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Maybe the B-field needs to change…..
B
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v
11
Maybe the B-field needs to change…..
I
B
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v
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Maybe the B-field needs to change…..
I
I
v
B
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I
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Faraday’s law of magnetic induction
In all of those experiment induced EMF is caused by a change in the
number of field lines through a loop. In other words,
The instantaneous EMF induced in a circuit equals the rate of change
of magnetic flux through the circuit.
E=
Lenz’s law
∆Φ
N
∆t
The number of loops matters
Lenz’s Law: The polarity of the induced emf is such that it produces a
current whose magnetic field opposes the change in magnetic flux
through the loop. That is, the induced current tends to maintain the
original flux through the circuit.
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Applications:
Ground fault interrupter
Electric guitar
SIDS monitor
Metal detector
…
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Example 1: EMF in a loop
A wire loop of radius 0.30m lies so that an external magnetic field
of strength +0.30T is perpendicular to the loop. The field changes
to -0.20T in 1.5s. (The plus and minus signs here refer to opposite
directions through the loop.) Find the magnitude of the average
induced emf in the loop during this time.
JG
B
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A wire loop of radius 0.30m lies so that an external magnetic field of strength
+0.30T is perpendicular to the loop. The field changes to -0.20T in 1.5s. (The
plus and minus signs here refer to opposite directions through the loop.) Find
the magnitude of the average induced emf in the loop during this time.
Given:
r = 0.30 m
Bi = 0.30 T
Bf = -0.20 T
∆t = 1.5 s
The loop is always perpendicular to the field, so the
normal to the loop is parallel to the field, so cosθ = 1.
The flux is then
Φ = BA = Bπ r 2
Initially the flux is
Φi = ( 0.30T ) π ( 0.30m ) =0.085 T ⋅ m 2
2
Find:
EMF=?
and after the field changes the flux is
Φ f = ( −0.20T ) π ( 0.30m ) =-0.057 T ⋅ m2
2
The magnitude of the average induced emf is:
∆Φ Φ f − Φi 0.085 T ⋅ m 2 -0.057 T ⋅ m 2
emf =
=
=
= 0.095V
1.5s
∆t
∆t
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Example 2: EMF of a flexible loop
The flexible loop in figure below has a radius of 12cm and is in a magnetic
field of strength 0.15T. The loop is grasped at points A and B and
stretched until it closes. If it takes 0.20s to close the loop, find the
magnitude of the average induced emf in it during this time.
X
X
X
A
X
X
X
X
X
X
X
X
X
X
X
X
X
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B
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20.3 Motional EMF
l
B
F
v
Let's consider a conducting bar moving perpendicular to a uniform
magnetic field with constant velocity v.
F = qvB sin θ
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This force will act on free charges in the conductor. It will tend to
move negative charge to one end, and leave the other end of the bar
with a net positive charge.
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Motional EMF
The separated charges will create an electric field which
will tend to pull the charges back together.
When equilibrium exists, the magnetic force, F=qvB, will
balance the electric force, F=qE, such that a free charge
in the bar will feel no net force.
Thus, at equilibrium, E = vB. The potential difference
across the ends of the bar is given by ∆V=El or
∆V = El = Blv
A potential difference is maintained across the conductor
as long as there is motion through the field. If the motion
is reversed, the polarity of the potential difference is also
reversed.
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Motional EMF – conducting rails
R
∆x
B
v
We can apply Faraday's law to the complete loop. The change of flux through
the loop is proportional to the change of area from the motion of the bar:
∆Φ = BA = Bl ∆x
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current
I=
or (Faraday’s law)
E Blv
=
R
R
∆Φ
∆x
E=
= Bl
= Blv
∆t
∆t
Motional EMF
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Example: wire in the magnetic field
Over a region where the vertical component of the Earth's magnetic field is
40.0µT directed downward, a 5.00 m length of wire is held in an east-west
direction and moved horizontally to the north with a speed of 10.0 m/s.
Calculate the potential difference between the ends of the wire, and
determine which end is positive.
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20.4 Lenz’s law revisited
Application of Lenz's law will tell
us the direction of induced
currents, the direction of
applied or produced forces,
and the polarity of induced
emf's.
Lenz's law says that the induced current will produce
magnetic flux opposing this change. To oppose an
increase into the page, it generates magnetic field which
points out of the page, at least in the interior of the loop.
Such a magnetic field is produced by a counterclockwise
current (use the right hand rule to verify).
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Lenz’s law: energy conservation
We arrive at the same conclusion from
energy conservation point of view
The preceding analysis found that the
current is moving ccw. Suppose that this
is not so.
„
If the current I is cw, the direction of the
magnetic force, BlI, on the sliding bar
„
„
„
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would be right.
This would accelerate the bar to the right,
increasing the area of the loop even more.
This would produce even greater force
and so on.
In effect, this would generate energy out of
nothing violating the law of conservation of
energy.
Our original
assertion that the
current is cw is
not right, so the
current is ccw!
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S
S
v
The induced
flux seeks to
counteract
the change.
N
change
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S
S
N v
N
change
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Example: direction of the current
Find the direction of the current induced in
the resistor at the instant the switch is
closed.
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Applications of Magnetic Induction
Tape / Hard Drive / ZIP Readout
„
Tiny coil responds to change in flux as the magnetic domains (encoding
0’s or 1’s) go by.
Question: How can your VCR display an image while paused?
Credit Card Reader
„
–
Must swipe card
Æ generates changing flux
Faster swipe Æ bigger signal
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