# Chapter 29

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18 Οκτ 2013 (πριν από 4 χρόνια και 6 μήνες)

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Chapter 29
:

Objectives

How can
Motion and Magnetism

produce an electric current and how
does

explain this?

What does
Lenz's Law

and Murphy's Laws have in common?

Changing Magnetic Fl
ux

is the answer. What is the question?

Electric Generators

how do they work and what do they do?

Sending Electric Power

from here to there

how do
Transformers

make that practical?

Counter EMF
, what is it and what are its consequences?

I have heard of
S
eismographs
, how do they work?

Lecture Note:

Electromagnetic Induction

1. Motional emf

Figure 1 shows a rod, made of conducting material, being moved with
a velocity v in a uniform magnetic field B. The magnetic force acting
on a free electron in the rod

will be directed upwards and has a
magnitude equal to

(1)

Figure 1. Moving conductor in magnetic field.

As a result of the magnetic force electrons will start to accumulate at
the top of the rod. The charge distribution of the rod will therefore
change, and

the top of the rod will have an excess of electrons
(negative charge) while the bottom of the rod will have a deficit of
electrons (positive charge). This charge distribution will produce an
electric field in the rod. The strength of this electric field w
ill increase
until the electrostatic force produced by this field is equal in magnitude
to the magnetic force. As this point the upward flow of electrons will
stop and

(2)

or

(3)

The induced electric field will generate a potential difference [Delta]V
betw
een the ends of the rod, equal to

(4)

where L is the length of the rod. If the ends of the rod are connected
with a circuit prov
iding a return path for the accumulated charge, the
rod will be a source of emf. Since the emf is associated with the
motion of the rod through the magnetic field it is called
motional
emf
. Equation (4) shows that the magnitude of the emf is proportional
t
o the velocity v. Looking at Figure 1 we observe that vL is the area
swept across by the rod per second. The quantity BvL is the magnetic
flux swept across by the rod per second. Thus

(5)

Although this formula was derived for the special case shown in Figure
1, it is valid in general. It holds for rods and wires of arbitrary shape
moving through arbitrary magnetic fields.

Equation

(5) relates the induced emf to the rate at which the enclosed
magnetic flux changes. In the system shown in Figure 1 the enclosed
flux changes due to the motion of the rod. The enclosed magnetic flux
can also be changed if the strength of the enclosed mag
netic field
changes. In both cases the result will be an induced emf. The relation
between the induced emf and the change in magnetic flux is known as

" The induced emf along a moving or changing mathematical path in a
constant

or changing magnetic field equals the rate at which magnetic
flux sweeps across the path. "

If we consider a closed path, Faraday's law can be stated as follows:

" The induced emf around a closed mathematical path in magnetic field
is equal to the rate o
f change of the magnetic flux intercepted by the
area within the path "

or

(6)

f the induced emf is
related to the sign of the flux and to the rate of change of flux. The
sign of the flux is fixed by the right
-
hand rule:

" Curl the fingers of your right hand in the direction in which we are
reckoning the emf around the path; the mag
netic flux is then positive
if the magnetic field lines point in the direction of the thumb, and
negative otherwise. "

Example Problem: Metal Rod in Magnetic Field

A metal rod of length L and mass m is free to slide, without friction, on
two parallel me
tal tracks. The tracks are connected at one end so that
they and the rod form a closed circuit (see Figure 2). The rod has a
resistance R, and the tracks have a negligible resistance. A uniform
magnetic field is perpendicular to the plane of this circuit.
The
magnetic field is increasing at a constant rate dB/dt. Initially the
magnetic field has a strength B
0

and the rod is at rest at a distance x
0

from the connected end of the rails. Express the acceleration of the
rod at this instant in terms of the given

quantities.

Figure 2. Metal Rod in Magnetic Field.

The magnetic flux [Phi] enclosed by the rod and the tracks at time t =
0 s i
s given by

(7)

The magnetic field is increasing with a constant rate, and consequently
the enclosed magnetic flux is also increa
sing:

(8)

Faraday's law of induction can now be used to determine the induced
emf:

(9)

As a result of the induced emf a current will flow through the rod with
a magnitude equal to

(10)

The direction of the current is along the wire, and therefore
perpendicular to the magnetic field. The force exerted by the magnetic
field on the rod is given by

(11)

(see Chapter 31). Combining eq.(10) and (11) we obtain for the force
on the wire

(12)

The acceleration of the rod at time t = 0 s is therefore equal to

(13)

F

The fact that electric currents can be induced by changing magnetic fields
was discovered in the early 1830s by Michael Faraday and simultaneously by
Joseph Henry. The results of many experiments by these men, and others can
be expressed by
a single relation known as Faraday's law.

Let us define the
magnetic flux

as the number of magnetic lines that pass
through a given surface. Symbolically, we can write this as

(82)

where

A

is the area of the
surface through which the flux passes, and I have
assumed that the magnetic field is constant over the surface. From this
equation, we see that the SI unit for magnetic flux is the
weber
. In terms of basic
units, we define a weber (Wb) as

1 Wb = 1 T m
2
.

If we allow the flux to vary with time, we find that this variation gives rise to an
induced emf

in a circuit crossed by the flux. We write this as

(83)

This is
.

How can we cause a change in the magneti
c flux? There are two ways.
The first way is to actually vary the magnetic field over time. Then the field
strength will vary and the flux will vary in the same manner. We can also cause
the flux to change by varying the cross sectional area of the cir
cuit through which
the magnetic field flows. If this area decreases, the flux will decrease
proportionally.

Lenz’s Law

Let us apply the conservation of energy principle to (83). Assume we
have a magnet with the north pole facing a closed loop. By conve
ntion, the lines
of force on a magnet flow from the north pole into the south pole. Thus, we see
that the north pole has an excess of positive charge. This positive charge
causes a current to be set up in the loop, creating a magnetic dipole. But, we
kn
ow that a current generates thermal energy via
P

=
i
2
R
, where
R

is the
resistance inherent in the loop. By conservation of energy, we see that the only
source of this energy must be the work done by the magnet on the loop dipole.
Thus, we see that the lo
op dipole must be oriented in such a way as to oppose
the motion of the magnet. This is known as
Lenz's law
. It can be stated more
generally as
the induced current will appear in such a direction that it
opposes the change that produced it
. Lenz's law is

incorporated into (83)
already. It is the reason for the minus sign on the right side.

Example
:

Consider the following setup.

This has a rectangular loop of wire of length
l
, one end of which is in a uniform
field
B

pointing at right angles to the plane of the loop. The dashed lines show
the assumed limits of the magnetic field. The flux enclosed by the loop is then

B

=
Blx

where
lx

is the area of that part of the loop in which
B

is not zer
o.

Suppose the loop were pulled out of the field with a steady velocity
v
. The
induced emf is then

This emf sets up a current in the loop which is equal to

where
R

is the loop resistance. From Lenz's law, we k
now that the current will
flow in a direction that creates a force that opposes the loop motion. The force
on the loop is given by

F

=
i
l

x

B
.

By symmetry, the only force that does not cancel out is that on side

1. By
requiring that the force point opp
osite to
v
, we see that the current must flow in a
clockwise manner. This force has a magnitude of

The power required to pull the loop at a constant velocity is given by

We can combine equations (82) and (83) t
o get

(84)

Let us restrict ourselves to the case where it is the magnetic field
B

that is
varying. Recall that we saw that we could view emfs as potential gains in a
circuit. This implies that

-
V

around a

circuit. Using the relation between the
potential and the electric field, this allows us to write

(85)

Combining (83) and (85), we thus see that

(86)

Notice that this is just the third equation in Maxwell's e
quations of
electrodynamics.

If we look at the physical interpretation of (86), we see that,
by itself
, the
electric field no longer is a conserved field. To see this, recall that the definition
of a conserved field is one for which we can define a poten
tial that varies from
point to point in such a way that the value of the potential difference is path
independent. This implies that, for a conserved field, the summation around a
closed loop must be

But, we just saw in (86) that this

is no longer true. Thus, the electric
field alone is no longer conserved when there are induced charges
present. However, we shall find that the combined electromagnetic
field is still conserved.

Example Problem: Induced EMF in a Solenoid

a) A long sol
enoid has 300 turns of wire per meter and has a radius of
3.0 cm. If the current in the wire is increasing at a rate of 50 A/s, at
what rate does the strength of the magnetic field in the solenoid
increase ?

b) The solenoid is surrounded by a coil with 12
this coil is 6.0 cm. What induced emf will be generated in this coil
while the current in the solenoid is increasing ?

a) The magnetic field in a solenoid was discussed in Chapter 31. If the
solenoid has n turns per meter and if I i
s the current through each coil
than the field inside the solenoid is equal to

(14)

Therefore,

(15)

In this problem n = 300 turns/meter and dI/dt = 50 A/s. The change
in the magnetic field is thus equal to

(16)

This equation shows that the magnetic field is increasing at a rate of
0.019 T/s.

b) Since the magnetic field in the solenoid is changing, the magnetic
flux enclosed by the

surrounding coil will also change. The flux
enclosed by a single winding of this coil is

(17)

where r
in

= 3.0 cm is the radius
of the solenoid. Here we have
assumed that the strength of the magnetic field outside the solenoid is
zero. The total flux enclosed by the outside coils is equal to

(18)

The rate of change of the magnetic flux due to that change in
magnetic field is given by

(19)

As a result of the change in the current in the solenoid an emf will be
induced in the outer coil, with a value equal to

(20)

If the ends of the coil are connected, a current will flow through the
conductor. The direction of the current in the coil can be determined
using
Lenz' law

which states that

" The induced emfs are always of such a polarity as to oppose the
change t
hat generates them "

Let us apply Lenz' law to problem 12. The direction of the magnetic
field can be determined using the right hand rule and is pointed to the
right. If the current in the solenoid increases the flux will also increase.
The current in th
e external coil will flow in such a direction as to
oppose this change. This implies that the current in this coil will flow
counter clock wise (the field generated by the induced current is
directed opposite to the field generated by the large solenoid).

2. The Induced Electric
-
Field

A rod moving in a magnetic field will have an induced emf as a result
of the magnetic force acting on the free electrons. The induced emf
will be proportional to the linear velocity v of the rod. If we look at the
rod from

a reference frame in which the rod is at rest, the magnetic
force will be zero. However, there must still be an induced emf. Since
this emf can not be generated by the magnetic field, it must be due to
an electric field which exists in the moving referenc
e frame. The
magnitude of this electric field must be such that the same induced
emf is created as is generated in the reference frame in which the rod
is moving. This requires that

(21)

The electric field E' that exists in the reference frame of the moving
rod is called the
induced electric field
. The emf generated between
the ends of the rod is equal to

(22)

which is equivalent to eq.(4). If the induced electric field is position
dependent, then we have to replace eq.(22) with an integral
expression

(23)

where the integral extends from one end of the rod to the other end of
the rod.

The difference between the induced electric fie
ld and the electric field
generated by a static charge distribution is that in the former case the
field is not conservative and the path integral along a closed path is
equal to

(24)

which is non
-
zero if the magnetic flux is time dependent.

3 Inductance

A changing current in a conductor (like a coil) produces a changing
magnetic field. This time
-
dependent magnetic field can in
duce a
current in a second conductor if it is placed in this field. The emf
induced in this second conductor, [epsilon]
2
, will depend on the
magnetic flux through this conductor:

(25)

The flux [Phi]
B1

depends on the strength of the magnetic field
generated by conductor 1, and is therefore proportional to the current
I
1

through this conductor:

(26)

Here, the constant L
21

depends on the size of the two coils, on their
separation distance, and on the number of turns in each coil. The
constant L
21

is called the

mutual inductance of the two coils. Using this
constant, eq.(25) can be rewritten as

(27)

The unit of inductance is the Henry (
H) and from eq.(27) we conclude
that

(28)

When the magnetic field generated by a coil changes (due to a change
in current) the m
agnetic flux enclosed by the coil will also change. This
change in flux will induce an emf in the coil, and since the emf is due
to a change in the current through the coil it is called the self
-
induced
emf. The self
-
induced emf is equal to

(29)

In equation (29) L is called the
self inductance

of the coil. The self
-
induced emf will act in such a direction to oppose the change in th
e
current.

Applications of electromagnetic induction

Electromagnetic induction is an incredibly useful phenomenon with a
wide variety of applications. Induction is used in power generation and
power transmission, and it's worth taking a look at how that'
s done.
There are other effects with some interesting applications to consider,
too, such as eddy currents.

Eddy currents

An eddy current is a swirling current set up in a conductor in response
to a changing magnetic field. By Lenz¹s law, the current swir
ls in such
a way as to create a magnetic field opposing the change; to do this in
a conductor, electrons swirl in a plane perpendicular to the magnetic
field.

Because of the tendency of eddy currents to oppose, eddy currents
cause energy to be lost. More
accurately, eddy currents transform
more useful forms of energy, such as kinetic energy, into heat, which
is generally much less useful. In many applications the loss of useful
energy is not particularly desirable, but there are some practical
applications
. One is in the brakes of some trains. During braking, the
metal wheels are exposed to a magnetic field from an electromagnet,
generating eddy currents in the wheels. The magnetic interaction
between the applied field and the eddy currents acts to slow the

wheels down. The faster the wheels are spinning, the stronger the
effect, meaning that as the train slows the braking force is reduced,
producing a smooth stopping motion.

An electric generator

A electric motor is a device for transforming electrical ene
rgy into
mechanical energy; an electric generator does the reverse, using
mechanical energy to generate electricity. At the heart of both motors
and generators is a wire coil in a magnetic field. In fact, the same
device can be used as a motor or a generat
or.

When the device is used as a motor, a current is passed through the
coil. The interaction of the magnetic field with the current causes the
coil to spin. To use the device as a generator, the coil can be spun,
inducing a current in the coil.

An AC (a
lternating current) generator utilizes Faraday's law of
induction, spinning a coil at a constant rate in a magnetic field to
induce an oscillating emf. The coil area and the magnetic field are kept
constant, so, by Faraday's law, the induced emf is given b
y:

If the loop spins at a constant rate,
. Using calculus, and taking the
derivative of th
e cosine to get a sine (as well as bringing out a factor
of
), it's easy to show that the emf can be expressed as:

The combination
represents the maximum value of the generated
voltage (i.e., emf) and can be shortened to
. This reduces the
expression for the emf to:

In other words, a coil of wire spun in a magnetic field at a constant
rate will produce

AC electricity. In North America, AC electricity from a
wall socket has a frequency of 60 Hz.

A coil turning in a magnetic field can also be used to generate DC
power. A DC generator uses the same kind of split
-
ring commutator
used in a DC motor. Unlike
the AC generator, the polarity of the
voltage generated by a DC generator is always the same. In a very
simple DC generator with a single rotating loop, the voltage level
would constantly fluctuate. The voltage from many loops (out of synch
with each other
voltage.

Rather than using a spinning coil in a constant magnetic field, another
way to utilize electromagnetic induction is to keep the coil stationary
and to spin permanent magnets (providing the

magnetic field and flux)
around the coil. A good example of this is the way power is generated,
such as at a hydro
-
electric power plant. The energy of falling water is
used to spin permanent magnets around a fixed loop, producing AC
power.

Back EMF in el
ectric motors

You may have noticed that when something like a refrigerator or an air
conditioner first turns on in your house, the lights dim momentarily.
This is because of the large current required to get the motor inside
these machines up to operating
speed. When the motors are turning,
much less current is necessary to keep them turning.

One way to analyze this is to realize that a spinning motor also acts
like a generator. A motor has coils turning inside magnetic fields, and
a coil turning inside a
magnetic field induces an emf. This emf, known
as the back emf, acts against the applied voltage that's causing the
motor to spin in the first place, and reduces the current flowing
through the coils. At operating speed, enough current flows to
overcome an
y losses due to friction and to provide the necessary
energy required for the motor to do work. This is generally much less
current than is required to get the motor spinning in the first place.

If the applied voltage is V, then the initial current flowin
g through a
motor with coils of resistance R is I = V / R. When the motor is
spinning and generating a back emf, the current is reduced:

Transformers

Electricity is often g
enerated a long way from where it is used, and is
transmitted long distances through power lines. Although the
resistance of a short length of power line is relatively low, over a long
distance the resistance can become substantial. A power line of
resista
nce R causes a power loss of I
2
R ; this is wasted as heat. By
reducing the current, therefore, the I
2
R losses can be minimized.

At the generating station, the power generated is given by P = VI. To
reduce the current while keeping the power constant, the
voltage can
be increased. Using AC power, and Faraday's law of induction, there is
a very simple way to increase voltage and decrease current (or vice
versa), and that is to use a transformer. A transformer is made up of
two coils, each with a different nu
mber of loops, linked by an iron core
so the magnetic flux from one passes through the other. When the flux
generated by one coil changes (as it does continually if the coil is
connected to an AC power source), the flux passing through the other
will chang
e, inducing a voltage in the second coil. With AC power, the
voltage induced in the second coil will also be AC.

In a standard transformer, the two coils are usually wrapped around
the same iron core, ensuring that the magnetic flux is the same
through bo
th coils. The coil that provides the flux (i.e., the coil
connected to the AC power source) is known as the primary coil, while
the coil in which voltage is induced is known as the secondary coil. If
the primary coil sets up a changing flux, the voltage in

the secondary
coil depends on the number of turns in the secondary:

Similarly, the relationship for the primary coil is:

Combining these gives the relationship between the primary and
secondary voltage:

Energy (or, equivalently, power) has to be conserved, so
:

If a transformer takes a high primary voltage and converts it to a low
secondary voltage, the current in the secondary will be higher than
that in the primary to compensa
te (and vice versa). A transformer in
which the voltage is higher in the primary than the secondary (i.e.,
more turns in the primary than the secondary) is known as a step
-
down transformer. A transformer in which the secondary has more
turns (and, therefor
e, higher voltage) is known as a step
-
up
transformer.

Power companies use step
-
up transformers to boost the voltage to
hundreds of kV before it is transmitted down a power line, reducing
the current and minimizing the power lost in transmission lines. Ste
p
-
down transformers are used at the other end, to decrease the voltage
to the 120 or 240 V used in household circuits.

Transformers require a varying flux to work. They are therefore perfect
for AC power, but do not work at all for DC power, which would k
eep
the flux constant. The ease with which voltage and current can be
tranformed in an AC circuit is a large part of the reason AC power,
rather than DC, is distributed by the power companies.

Although transformers dramatically reduce the energy lost to I
2
R
heating in power line, they don't give something for nothing.
Transformers will also dissipate some energy, in the form of:

1.

flux leakage
-

not all the magnetic flux from the primary passes
through the secondary

2.

self
-
induction
-

the opposition of the c
oils to a changing flux in
them

3.

heating losses in the coils of the transformer

4.

eddy currents

In the iron core of a transformer, electrons would swirl in cross
-
sectional planes. This current would heat up the transformer, wasting
power as heat. To minimi
ze power losses due to eddy currents, the
iron core is usually made up of thin laminated slices, rather than one
solid piece. Current is then confined within each laminated piece,
significantly reducing the swirling tendency as well as the losses by
heatin
g.