Chapter 29
Electromagnetic Induction
In this chapter we investigate we investigate how changing the magnetic flux in a
circuit induces an emf and a current. We learned in Chapter 25 that an electromotive force (E) is required for a current to flow in a circuit. Up until now, we used a
battery as our emf source. However, most of our power comes from electric generating stations where the source of power is gravitational potential energy, chemical
energy, and nuclear energy. How are these forms of energy converted into electrical
energy? This is one by electromagnetic induction. In all power=generating stations, magnets move relative to coils of wire to produce a changing magnetic flux
in the coils and hence an emf.
In this chapter we will study Faraday’s law. This relates the induced emf to changing magnetic flux in any loop. We also discuss Lenz’s law, which helps predict
the direction of the induced emfs and currents. These principles are at the heart
of electrical energy conversion devices such as motors, generators, and transformers.
Finally, electromagnetic induction tells us that a time-varying magnetic field can
act as a source of electric field. We will also see how a time-varying electric field
can act as a source of magnetic field. These are the principle concepts embodied in
what has been called Maxwell’s equations. Just as Newton’s laws describe classical
kinematics and dynamics, Maxwell’s equations provide the fundamental framework
for electrodynamics.
1
Induction Experiments
Magnetically induced emf was first of observed in the 1830’s by Michael Faraday
(England), and by Joseph Henry (United States). The changing flux of magnetic
field passing through a closed loop (or circuit) was observed to produce an induced current and the corresponding emf required to cause this current is call
an induced emf (E).
1
Figure 1: This figure shows the measurements observed when a changing magnetic flux occurs in a
coil of wire connected to a galvanometer.
The following induction experiments were performed using the apparatus shown in Fig. 2 below.
2
Figure 2: This figure is used to describe the observations made below. The figure shows a coil
placed inside an electromagnetic and connected to a galvanometer. The magnetic field shown with
the N and S pole faces can be adjusted by varying the current into the device.
What was observed?
~ = ~0, the galvanome1. When there is no current in the electromagnet, so that B
ter show no current.
2. When the electromagnetic is turned on, there is momentary current through
~ increases.
the meter as B
~ levels off at a steady value, the current drops to zero.
3. When B
4. With the coil in a horizontal plane, we squeeze it so as to decrease the crosssectional area of the coil. The meter detects current only during the deformation, not before or after. When we increase the area to return the coil to its
original shape, there is current in the opposite direction, but only while the
area of the coil is changing.
5. If we rotate the coil a few degrees about a horizontal axis, the meter detects
current during the rotation, in the same direction as when we decreased the
area. When we rotate the coil back, there is a current in the opposite direction
during this rotation.
6. If we decrease the number of turns in the coil by unwinding one or more turns,
3
there is a current during the unwiding, in the same direction as when we
decreased the area. If we wind more turns onto the coil, there is a current in
the opposite direction during the winding.
7. When the magnet is turned off, there is a momentary current in the direction
opposite to the current when it was turned on.
8. The faster we carry out any of these changes, the greater the current.
9. If all these experiments are repeated with a coil that has the same shape but
different material and different resistance, the current in each case is inversely
proportional to the total circuit resistance. This shows that the induced emfs
that are causing the current do not depend on the material of the coil but only
on its shape and the magnetic field.
4
2
Faraday’s Law
The common feature in all induction effects is changing magnetic flux through a
~ in a magnetic field B,
~ the magnetic
circuit. For an infinitesimal-area elements dA
flux dΦB through the area is:
~ · dA
~ = B⊥ dA = B dA cos φ
dΦB = B
Figure 3: This figure shows how the magnetic flux is calculated as it passes through a surface.
Figure 4: Calculating the flux of a uniform magnetic field through a flat area. Different orientations
of the area result in different amounts of flux ΦB .
~ ·A
~ = BA cos φ
ΦB = B
5
Faraday’s Law of Induction states:
E = −
E = −
dΦ
dt
(Faraday’s Law)
(1)
dB
dA
dφ
A cos φ + B
cos φ − BA sin φ
dt
dt
dt
There are 3 ways to induce an emf E:
~
1. Change the strength of the magnetic field B,
2. Change the area of the loop or circuit, and
~ field.
3. Change the angle the area makes with respect to the B
2.1
Direction of Induced emf
Figure 5: The magnetic flux ΦB is increasing in (a) and (d) thus inducing an negative emf. In
figures (b) and (c) the magnetic flux ΦB is decreasing, thereby inducing a positive emf
.
6
2.2
Magnitude and Direction of an Induced EMF
Figure 6: In this figure the uniform magnetic field is decreased at a rate of 0.200 T/s. What is the
induced emf in this 500-loop wire coil of radius 4.00 cm?
2.3
A Simple Alternator
Figure 7: A schematic diagram of an alternator is shown (a). The conducting loop rotates in a
magnetic field, producing an emf shown in (b). Connections from each end of the loop to the
external circuit are made by means of that end’s slip ring. One cycle of the generated emf is shown
in the figure.
E = −
dΦ
d
d
= −BA cos φ = − BA (cos ωt) = BAω sin(ωt)
dt
dt
dt
This is one of the starting points for producing AC power.
7
A DC Generator
Figure 8: This figure is used to describe the observations made on the following page.
The time-averaged back-emf for a coil having N loops is:
Eav =
2.4
2N ωBA
π
The Slidewire Generator
Figure 9: This figure is used to describe the observations made on the following page.
In this case the magnetic flux changes due to the changing area dA/dt.
E = −
dΦ
dA
(Lv dt)
= −B
= −B
= −BLv
dt
dt
dt
8
The power required to move the bar is equal to the power dissipated by the I 2 R
loss in the current loop.
Papplied
3
B 2 L2 v 2
=
R
Lenz’s Law
The direction of any magnetic induction effect is such as to oppose the
cause of the effect
Example: Lenz’s Law and the Slidewire Generator
3.1
Lenz’s Law and the direction of Induced Current
Figure 10: This figure shows how to visualize the induced emf, and current, when the strength of
~ induced )
the external B-field is increased. Lenz’s law says that there will be an opposing B-field, (B
is produced in response to the increasing external field.
9
Figure 11: This figure shows four possible ways of increasing or decreasing the flux with two
magnetic poles. In all cases, the induced B-field is created in a manner that opposes the change to
the electric field. The induced B-field is produced as a result of the induced emf and its associated
current.
10
4
Motional Electromotive Force
Figure 12: This figure shows a conducting rod moving in a uniform magnetic field. The velocity
of the rod and the field are mutually perpendicular. In part (a) an electric field is induced in the
~ field, and positive charges migrate upward. In part (b) of the figure
bar as it moves through the B
shows the direction of the induced current in the circuit when the bar is connected to a conducting
loop.
The magnitude of the potential difference Vab − Va − Vb is equal to the electric-field
magnitude E multipled by the length L of the rod. Since E = vB, we have:
Vab = EL = vBL
The motional emf generated in part (b) of the figure is:
E = vBL
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4.1
Motional emf: General Form
Figure 13: This figure shows how the motional emf arises for a current loop moving in a static
magnetic field. The velocity ~v can be different for different elements if the loop is rotating or
~ can also have different values at different points around the
changing shape. The magnetic field B
loop.
The motional emf produced for a segment of the loop is:
~ · d~`
dE = (~v × B)
For any closed conducting loop, the total emf is:
I ~ · d~`
E =
~v × B
I
~ nc · d~` 6= 0
E
~ field)
(non-conservative E
12
(2)
5
Induced Electric Fields
Figure 14: This figure shows the winding on a solenoid carrying current I that is increasing at a
rate dI/dt. The magnetic flux in the solenoid is increasing at a rate dΦ/dt, and this changing flux
passes through a wire loop. An emf E = −dΦ/dt is induced in the loop, inducing a current I 0 that
is measured by the galvanometer G.
The magnetic flux created by the solenoid is:
Φ = BA = µo nIA
When the solenoid current changes, so does the flux Φ.
E = −
dΦ
dI
= µo nA
dt
dt
13
There is an induced electric field in the conductor caused by the changing magnetic flux.
14
5.1
Nonelectrostatic Electric Fields
Figure 15: Here are some applications of induced electric fields. (a) This hybrid automobile has
both a gasoline engine and an electric motor. As the car comes to a halt, the spinning wheels
run the motor backward so that it acts as a generator. The induced current is used to recharge
the car’s batteries. (b) The rotating crankshaft of a piston-engine airplane spins a magnet (i.e., a
magneto) inducing an emf in an adjacent coil and generating the spark that ignites fueld in the
engine cylinders. This process continues even if the airplane’s other electrical systems fail.
15
6
Eddy Currents
In previous examples we observed the induction effects and the induced currents
they produced. However, many pieces of electrical equipment contain masses of
metal moving in magnetic fields or located in changing magnetic fields (e.g., transformers). In these situations we have induced currents that circulate throughout
the volume of the material. Because their flow patterns resemble swirling eddies in
a river, we call them eddy currents.
Figure 16: These figures (a and b) shows the eddy currents induced in a rotating metal disk as it
rotates through a magnetic field. In figure (b) you can see the braking force F~ that occurs from the
~ ×B
~ opposing the rotation of the disk.
IL
16
Figure 17: Eddy currents can be used in (a) metal detectors at airport security checkpoints. When
~ o , this induces eddy currents in a
the transmitting coil generates an alternating magnetic field B
conducting object carried through the detector. The eddy currents in turn produce an alternating
~ 0 , which induces a current in the detector’s receiver coil. The same principle is used
magnetic field B
for portable metal detectors shown in fig. (b).
7
Displacement Current and Maxwell’s Equations
In the previous sections we have seen how varying magnetic fields give rise to an
induced electric field. One of the more remarkable symmetries of nature is that a
varying electric field gives rise to a magnetic field. This symmetric phenomenon
explains the existence of radio waves, gamma rays, and visible light, as well as all
other forms of electromagnetic waves.
In order to illustrate this symmetric relationship between varying electric fields and
magnetic fields, we return to Ampère’s law:
I
~ · d~` = µo Iencl
B
However, when we apply this law to the charging of a capacitor (see the figure below), we see that this equation is incomplete. No conduction current iC is observed
between the capacitor plates; however, there is current iC coming into the positiely charged plate and current iC leaving the negatively charged plate. While the
17
right-hand side of Ampère’s law is equal to µo iC for the planar surface, it’s equal
to zero if we use the bulging surface. However, it cannot be both, so, something is
missing.
However, something else is happening at the bulging surface. While the charge is
~
building up on the capacitor plates, the E-field
is changing between the plates. In
~
fact, the flux of the E-field ΦE is increasing. Let’s try to find a relationship between
the conducting current iC and the changing electric field dE/dt. The instantaneous
charge on the capacitor q is related to the instantaneous voltage v by the following
equation: q = vC where C is the capacitance /Ad.
Figure 18: This figure shows a current charging a parallel-plate capacitor. The conduction current
through the plane surface is iC , but there is no conduction current through the surface bulging out
between the plates. The two surfaces have a common boundary, so this difference in iencl leads to
an apparent contradiction in applying Ampère’s law.
q = Cv =
A
(Ed) = EA = ΦE
d
(3)
By taking a time derivative of both sides of this equation, we can relate the current coming in to the capacitor to the changing E-field building up between the
plates
iC =
dq
dΦE
= dt
dt
where
18
iD = dΦE
dt
is called the displacement current.
We include this fictitious current, along with the real conduction current iC , to
rewrite Ampère’s law as:
I
~ · d~` = µo (iC + iD )
B
encl
(4)
Figure 19: This figure shows a capacitor being charged by a current iC with a displacement current
equal to iC between the plates, with displacement current density jD = dE/dt. This can be
regarded as the source of magnetic field between the plates.
The displacement current, along with the magnetic field it produces, is more
than an artifice; it is “real” and a fundamental property of nature.
19
8
Maxwells’ Equations of Electromagnetism
We can now summarize all the relationships between electric and magnetic fields
into a package of four equations, called Maxwell’s equations. Two of Maxwell’s
~ or B
~ over a closed surface.
equations involve an integral of E
~ or B
~ around a closed
The third and fourth equations involve a line integral of E
path. Faraday’s law states that a changing magnetic flux acts as a source of electric
field.
1.
I
2.
3.
~ · dA
~ = Qencl
E
o
I
I
~ · dA
~ = 0
B
~ · d~` = − dΦB
E
dt
4.
I
(Faraday’s Law)
dΦ
E
~ · d~` = µo iC + o
B
dt
The electric field in Maxwell’s equations includes both conservative and non~ field:
conservative components of the E
~ = E
~c + E
~ nc
E
N.B. These four equations are highly symmetric in empty space. The third and
fourth equations can be rewritten in empty space (iC = 0 and Qencl = 0) as
follow:
I
I
~ · d~` = − d
E
dt
Z
~ · d~` = µo o d
B
dt
20
Z
~ · dA
~
B
~ · dA
~
E
(5)
(6)