Unformatted text preview: 4/6/10 CHAPTER 29
ELECTROMAGNETIC
INDUCTION
Vahé Peroomian LET’S BEGIN WITH AN EXPERIMENT!
What happens when a magnet is moved in
and out of a coil? MORE EXPERIMENTAL DATA
When B = 0, the galvanometer shows
no current.
When electromagnet is turned on,
increasing B, there is a momentary
current.
Any changes in the orientation of
the coil, or its position, etc., will
cause momentary jumps in the
needle, but only as long as the
motion is taking place.
What is changing in all these
experiments? 1 4/6/10 MAGNETIC FLUX
The magnetic flux dΦB through an
infinitesimalarea element dA in a
magnetic field B d Φ B = B ⋅ dA
= B⊥ dA = BdA cos φ
Choice of direction
of dA is up to us! FARADAY’S LAW OF INDUCTION
The induced emf in a closed loop equals
the negative of the time rate of change of
the magnetic flux through the loop.
ε=− dΦ B
dt Where did that minus sign come from??? EXAMPLE 1
A coil consists of 200 turns of wire. Each turn is a square
of side d = 18 cm, and a uniform magnetic field directed
perpendicular to the plane of the coil is turned on. If the
field changes linearly from 0 to 0.50 T in 0.80 s, what is the
magnitude of the induced emf in the coil while the field is
changing? 2 4/6/10 DIRECTION OF INDUCED EMF .  Faraday’s Law of Induction
Choose a direction for A From the directions of A c ont.
and B , find the sign of
Φ and dΦ/dt.
ION
3. Find the sign of the
tualize From the description in the problem, imagine magnetic field lines passing through the coil. Because the
induced current:
t ic field is changing in magnitude, an emf is induced in the coil.
If flux is increasing,
rize We will evaluate the emf using Faraday’s law from this section, so we categorize this example as a substitution
dΦ/dt is positive,
.
and induced emf or
Bf 2 Bi
DFB
D 1 BA 2
DB
0e0 5
te Equation 31.2 for the situation described here, negative. N
current is
5N
5 NA
5 Nd 2
Dt
Dt
Dt
Dt
t hat the magnetic field changes linearlyUse right hand rule
with
4.
to find direction of
induced emf. 0 e 0 5 1 200 2 1 0.18 m 2 2 1 0.50 T 2 0 2 5 4.0 V
ute numerical values: 1.
2. 0.80 s IF? W hat if you were asked to find the magnitude of the induced current in the coil while the field is changing?
u answer that question? 0 e0 r I f the ends of the coil are not connected to a circuit, the answer to this question is easy: the current is zero!
es move within the wire of the coil, but they cannot move into or out of the ends of the coil.) For a steady current
, t he ends of the coil must be connected to an external circuit. Let’s assume the coil is connected to a circuit and
al resistance of the coil and the circuit is 2.0 V. Then, the magnitude of the induced current in the coil is
4.0 V 5
I5
EXAMPLE 2 5 2.0 A
R
2.0 V A loop of wire enclosing an area A is placed in a
region where the magnetic field is perpendicular to
the plane of the loop. The magnitude of B varies in
a m pl e .
An Exponentially Decaying Magnetic Field Bmaxe–at,
time according to the expression B =
where a is some constant. That is, at t = 0, the field
of wire enclosing an area A is placed in a
B
is Bmax,
where the magnetic field is perpendicular and for t > 0, the field
S
plane of the loop. The magnitudedof B var exponentially (see
ecreases
2at
ime according to the expression B 5igure)., Find the induced emf
Bmax
f B maxe
a is some constant. That is, at t 5 0, the field
iexponen as a function of time.
n the loop
, and for t . 0, the field decreases
Fig. 31.6). Find the induced emf in the loop
ction of time.
ION Figure . (Example 31.2)
Exponential decrease in the
magnitude of the magnetic
f ield with time. The induced
emf and induced current vary
w ith time in the same way. t tualize The physical situation is similar to
Example 31.1 except for two things: there is only one loop, and the field varies exponentially with time rather
nearly.
d FB
d
d 2at
5 2 1 ABmax e 2at 2 5 2ABmax
e 5 aABmax e 2at
dt
dt
dt rize We will evaluate the emf using Faraday’s law from this section, so we categorize this example as a substitution
.
te Equation 31.1 for the situation
ed here: e52 EXAMPLE 29.4 pression indicates that the induced emf decays exponentially in time. The maximum emf occurs at t 5 0, where
aAB max. The plot of e versus t is similar to the B versust curve shown in Figure 31.6. The figure below shows a simple version of an alternator, a
device that generates an emf. A rectangular loop is made to
rotate with constant angular speed ω about the axis shown.
The magnetic field B is uniform and constant. At time t =
0, φ = 0. Determine the induced emf. 897
897 10/6/09 8:46:17 AM
10/6/09 8:46:17 AM 3 4/6/10 EXAMPLE 29.5
Instead of the alternating current (ac) emf produced by the
generator in Example 29.4, we can use a similar scheme to
create a direct current (dc) generator that produces an emf
that always has the same sign. Consider a motor with a
square coil 10.0 cm on a side, with 500 turns of wire. If the
magnetic field has magnitude 0.200 T, at what motion
speed is the average back emf of the motor equal to 112 V? EXAMPLE 29.6
The figure shows a Ushaped conductor in a uniform
magnetic field B perpendicular to the plane of the figure,
directed into the page. We lay a metal rod with length L
across the two arms of the conductor, forming a circuit, and
move the rod to the right with constant velocity v. This
induces an emf and a current, which is why the device is
called a slidewire generator.
Find the magnitude and
direction of the reducing
induced emf. EXAMPLE 29.7
In the slidewire generator, energy is dissipated in the circuit
owing to its resistance. Let the resistance of the circuit at a
given point in the slidewire’s motion be R. Show that the
rate at which work must be done to move the rod through
the magnetic field. 4 4/6/10 PROBLEM 29.10
A rectangle measuring 30.0 cm by 40.0 cm is located inside
a spatially uniform magnetic field of 1.25 T, with the field
perpendicular to the plane of the coil. The coil is pulled
out at a steady rate of 2.00 cm/s traveling perpendicular to
the field lines. The region of the field ends abruptly. Find
the emf induced in this coil when it is
(a) All inside the field
(b) Partly inside the field
(c) All outside the field. LENZ’S LAW
The direction of any magnetic induction
effect is such as to oppose the cause of the
effect (from our textbook).
The induced current in a loop is in the
direction that creates a magnetic field that
opposes the change in magnetic flux
through the area enclosed by the loop
(from Serway and Jewett).
LENZ’S LAW AND THE SLIDEWIRE .  Lenz’s Law Figure . (a) Lenz’s law can be
used to determine the direction of
t he induced current. (b) When the
bar moves to the left, the induced
current must be clockwise. Why? As the conducting bar slides to the
right, the magnetic flux due to the
external magnetic field into the
page through the area enclosed by
the loop increases in time.
S S Bin Bin By Lenz’s law, the
induced current
must be
counterclockwise
to produce a
counteracting
magnetic field
directed out of
the page. a I R S v R I S v b the area enclosed by the circuit increases with time because the area increases.
Lenz’s law states that the induced current must be directed so that the magnetic
f ield it produces opposes the change in the external magnetic flux. Because the
magnetic flux due to an external field directed into the page is increasing, the
induced current—if it is to oppose this change—must produce a field directed out
of the page. Hence, the induced current must be directed counterclockwise when
the bar moves to the right. (Use the righthand rule to verify this direction.) If the
bar is moving to the left as in Figure 31.11b, the external magnetic flux through the
area enclosed by the loop decreases with time. Because the field is directed into
the page, the direction of the induced current must be clockwise if it is to produce a
f ield that also is directed into the page. In either case, the induced current attempts
to maintain the original flux through the area enclosed by the current loop.
Let’s examine this situation using energy considerations. Suppose the bar is
g iven a slight push to the right. In the preceding analysis, we found that this motion 5 4/6/10 LENZ’S LAW AND
A CURRENT LOOP MOTIONAL ELECTROMOTIVE FORCE
Motional emf for a closed
conducting loop is given by
ε= ( v × B ) ⋅ dl
∫ EXAMPLE 3
CHAPTER  Faraday’s Law The conducting bar illustrated in the figure moves on two
frictionless, parallel rails in the presence of a uniform
magnetic field directed into the page. The bar has mass m,
pl e .
Magnetic Force Acting on a Sliding Bar initial velocity v to
and its length is l. The bar is given an
i
the right and is released at t = 0.
ting bar illustrated in Figure 31.9 moves on two frictionless,
(a) Using Newton’s into find
in the presence of a uniform magnetic field directed laws,the the
velocity given an as a function
ar has mass m , and its length is ,. The bar is of the bar initial
t he right and is released at t 5 0. of time.
S
R
(b) Show that the same result is
FB
ewton’s laws, find the velocity of the bar as a function of
found by using an energy
approach. S Bin S vi I ze A s the bar slides to the right in Figure 31.9, a counterclockt is established in the circuit consisting of the bar, the rails,
tor. The upward current in the bar results in a magnetic force
t he bar as shown in the figure. Therefore, the bar must slow
r mathematical solution should demonstrate that. x Figure . (Example 31.3) A conducting
bar of length , on two fixed conducting rails
S
is given an initial velocity v i to the right. The text already categorizes this problem as one that uses
s. We model the bar as a particle under a net force.
m Equation 29.10, the magnetic force is FB 5 2I,B , where the negative sign indicates that the force is to the
g netic force is the only horizontal force acting on the bar.
n’s second law to the bar in the horizontal 5 B ,v/R from Equation 31.6: dv
5 2I ,B
Fx 5 ma 5 m
dt
m dv 52 B 2 ,2 v 6 4/6/10 EXAMPLE 4
A conducting bar of length l rotates with a constant angular
speed ω about a pivot at one end. A uniform magnetic field
B is directed perpendicular to the plane of rotation as
shown in the figure. Find the motional emf induced
between the ends of the bar. PROBLEM 29.49
A loop is being pulled to the right at constant speed v away from
an infinitely long straight wire with current I.
(a) Calculate the magnitude of the net emf induced in the loop.
Do this two ways: (i) by using Faraday’s law of induction and
(ii) by looking at the emf induced in each segment of the loop
due to its motion.
(b) Find the direction of the current induced in the loop. Do this
two ways: (i) using Lenz’s law, and (ii) using the magnetic force
on the charges in the loop.
(c) Check your answer to part (a) in the following special cases
to see whether it is physically reasonable: (i) the loop is
stationary, (ii) The loop is very thin, so a ~ 0, and (iii) The loop
gets very far from the wire. 7 ...
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 Spring '10
 peroomian
 Physics, Current, Magnetic Field, Lenz

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