CHAPTER 22 ELECTROMAGNETIC INDUCTION
Induced emf and Induced Current
When a magnet moves relative to a coil, or if the coil moves
relative to the magnet, an emf is generated. The fact that the
coil is in a changing magnetic field is what causes the emf to
appear.
As you can see from the drawing, the motion of the magnet
causes a change in the magnetic field strength and therefore
the magnetic flux through the coil.
This emf that is generated by the changing flux is called an
induced emf and results in an induced current.
Another way to induce a current in a coil is to keep both the
coil and the magnet stationary but change the area of the coil.
This changes the magnetic flux through the coil resulting in an
induced emf.
A third possibility is to keep two coils stationary near
each other and vary the current in one of them. This
causes a changing magnetic field to exist in the other
coil and generates an emf. This effect is used in
transformers.
Electromagnetic induction is the process of producing an emf
and the resulting current through the interaction of a
conductor and a magnetic field.
Motional emf
When a conductor moves in a magnetic field, the charges in the
conductor experience a force given by the equation F =
qvBsinθ where θ is the angle between the velocity vector and
the magnetic field vector.
This force causes negative charge to move to one end of the
conductor leaving a net positive charge at the other end.
This separation of charge results in an induced emf called a
motional emf since it exists because of the motion of charges
through a magnetic field.
An external circuit with a device like a light bulb can be
connected and powered as long as the rod moves.
The magnitude of the emf generated depends on the length of
the wire, the velocity of the wire and the magnetic field
strength. If they are all perpendicular to each other, the
equation is:
emf = vBL
Example
The drawing shows a type of flow meter that can be used to
measure the speed of blood in a blood vessel when it is exposed.
Blood can be treated as a moving conductor. Electrodes can be
used to measure the small voltage that develops across the
vessel. If the vessel has a diameter of 5.6 mm and is placed in a
0.60T magnetic field, find the blood velocity that results in a
1.5 mV reading.
When a current exists in a wire moving in a magnetic field,
another force is generated by the interaction of the current and
the magnetic field. The equation for the force is F = ILBsinθ
where θ is the angle between the magnetic field vector and the
direction of the current.
If we use the right hand rule to find the direction of the force,
we see that it opposes the velocity of the wire.
This means that the moving wire will come to rest unless a
force acts to the right to put the wire in equilibrium.
This means that the electrical energy used by the light bulb
comes from the kinetic energy of the wire which is supplied by
the work done by the force which keeps it moving.
This is consistent with the principle of conservation of energy.
In the drawing below, three identical rods, A, B, and C move in
different planes. A constant magnetic field of 0.45 T is directed
along the positive y axis. The length of each rod is 1.3 meters
and the speeds are all 2.7 m/s. Find the magnitude of the
motional emf for each rod and determine which end of each
rod is positive.
Magnetic Flux
Magnetic flux is defined the same way as electric flux, that is,
the field strength multiplied by the area through which it
passes. The difference is that we use magnetic field strength for
magnetic flux.
The equation is:
Φ = BA
The motional emf can then be calculated in terms of changing
magnetic flux.
emf = vBL
emf = (x - x0)BL
(t - t0)
emf = (xL - x0L)B
(t - t0)
emf = (A - A0)B
(t - t0)
emf = (BA - BA0)
(t - t0)
emf = (Φ - Φ0) = ΔΦ/Δt
(t - t0)
This means that the emf is equal to the time rate of change of
the magnetic flux.
Only the component of the magnetic field perpendicular to the
surface experiencing the flux actually passes through the
surface. This component generates the flux and the equation
must include the cosine of the angle between the magnetic field
vector and the normal to the surface.
Φ = Bacosφ
Example
A loop of wire bent in the shape of a semicircle with radius 0.20
m is rotated in a constant magnetic field of 0.75 T. The normal
to the plane of the loop is initially parallel to the direction of
the magnetic field and it is rotated through half of a revolution.
Find the change in magnetic flux.
Magnetic flux is sometimes expressed in webers(Wb) which has
the dimensions of 1 Tm2. In the diagram below, the number of
webers is 3 times as much in figure (a) since the magnetic field
strength is 3 times as much through the same area.
Faraday's Law of Electromagnetic Induction
Faraday's Law states that a change in magnetic flux per unit of
time causes an emf to exist in the coil that experiences this
change. The magnitude of the emf is directly proportional to
the time rate of change of flux and the number of turns in the
coil. The equation is:
emf = -N(Φ - Φ0)/(t - t0)
emf = -N(ΔΦ)/Δt
Example
During an MRI, the patient is placed in a strong magnetic
field. One safety concern is the production of a relatively large
induced current in the patient during an equipment failure
which results in a rapidly disappearing magnetic field. Suppose
the largest body surface through which the magnetic flux
passes has an area of 0.032 m2 and its normal is parallel to a
magnetic field of 1.5 T. Determine the smallest time period
over which the field can vanish if the maximum induced emf is
to be 0.010 V.
Lenz's Law
The induced emf caused by a changing magnetic flux has a
polarity that leads to an induced current whose direction is
such that the induced magnetic field opposes the original flux
change.
The bar magnet moving to the right causes an increase in
magnetic flux in the loop since the field strength relative to the
loop is increasing.
The direction of the current in the loop shown above must be
from B to A to produce an induced magnetic field that opposes
the original flux change. The resulting induced magnetic field
causes a reduction in field strength and therefore a reduction
in flux.
Remember that the induced field opposes the change in flux,
not the flux itself. When the flux is decreasing, the induced
current will cause an induced magnetic field in the same
direction as the flux.
Lenz's Law is a result of the law of conservation of energy. If
Lenz's law was reversed, so that the induced field increased the
rate of change of magnetic flux, induced voltages and currents
would provide more energy than is available due to the motion
of the system.
Electromagnetic induction
Electric Generators
Electric generators convert energy of motion into electricity by
rotating a conducting coil in a magnetic field. The magnitude
of the emf produced depends on the number of turns in the
coil, the magnetic field strength, the area of the coil, and the
rate at which the coil turns.
emf = NABωsinωt
Since NABω gives us the maximum voltage, we can write the
equation as:
emf = emf0sin2πft
in which emf0 is the maximum voltage.
A generator that produces a voltage that varies in this manner
is called an AC generator. The graph of the emf looks like this.
Example
The coil of the device with the voltage output pictured above
has a cross sectional area of 0.020 m2 and contains 150 turns.
Find (a) the frequency of the generator in Hz, (b) the angular
speed in rad/sec and (c) the magnitude of the magnetic field.
ω = 2πf
An electric motor operates on the same principles as a
generator. The difference is that the motor converts electrical
energy into mechanical energy plus some heat energy due to
friction.
Since the construction is nearly identical, the motor acts like a
generator as it turns to produce a voltage that opposes the
voltage supplied to it.
This opposing voltage is called a back emf and will nearly
equal the supplied voltage while the motor is operating at full
speed.
The current flowing through the motor is large initially but
becomes relatively small when the back emf reaches its full
value.
Mutual Inductance and Self Inductance
Mutual induction is the process of generating an emf in a
second coil due to a changing current in a primary coil.
Mutual inductance is a proportionality constant for two coils
determined by the number of turns in the secondary and
magnetic flux in the secondary and inversely proportional to
the current in the primary.
M = NsΦs/Ip
This allows us to derive from Faraday's Law the expression for
the emf induced in the secondary.
emfs = -M(ΔIp/Δt)
This means the emf induced in the secondary is proportional to
the mutual inductance and the time rate of change of current
in the primary.
The unit for inductance is called the Henry and is equal to 1
V·s/A.
Example
The average emf induced in the secondary coil is 0.12 V when
the current in the primary changes from 3.4 to 1.6 A in 0.14 s.
Find the mutual inductance of the coils.
When there is only one coil present with a changing electric
current through it, the changing magnetic field produces an
emf in the coil itself. This is called self induction.
The self inductance of a coil is proportional to the number of
turns and the net flux through one turn and inversely
proportional to the current passing through the coil.
L = NΦ/I
emf = -L(ΔI/Δt)
The equation for the emf comes from Faraday's Law that
states the emf generated depends on the time rate of change of
magnetic flux.
Example
The current through a 3.2 mH inductor varies with time
according to the graph. What is the average induced emf
during each of the three time intervals?
Transformers
A transformer is a device used to increase or decrease AC
voltage. It will not work with DC voltage since the emf induced
in the secondary coil depends on changing magnetic flux.
Two coils of wire are wound on an iron core. The primary coil
is connected to the original source of current. Changing
current in the primary coil causes a change in magnetic flux
through the secondary coil producing an induced emf in the
secondary coil.
Since the same rate of flux change is present in both coils, the
voltage in both coils is related to the number of turns in each
coil.
Vs/Vp = Ns/Np
If the secondary has more turns than the primary, the
transformer is called a step-up transformer since it has a
higher voltage output than input.
If the voltage output is lower than the voltage input, it is called
a step-down transformer.
Example
A neon sign requires 12,000 V to operate. It gets power from a
220 V outlet. Is a step-up or step-down transformer needed?
What is the turns ratio Ns/Np of the transformer?
If a transformer is 100% efficient, the power in the secondary
must equal the power in the primary. Most are less than 100%
efficient with some of the energy being converted to heat. For a
100% efficient transformer,
Ps = P p
VsIs = VpIp
Is/Ip = Vp/Vs = Np/Ns
This tells us that the current in a transformer coil is inversely
proportional to the number of turns in the coil. If the voltage
increases, the current must decrease and vice-versa.
Example
A step-up transformer provides the voltage to operate an
electrostatic air filter and has a turns ratio of 50 to 1. The
primary is connected to a 120 V outlet and the current in the
secondary is 0.0017 A. Find the power consumed by the filter.
P 717 Questions 1, 2, 3, 4, 6, 7, 8, 12, 13
P 718 Problems 1, 4, 5, 10, 14, 16, 17, 19, 22, 24, 29, 35, 37, 44,
46, 54, 55, 57, 63, 64, 66