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General Physics 2
Quarter 4 – Week 1
Module 1 – Magnetic Induction and
Faraday’s Law
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General Physics 2
Grade 12 Quarter 4: Week 1 - Module 1 – Magnetic Induction and
Faraday’s Law
First Edition, 2021
Copyright © 2021
La Union Schools Division
Region I
All rights reserved. No part of this module may be reproduced in any form
without written permission from the copyright owners.
Development Team of the Module
Author: DARRYL G. BERSALONA, SST-I
Editor: SDO La Union, Learning Resource Quality Assurance Team
Illustrator: Ernesto F. Ramos Jr., P II
Management Team:
Atty. Donato D. Balderas, Jr.
Schools Division Superintendent
Vivian Luz S. Pagatpatan, Ph.D
Assistant Schools Division Superintendent
German E. Flora, Ph.D, CID Chief
Virgilio C. Boado, Ph.D, EPS in Charge of LRMS
Rominel S. Sobremonte, Ed.D, EPS in Charge of Science
Michael Jason D. Morales, PDO II
Claire P. Toluyen, Librarian II
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Target
Almost every modern device or machine, from a computer to a washing machine to
a power drill, has electric circuits at its heart. We learned from the previous lessons that
an electromotive force (emf) is required for a current to flow in a circuit; and we almost
always took the source of emf to be a battery.
In the preceding discussion, you have learned about Biot-Savart Law and Ampere’s
Law. In this section, we would be going to deal with magnetic induction and Faraday’s Law.
After going through this module, you are expected to:
1. Identify the factors that affect the magnitude of the induced emf and the magnitude
2.
3.
and direction of the induced current (Faraday’s Law) (STEM_GP12EMIVa-1);
Compare and contrast electrostatic electric field and non-electrostatic/induced
electric field (STEM_GP12EMIVa-3);
Calculate the induced emf in a closed loop due to a time-varying magnetic flux using
Faraday’s Law (STEM_GP12EMIVa-4).
Before going on, check how much you know about this topic. Answer the
pretest on the next page in a separate sheet of paper.
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Jumpstart
For you to understand the lesson well, do the following activities.
Have fun and good luck!
Direction: Write the letter of the term or phrase that best completes the statement or
answers the question.
1. Electromagnetic induction is change in ___________________.
A. surface area
B. magnetic flux C. magnetic poles D. electric field
2. What would happen if I move a bar magnet in and out of a coil of copper wire?
A. Electric current would disappear
B. It would produce a gravitational field
C. Electric current will flow through the wire
D. The magnet would explode
3. ___________ law says that the Induced current is proportional to the change of
magnetic flux.
A. Lenz's
B. Ampere's
C. Biot-Savart’s
D. Faraday's
4. Where is the strongest attraction force of the magnet?
A. at the poles B. above the magnet C. in the middle
D. below the magnet
5. What type of current is produced by a battery?
A. parallel current B. direct current C. alternating current D. potential current
6. What creates a magnetic field?
A. charged particles that do not move
C. moving electric charges
B. gravity
D. an isolated magnetic pole
7. Voltage can be induced in a wire by _______________.
A. moving the wire near a magnet
B. moving a magnet near the wire
C. changing the current in a nearby wire
D. all of these
8. A magnet can move in a coil of wire to produce electricity in which system?
A. Generator
B. Magnet
C. Motor
D. Transformer
9. Magnetic Field lines around a bar magnet ____________.
A. are perpendicular the magnet
B. cross back and forth over each other
C. spread out from north pole and curve to south
D. are perfectly straight
10. How do Maglev trains go up to 311 MPH?
A. a train is pulled by a big magnet at the end of the tracks
B. magnetized coils repel magnets on the train which moves it
C. electric motors push the train and cause it to levitate
D. a generator creates electricity which fuels the train
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Discover
In the previous lesson we almost always took the source of emf to be a
battery. But for the vast majority of electric devices that are used in industry and
in the home (including any device that you plug into a wall socket), the source of
emf is not a battery but an electric generating station. Such a station produces
electric energy by converting other forms of energy: gravitational potential energy
at a hydroelectric plant, chemical energy in a coal- or oil-fired plant, nuclear energy
at a nuclear plant. But how is this energy conversion done?
The answer is a phenomenon known as electromagnetic induction: If the
magnetic flux through a circuit changes, an emf and a current are induced in the
circuit. In a power-generating station, magnets move relative to coils of wire to
produce a changing magnetic flux in the coils and hence an emf. Other key
components of electric power systems, such as transformers, also depend on
magnetically induced emfs.
The central principle of electromagnetic induction, and the keystone of this
chapter, is Faraday’s law. This law relates induced emf to changing magnetic flux
in any loop, including a closed circuit.
Induction Experiments
During the 1830s, several pioneering experiments with magnetically
induced emf were carried out in England by Michael Faraday and in the United
States by Joseph Henry (1797–1878), later the first director of the Smithsonian
Institution. Figure 1 shows several examples. In Fig.1a, a coil of wire is connected
to a galvanometer. When the nearby magnet is stationary, the meter shows no
current. This isn’t surprising; there is no source of emf in the circuit. But when we
move the magnet either toward or away from the coil, the meter shows current in
the circuit, but only while the magnet is moving (Fig.1b). If we keep the magnet
stationary and move the coil, we again detect a current during the motion. We call
this an induced current, and the corresponding emf required to cause this current
is called an induced emf.
In Fig.1c we replace the magnet with a second coil connected to a battery.
When the second coil is stationary, there is no current in the first coil. However,
when we move the second coil toward or away from the first or move the first toward
or away from the second, there is current in the first coil, but again only while one
coil is moving relative to the other.
Finally, using the two-coil setup in Fig.1d, we keep both coils stationary and
vary the current in the second coil, either by opening and closing the switch or by
changing the resistance of the second coil with the switch closed (perhaps by
changing the second coil’s temperature). We find that as we open or close the
switch, there is a momentary current pulse in the first circuit. When we vary the
resistance (and thus the current) in the second coil, there is an induced current in
the first circuit, but only while the current in the second circuit is changing.
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Figure 1. Demonstrating the phenomenon of induced current.
Photo credit: University Physics with Modern Physics 13th Edition
The common element in all these experiments is changing magnetic flux
𝚽𝑩 through the coil connected to the galvanometer. In each case the flux changes
either because the magnetic field changes with time or because the coil is moving
through a nonuniform magnetic field. Faraday’s law of induction, the subject of
the next section, states that in all of these situations the induced emf is
proportional to the rate of change of magnetic flux Φ𝐵 through the coil. The direction
of the induced emf depends on whether the flux is increasing or decreasing. If the
flux is constant, there is no induced emf.
Induced emfs are not mere laboratory curiosities but have a tremendous
number of practical applications. If you are reading these words indoors, you are
making use of induced emfs right now! At the power plant that supplies your
neighborhood, an electric generator produces an emf by varying the magnetic flux
through coils of wire. This emf supplies the voltage between the terminals of the
wall sockets in your home, and this voltage supplies the power to your reading
lamp. Indeed, any appliance that you plug into a wall socket makes use of induced
emfs.
Magnetic Flux
The magnetic flux (often denoted Φ or ΦB) through a surface is the
component of the magnetic field passing through that surface. The magnetic flux
through some surface is proportional to the number of field lines passing through
that surface. The magnetic flux passing through a surface of vector area A is
⃗ • 𝐴 = 𝐵 𝐴 𝑐𝑜𝑠𝜃
Φ𝐵 = 𝐵
where B is the magnitude of the magnetic field (having the unit of Tesla, T), A is
the area of the surface, and θ is the angle between the magnetic field lines and the
normal (perpendicular) to A.
From the definition of magnetic flux, we see that its SI unit is the tesla–
square meter (𝑇 ∙ 𝑚2 ), which is called the weber (abbreviated Wb):
1 𝑤𝑒𝑏𝑒𝑟 = 1 𝑊𝑏 = 1 𝑇 ∙ 𝑚2 .
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Faraday’s Law
The common element in all induction effects is changing magnetic flux
through a circuit. Before stating the simple physical law that summarizes all of the
kinds of experiments described in Induction Experiments part, let’s first review the
concept of magnetic flux Φ𝐵 . For an infinitesimal area element 𝑑𝐴 in a magnetic
⃗ (Fig. 2), the magnetic flux 𝑑Φ𝐵 through the
field 𝐵
area is
⃗ • 𝑑𝐴 = 𝐵⊥ 𝑑𝐴 = 𝐵 𝑑𝐴 𝑐𝑜𝑠𝜃
𝑑Φ𝐵 = 𝐵
𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛 1
Figure 2. Calculating the magnetic flux through an area
element.
Photo credit: University Physics with Modern Physics (13th Edition)
⃗ perpendicular to the surface of the area element
where 𝐵⊥ is the component of 𝐵
⃗
and 𝜃 is the angle between 𝐵 and 𝑑𝐴.
Figure 3. Calculating the flux of a uniform magnetic field through a flat area.
Photo credit: University Physics with Modern Physics (13th Edition)
The total magnetic flux Φ𝐵 through a finite area is the integral of this
expression over the area:
⃗ • 𝐴 = 𝐵𝐴 𝑐𝑜𝑠𝜃
Φ𝐵 = 𝐵
𝐸𝑞. 2
Faraday’s law of induction states:
The induced emf in a closed loop equals the negative of the time rate of
change of magnetic flux through the loop.
In symbols, Faraday’s law is
𝒅𝚽𝑩
(𝐹𝑎𝑟𝑎𝑑𝑎𝑦 ′ 𝑠 𝐿𝑎𝑤 𝑜𝑓 𝐼𝑛𝑑𝑢𝑐𝑡𝑖𝑜𝑛)
𝜺=−
𝒅𝒕
6
𝐸𝑞. 3
ΔΦ
(We can use this formula from other books and references: 𝜀 = − 𝐵)
Δ𝑡
As you will see the formula, the induced emf tends to oppose the flux change, so
Faraday’s law is formally written as seen above with the minus sign indicating that
opposition. We often neglect the minus sign, seeking only the magnitude of the
induced emf.
(The minus sign is there to remind us in which direction the induced emf
acts. Experiments shows that a current produced by an induced emf moves in
a direction so that its magnetic field opposes the original change in flux. This
is known as Lenz’s Law.)
If we change the magnetic flux through a coil of N turns, an induced emf
appears in every turn and the total emf induced in the coil is the sum of these
individual induced emfs. If the coil is tightly wound (closely packed), so that the
same magnetic flux
passes through all the turns, the total emf induced in the
coil is
𝜀 = −𝑁
𝑑Φ𝐵
𝑑𝑡
(𝑐𝑜𝑖𝑙 𝑜𝑓 𝑁 𝑡𝑢𝑟𝑛𝑠)
𝐸𝑞. 4
Here are the general means by which we can change the magnetic flux
through a coil:
1. Change the magnitude B of the magnetic field within the coil.
2. Change either the total area of the coil or the portion of that area that lies
within the magnetic field (for example, by expanding the coil or sliding it into
or out of the field).
⃗ and the plane
3. Change the angle between the direction of the magnetic field 𝐵
⃗ is first
of the coil (for example, by rotating the coil so that field 𝐵
perpendicular to the plane of the coil and then is along that plane).
Example 1: Emf and current induced in a loop
The magnetic field between the poles of the electromagnet in Fig. 4 is
uniform at any time, but its magnitude is increasing at the rate of 0.020 T/s. The
area of the conducting loop in the field
is 120 cm2, and the total circuit
resistance, including the meter, is 5.0
Ω.
Figure 4. A stationary conducting loop in an
increasing magnetic field.
Photo credit: University Physics with Modern
Physics (13th Edition)
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(a) Find the induced emf and the induced current in the circuit.
(b) If the loop is replaced by one made of an insulator, what effect does this have
on the induced emf and induced current?
Solution
IDENTIFY and SET UP: The magnetic flux Φ𝐵 through the loop changes as
the magnetic field changes. Hence there will be an induced emf 𝜀 and an induced
current
I
in
the
loop.
We
calculate
Φ𝐵
using
Eq.
⃗
(Φ𝐵 = 𝐵 • 𝐴 = 𝐵𝐴 𝑐𝑜𝑠𝜃), then find 𝜀 using Faraday’s law. Finally, we calculate I using
𝜀 = 𝐼𝑅 where R is the total resistance of the circuit that includes the loop.
EXECUTE: (a) The area 𝐴 vector for the loop is perpendicular to the plane of
⃗ are parallel, and because
the loop; we take 𝐴 to be vertically upward. Then 𝐴 and 𝐵
⃗ is uniform the magnetic flux through the loop is Φ𝐵 = 𝐵
⃗ • 𝐴 = 𝐵𝐴 𝑐𝑜𝑠𝜃 = 𝐵𝐴. The
𝐵
2
area 𝐴 = 0.012𝑚 is constant, so the rate of change of magnetic flux is
𝑑Φ𝐵 𝑑(𝐵𝐴) 𝑑𝐵
𝑇
=
=
𝐴 = (0.020 ) (0.012𝑚2 )
𝑑𝑡
𝑑𝑡
𝑑𝑡
𝑠
= 2.4 𝑥 10−4 𝑉 = 0.24 𝑚𝑉
This, apart from a sign that we haven’t discussed yet, is the induced emf 𝜀.
The corresponding induced current is
𝜀 2.4 𝑥 10−4 𝑉
𝐼= =
= 4.8 𝑥 10−5 𝐴 = 0.048 𝑚𝐴
𝑅
5.0 Ω
(b) By changing to an insulating loop, we’ve made the resistance of the loop
𝑑Φ
very high. Faraday’s law, Eq. (𝜀 = − 𝐵 ), does not involve the resistance of the
𝑑𝑡
circuit in any way, so the induced emf does not change. But the current will be
𝜀
smaller, as given by the equation 𝐼 = . If the loop is made of a perfect insulator
𝑅
with infinite resistance, the induced current is zero. This situation is analogous to
an isolated battery whose terminals aren’t connected to anything: An emf is
present, but no current flows.
EVALUATE: Let’s verify unit consistency in this calculation. One way to do
⃗⃗ implies that the units
this is to note that the magnetic-force relationship 𝐹 = 𝑞𝑣 𝑥 𝑩
⃗⃗ are the units of force divided by the units of (charge times velocity): 1𝑇 =
of 𝑩
(1𝑁)/1𝐶 ∙ 𝑚/𝑠). The units of magnetic flux are then (1𝑇)(1𝑚2 ) = 1𝑁 ∙ 𝑠 ∙ 𝑚/𝐶, and the
𝑚
𝐽
𝑑Φ𝐵
rate of change of magnetic flux is 1𝑁 ∙ = 1 = 1𝑉. Thus the unit of
is the volt,
𝑑Φ
− 𝐵 ).
𝑑𝑡
𝐶
𝐶
𝑑𝑡
as required by Eq. (𝜀 =
Also recall that the unit of magnetic flux is the weber
2
(Wb): 1 𝑇 ∙ 𝑚 = 1𝑊𝑏, so 1 𝑉 = 1𝑊𝑏/𝑠.
Direction of Induced emf
We can find the direction of an induced emf or current by using Eq. (𝜀 = −
together with some simple sign rules. Here’s the procedure:
1.
Define a positive direction for the vector area 𝐴.
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𝑑Φ𝐵
𝑑𝑡
)
⃗ , determine the sign of the
From the directions of 𝐴 and the magnetic field 𝐵
𝑑Φ𝐵
magnetic flux Φ𝐵 and its rate of change
. Figure 5 shows several examples.
𝑑𝑡
3. Determine the sign of the induced emf or current. If the flux is increasing, so
𝑑Φ𝐵
is positive, then the induced emf or current is negative; if the flux is
2.
𝑑𝑡
𝑑Φ
decreasing, 𝐵 is negative and the induced emf or current is positive.
𝑑𝑡
4. Finally, determine the direction of the induced emf or current using your right
hand. Curl the fingers of your right hand around the vector 𝐴, with your right
thumb in the direction of 𝐴. If the induced emf or current in the circuit is
positive, it is in the same direction as your curled fingers; if the induced emf or
current is negative, it is in the opposite direction.
In Example 1, in which 𝐴 is upward, a positive 𝜀 would be directed counter⃗ are upward in
clockwise around the loop, as seen from the example. Both 𝐴 and 𝐵
𝑑Φ
this example, so Φ𝐵 is positive; the magnitude B is increasing, so 𝐵 is positive.
𝑑𝑡
Hence by Eq. (3), in Example 1 is negative. Its actual direction is thus clockwise
around the loop, as seen from above.
If the loop in Fig. 4 is a conductor, an induced current results from this emf;
this current is also clockwise, as Fig. 4 shows. This induced current produces an
additional magnetic field through the loop, and the right-hand rule shows that this
field is opposite in direction to the increasing field produced by the electromagnet.
This is an example of a general rule called Lenz’s law, which says that any
induction effect tends to oppose the change that caused it; in this case the change
is the increase in the flux of the electromagnet’s field through the loop.
Figure 5. The magnetic flux is becoming (a) more positive, (b) less positive, (c) more negative, and (d) less
negative. Therefore Φ𝐵 is increasing in (a) and (d) and decreasing in (b) and (c). In (a) and (d) the emfs are
negative (they are opposite to the direction of the curled fingers of your right hand when your right thumb
points along 𝐴). In (b) and (c) the emfs are positive (in the same direction as the curled fingers).
Photo credit: University Physics with Modern Physics (13th Edition)
Lenz’s Law
The minus sign in Faraday’s law of induction is very important. The minus
means that the EMF creates a current I and magnetic field B that oppose the
change in flux ΔΦ this is known as Lenz’ law. The direction (given by the minus
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sign) of the EMF is so important that it is called Lenz’ law after the Russian
Heinrich Lenz (1804–1865), who, like Faraday and Henry, independently
investigated aspects of induction. Faraday was aware of the direction, but Lenz
stated it, so he is credited for its discovery.
Figure 6. Magnet subjected to motion into a coil
Photo credit: Lumen Learning
Lenz’s Law: (a) When this bar magnet is thrust into the coil, the strength of
the magnetic field increases in the coil. The current induced in the coil creates
another field, in the opposite direction of the bar magnets to oppose the increase.
This is one aspect of Lenz’s law – induction opposes any change in flux. (b) and (c)
are two other situations. Verify for yourself that the direction of the induced B coil
shown indeed opposes the change in flux and that the current direction shown is
consistent with the right-hand rule.
Energy Conservation
Lenz’ law is a manifestation of the conservation of energy. The induced EMF
produces a current that opposes the change in flux, because a change in flux
means a change in energy. Energy can enter or leave, but not instantaneously.
Lenz’ law is a consequence. As the change begins, the law says induction opposes
and, thus, slows the change. In fact, if the induced EMF were in the same direction
as the change in flux, there would be a positive feedback that would give us free
energy from no apparent source—conservation of energy would be violated.
With so much things to consider in this lesson, here are the important things
to consider, or in other words, Electromagnetic Induction is the process of using
magnetic fields to produce voltage, and in a closed circuit, a current.
So how much voltage (emf) can be induced into the coil using just
magnetism? Well, this is determined by the following 3 different factors.
1) Increasing the number of turns of wire in the coil – By increasing the amount
of individual conductors cutting through the magnetic field, the amount of
induced emf produced will be the sum of all the individual loops of the coil,
so if there are 20 turns in the coil there will be 20 times more induced emf
than in one piece of wire.
2) Increasing the speed of the relative motion between the coil and the magnet
– If the same coil of wire passed through the same magnetic field but its
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speed or velocity is increased, the wire will cut the lines of flux at a faster
rate so more induced emf would be produced.
3) Increasing the strength of the magnetic field – If the same coil of wire is
moved at the same speed through a stronger magnetic field, there will be
more emf produced because there are more lines of force to cut.
Applications of Faraday’s Law
Following are the fields where Faraday’s law finds applications:
1. Electrical equipment like transformers works on the basis of Faraday’s law.
2. Induction cooker works on the basis of mutual induction which is the
principle of Faraday’s law.
3. By inducing an electromotive force into an electromagnetic flowmeter, the
velocity of the fluids is recorded.
4. Electric guitar and electric violin are the musical instruments that find an
application of Faraday’s law.
5. Maxwell’s equation is based on the converse of Faraday’s laws which states
that change in the magnetic field brings a change in the electric field.
Example 2: Magnitude and direction of an induced
emf
A 500-loop circular wire coil with radius 4.00 cm is placed between the poles
of a large electromagnet. The
magnetic field is uniform and makes
an angle of 60° with the plane of the
coil; it decreases at 0.200 T/s. What
are the magnitude and direction of
the induced emf?
Figure 7. Circular wire coil subjected to a
magnetic field
Photo credit: University Physics with Modern Physics (13th
Edition)
SOLUTION
IDENTIFY and SET UP: Our target variable is the emf induced by a varying
magnetic flux through the coil. The flux varies because the magnetic field decreases
in amplitude. We choose the area vector 𝐴 to be in the direction shown in Fig. 7.
With this choice, the geometry is similar to that of Fig. 5b. That figure will help us
determine the direction of the induced emf.
EXECUTE: The magnetic field is uniform over the loop, so we can calculate
the flux using Eq. (Φ𝐵 = 𝐵𝐴 cos 𝜃) where 𝜃 = 30°. In this expression, the only
𝑑Φ
𝑑𝐵
quantity that changes with time is the magnitude B of the field, so 𝐵 = ( ) 𝐴 𝑐𝑜𝑠𝜃.
𝑑𝑡
11
𝑑𝑡
*CAUTION: Remember how 𝜃 is defined
You may have been tempted to say that 𝜃 = 60° in this problem. If so,
⃗ , not the angle between 𝐵
⃗ and the
remember that 𝜃 is the angle between 𝐴 and 𝐵
plane of the loop.
From the faraday’s law equation, the induced emf in the coil of N=500 turns is
𝑑Φ𝐵
𝑑B
𝜀 = −𝑁
=𝑁
𝐴 𝑐𝑜𝑠𝜃
𝑑𝑡
𝑑𝑡
𝑇
= 500(−0.200 )𝜋(0.0400 𝑚)2 (cos 30°) = 𝟎. 𝟒𝟑𝟓 𝑽
𝑠
The positive answer means that when you point your right thumb in the
direction of the area vector 𝐴 (30° below the magnetic field in Fig. 7), the positive
direction for 𝜀 is in the direction of the curled fingers of your right hand. If you
viewed the coil from the left in Fig.7 and looked in the direction of 𝐴 the emf would
be clockwise.
EVALUATE: If the ends of the wire are connected, the direction of current in
the coil is in the same direction as the emf—that is, clockwise as seen from the left
side of the coil. A clockwise current increases the magnetic flux through the coil,
and therefore tends to oppose the decrease in total flux. This is an example of
Lenz’s law.
Induced Electric Field
So, we seen that a changing magnetic flux result in an induced emf.
Faraday’s law is
𝑑Φ𝐵
𝜀=−
𝑑𝑡
Faraday’s law states that induced emf is the negative rate of change of
magnetic flux. If magnetic flux changes over time, then there has to be an induced
emf. And if a closed conducting path is available, then charges can move and
produce a current. We sometimes refer to this current as an induced current
because of how it results from an induced emf.
The question now is: what is making the charges move? On electrostatics,
we learned that a charge placed in an electric field is pushed or pulled by that field.
So, if the charge is free to move along a wire, then we have a current.
Then, we learned that a moving charge placed in a magnetic field experience
a force exerted by the magnetic field itself. Which of these two forces, the one
exerted by the electric field or the one by the magnetic field, is responsible for the
induced current?
Consider a region in space where there is a time-varying magnetic field (see
Fig. 8). The magnetic field is going into the paper and a conducting loop is placed
perpendicular to the field lines. Let us suppose that this magnetic field is
increasing. There will be an induced magnetic field that will oppose this increase.
So, the induced magnetic field will be pointing out of the paper. Using the RH grip
rule, the induced current is counterclockwise.
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Figure 8. This conducting loop is placed in a
changing magnetic field in space
Photo credit: General Physics 2 (Santisteban-Cook 2018)
We know there is an induced emf
because Faraday’s law tells us so. But where
does the emf come from? What makes the
charges move? It cannot be the magnetic field
because the magnetic field only pushes on
charges that are moving already. If it is not
magnetic field, it has to be the electric field.
This gives us a new way of thinking about Faraday’s law. The changing
magnetic field creates an electric field on the loop, in the same direction as the
current. Imagine that you are walking around the loop. At every step you take (no
matter how small0, you will see an electric field directed tangent to the loop, and
this electric field exerts an electric force on the charges all along the loop (see Fig.
9).
The induced electric field is different from the electric field we learned before.
All the electric fields that are discussed before Faraday’s law are electrostatic fields.
This induced electric field is a non-electrostatic field. The equation
𝐹
𝐸=
𝑞
is still true, but there are some differences.
Electrostatic fields start at positive charges and end at negative charges.
They have a beginning and an end. Induced electric fields do not start or end at a
charge; they just go around in loops. They are also not fixed in time; these fields
are always changing. They appear when a changing magnetic flux is present and
disappear when there is no change.
Another difference is in how we get the potential. In an electrostatic field, we
can get the potential using the dot product 𝐸 ∙ 𝑑𝑙, and just like work, if you start
and end at the same point, the result is zero. If you travel around a closed loop in
an electrostatic field, the potential difference is zero. This is because an
electrostatic field is a conservative field, just like the Earth’s gravitational field.
If you go around the loop in either Fig. 8 or
Fig. 9, you are doing work all around the loop, and
when you reach the point where you started on the
loop, the work done is not zero. If you go around
the loop and calculate all the 𝐸 ∙ 𝑑𝑙′𝑠 and then get
Figure 9. There is an induced electric field all along
the loop as a result of the changing magnetic flux
Photo credit: General Physics 2 (Santisteban-Cook 2018)
13
the sum over the entire loop, you do not get zero. You get
result. This is the emf.
𝑑Φ𝐵
𝑑𝑡
which is an important
So, this is what Faraday’s law now look like: the left side is the emf, but
more importantly, it is the summation of all the 𝐸 ∙ 𝑑𝑙′𝑠 around a closed loop:
𝑑Φ𝐵
∮ 𝐸⃗ ∙ ⃗⃗⃗
𝑑𝑙 = −
𝑑𝑡
And because this equation is not equal to zero, we know that an induced
electric field is nonconservative, unlike an electrostatic field. One more thing needs
to be pointed out about the induced electric field: the conducting loop does not
even have to be in the given space for the field to exist. The only thing it requires
to exist is varying magnetic field. The electric field will exist with or without free
electrons moving around in a loop because the field is a property of the space, not
a property of the charges.
Let’s take a look at a more detailed explanation.
The fact that emfs are induced in circuits implies that work is being done on
the conduction electrons in the wires. What can possibly be the source of this
work? We know that it’s neither a battery nor a magnetic field, for a battery does
not have to be present in a circuit where current is induced, and magnetic fields
never do work on moving charges. The answer is that the source of the work is
an electric field 𝐸⃗ that is induced in the wires. The work done by 𝐸⃗ in moving a unit
charge completely around a circuit is the induced emf ε; that is,
𝜀 = ∮ 𝐸⃗ ∙ ⃗⃗⃗
𝑑𝑙,
where ∮ represents the line integral around the circuit. Faraday’s law can be
written in terms of the induced electric field as
𝑑Φ𝐵
∮ 𝐸⃗ ∙ ⃗⃗⃗
𝑑𝑙 = −
𝑑𝑡
There is an important distinction between the electric field induced by a
changing magnetic field and the electrostatic field produced by a fixed charge
distribution. Specifically, the induced electric field is nonconservative because it
does net work in moving a charge over a closed path, whereas the electrostatic field
is conservative and does no net work over a closed path. Hence, electric potential
can be associated with the electrostatic field, but not with the induced field. The
following equations represent the distinction between the two types of electric field:
⃗⃗⃗ ≠ 0
∮ 𝐸⃗ ∙ 𝑑𝑙
∮ 𝐸⃗ ∙ ⃗⃗⃗
𝑑𝑙 = 0
𝐼𝑛𝑑𝑢𝑐𝑒𝑑 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝐹𝑖𝑒𝑙𝑑
𝐸𝑙𝑒𝑐𝑡𝑟𝑜𝑠𝑡𝑎𝑡𝑖𝑐 𝐸𝑙𝑒𝑐𝑡𝑟𝑖𝑐 𝐹𝑖𝑒𝑙𝑑𝑠
Our results can be summarized by combining these equations:
𝑑Φ𝑚
⃗⃗⃗ = −
𝜀 = ∮ 𝐸⃗ ∙ 𝑑𝑙
𝑑𝑡
14
Explore
Direction: Read and analyze the following problems . Answer them properly.
Problem 1. Calculate the flux
A square loop of wire 10.0 cm on a side is in a 1.25 T magnetic field B. What are
the maximum and minimum values of flux that can pass through the loop?
Approach
The flux is given by Eq. 2 (Φ𝐵 = 𝐵𝐴 cos 𝜃). It is a maximum for 𝜃 = 0°, which occurs when
⃗ . The minimum value occurs when 𝜃 = 90° and
the plane of the loop is perpendicular to 𝐵
⃗.
the plane of the loop is aligned with 𝐵
Problem 2. Change in flux and induced emf
A coil of wire is situated in a 0.5 T uniform magnetic field. The area of the coil is
2.0 m2. (a) What is the magnetic flux if the angle between the magnetic field and
the normal to the surface of the coil is 60°? (b) After 5 s, the magnetic field is now
parallel to the normal to the surface. What is the induced emf?
Approach
(a) The angle between the magnetic field and the normal to the surface of the
coil is 60°
(b) The angle between the magnetic field and the normal to the surface of the
coil is 0°, because the magnetic field is now parallel to the normal surface
(see Fig. 3).
Deepen
Direction: Read and analyze the following problems . Answer them properly.
Activity: Calculating EMF: How great is the induced EMF?
Problem 1
A uniform magnetic field is directed at
an angle of 30° with the plane of a
circular coil of radius 2 cm and 2000
turns. If the magnetic field changes at
a rate of 4T per second, calculate the
induced emf.
15
Approach
We are given the angle 30°, but note that this is the angle of B with respect to the
plane of the coil. Thus, the angle with respect to the area vector is 60°.
We also know the radius of the coil. Thus, we can calculate its area:
𝐴 = 𝜋𝑟 2 = 𝜋(0.02𝑚)2 = 1.26𝑥10−3 𝑚2
The emf is induced because the flux is changing. In this case, the reason for the
change in flux is the increasing magnetic field (we know this because the rate of
𝑑𝐵
𝑇
change given is positive) at
= 4 . We can write this into the law of induction.
𝑑𝑡
𝑠
𝑑Φ𝐵
𝑁
𝜀 = −𝑁
= − 𝑑(𝐵𝐴𝑐𝑜𝑠60°)
𝑑𝑡
𝑑𝑡
The factor A cos 60° is not part of the change so we can take it out of the
parentheses.
𝑑𝐵
𝜀 = −𝑁𝐴𝑐𝑜𝑠60°
𝑑𝑡
At this point, we are ready to substitute the given.
Problem 2
Calculate the magnitude of the induced emf when the magnet is thrust into the
coil, given the following information: the single loop coil has a radius of 6.00 cm
and the average value of B cos θ (this is given, since the bar magnet’s field is
complex) increases from 0.0500 T to 0.250 T in 0.100s.
Strategy
To find the magnitude of emf, we use Faraday’s law of induction as stated by
ΔΦ𝐵
𝜀 = −𝑁
Δ𝑡
but without minus sign indicates direction:
ΔΦ𝐵
𝜀=𝑁
Δ𝑡
We are given that N=1 and ∆t=0.100s, but we must determine the change in flux
ΔΦ before we can find emf. Since the area of the loop is fixed, we see that
ΔΦ𝐵 = Δ(𝐵𝐴𝑐𝑜𝑠𝜃) = 𝐴 Δ(𝐵 cos 𝜃).
Now Δ(𝐵 cos 𝜃) = 0.200 𝑇, since it was given that 𝐵 cos 𝜃 changes from 0.500 to 0.250
T. The area of the loop is 𝐴 = 𝜋𝑟 2 = 𝜋(0.060𝑚)2 = 1.13𝑥10−2 𝑚2 . Thus,
ΔΦ = (1.13𝑥10−2 𝑚2 )(0.200 𝑇)
16
Gauge
Directions: Read carefully each item. Write only the letter of the best answer before
the number.
1. A vector quantity which defined as the dot product of the magnetic field and
the area vector.
A. Electric Field B. Magnetic Flux
C. Induction
D. Induced EMF
2. From the definition of magnetic flux, which of the following is the SI unit for
magnetic flux?
A. V
B. T
C. Wb
D. J
3. If the magnetic flux through an area bounded by a closed conducting loop
changes with time, a current and an emf are produced in the loop, what do
you call this process?
A. Induction
B. Intensity
C. Current
D. EMF
4. What device was used in conducting various experiments such as with
magnetically induced emf by Faraday?
A. Voltmeter
B. Ohmmeter
C. Ammeter
D. Galvanometer
5. “Moving the magnet toward or away from the coil.” “Moving a current
carrying coil toward or away from the coil.” Based from these actions, what
do they have in common?
A. All these actions do induce a current in a coil
B. All these actions were supported by stationary motion
C. All these actions are in a closed circuit
D. All these actions used magnets to move the galvanometer to another
place.
6. Faraday’s law of induction, the induced emf is proportional to the
___________ of magnetic flux through the coil.
A. Type of metal coil
B. Color of the coil
C. Quantity of magnets
D. Rate of change
7. Which of the following statements can describe the process of the law of
induction?
A. Increase in the number of coils decreases the magnetic flux
B. Increase in the number of turns in the coil increases the induced emf
C. Decrease in magnetic field decreases the induced current
D. Decrease in the speed of relative motion between coil and magnet will
result in increased flux
8. The minus sign in Faraday’s law of induction is very important. The minus
means that the EMF creates a current I and magnetic field B that oppose
the change in flux ΔΦ this is known as ________________.
A. Ampere’s Law
B. Lenz’ law
C. Magnetic flux density
D. Induced electric field
17
9. What would be the implication if your curled fingers have the same direction
with the induced current or emf in the circuit?
A. Positive I and EMF
B. No implication was given
C. Negative I and EMF
D. Both A and B
10. Based on the following statements, which is incorrect about the sign of the
induced emf or current?
𝑑Φ𝐵
I.
If the flux is increasing, so
is positive, then the induced emf or
𝑑𝑡
current is negative
𝑑Φ𝐵
II.
If the flux is decreasing,
is negative and the induced emf or
𝑑𝑡
current is positive
𝑑Φ𝐵
III.
If the flux is increasing,
is negative and the induced emf or
𝑑𝑡
current is negatives
A. I
B. II
C. III
D. I and II
11. Which of the following is a nonconservative field where it does net work in
moving a charge over a closed path?
A. Magnetic field
B. Electrostatic electric field
C. Induced emf
D. Induced Electric field
12. The fact that emfs are induced in circuits implies that work is being done
on the conduction electrons in the wires. What can possibly be the source
of this work?
A. Electric flux
B. Faraday’s law
C. Magnetic flux
D. Lenz’s Law
For numbers 13-14.
A circular coil 50 cm in diameter
is rotating in a magnetic field
directed upward with a magnitude
of 65 mT. Calculate the magnetic
flux through the coil at the
positions shown below.
13. (a)
A. 10.0 T∙m2
B. 0.10 Wb
14. (b)
A. 5.00 Wb
B. 50.0 T∙m2
C. 1.00 T∙m2
D. -0.01 Wb
C. 0.05 Wb
D. 0.50 T∙m2
15. A magnetic field B= 0.6T is directed upward through a circular loop of
diameter 7 cm and 500 turns. The loop is initially horizontal, so it is
perpendicular to the magnetic field. It rotates through a horizontal axis so
that the plane of the loop is at 74° with the horizontal axis within 1 second.
What is the magnitude of the induced emf?
A. -15.3 V
B. 17.6 V
C. 16.4 V
D. -18.18 V
18
Jumpstart
1. B
2. C
3. D
4. A
5. D
6. C
7. D
8. A
9. C
10. B
19
Gauge
1. B
2. C
3. A
4. D
5. A
6. D
7. B
8. B
9. A
10. C
11.
12.
13.
14.
15.
D
A
B
C
D
Explore
Problem 1. Calculate the flux
⃗ = 1.25𝑇
Given: r=10.0 cm, 𝐵
Unknown: Φ𝑚𝑎𝑥 =? 𝑎𝑛𝑑 Φ𝑚𝑖𝑛 =?
Formula: Φ𝐵 𝑚𝑎𝑥 = 𝐵𝐴 cos 𝜃 𝑎𝑛𝑑 Φ𝐵 𝑚𝑖𝑛 = 𝐵𝐴 cos 𝜃
Solution:
From Eq. 2, the maximum value is
Φ𝐵 = 𝐵𝐴 cos 𝜃 = (1.25 𝑇)(0.100 𝑚)(0.100𝑚) cos 0° = 0.0125 𝑊𝑏.
The minimum value is 0 Wb when 𝜃 = 90° and cos 90° = 0.
Answer: Φ𝐵 𝑚𝑎𝑥 = 0.0125 𝑊𝑏 and Φ𝐵 𝑚𝑖𝑛 = 0 𝑊𝑏
Problem 2. Change in flux and induced emf
⃗ = 0.5𝑇, 𝜃 = 0°, A=2.0m2
Given: 𝐵
Unknown: (a) Φ =? 𝑎𝑛𝑑 𝜀 =?
ΔΦ
Formula: (a) Φ𝐵 = 𝐵𝐴 cos 𝜃 𝑎𝑛𝑑 𝜀 = −𝑁 Δ𝑡𝐵
Solution: (a) Φ𝐵 = 𝐵𝐴 cos 𝜃 = (0.5𝑇)(2𝑚2 ) cos 60° = 0.5 𝑊𝑏
(b) After 5 s, the flux is Φ𝐵 = 𝐵𝐴 cos 𝜃 = (0.5𝑇)(2𝑚2 ) cos 0° = 1 𝑊𝑏
Solving for the induced emf
(1𝑊𝑏−0.5𝑊𝑏)
ΔΦ
Answer: 𝜀 = −𝑁 Δ𝑡𝐵 = −
= −0.1 𝑉
5𝑠
Deepen
Activity: Calculating EMF: How great is the induced EMF?
Problem 1
Given: θ=30°, r=2cm or 0.02m, N=2000 turns, 4𝑇/𝑠
Unknown: 𝜀 =?
𝑑Φ
Formula: 𝐴 = 𝜋𝑟 2 , Φ𝐵 = 𝐵𝐴𝑐𝑜𝑠𝜃, 𝜀 = −𝑁 𝐵
𝑑𝑡
Solution:
4𝑇
Answer: 𝜀 = −(2000)(1.26𝑥10−3 𝑚2 )𝑐𝑜𝑠60 ( 𝑠 ) = −5.03 𝑉
Problem 2
Given: r=6.00 cm, Δ𝐵𝑖 = 0.0500 𝑇, Δ𝐵𝑓 = 0.250 𝑇, t=0.100 s
Unknown: 𝜀 =?
ΔΦ
Formula: 𝜀 = 𝑁 Δ𝑡𝐵
Solution:
Entering the determined values into the expression for emf gives
(0.200𝑇)(1.13𝑥10−2 𝑚2 )
ΔΦ
Δ𝐵𝐴
𝐵
𝜀=𝑁
=𝑁
= (1)
= 0.0226 𝑉 𝑜𝑟 22.6 𝑚𝑉
Δ𝑡
Δ𝑡
0.100𝑠
Answer: 0.0226 𝑉 𝑜𝑟 22.6 𝑚𝑉
Answer Key
References
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