Electromagnetic Induction

1
Chapter 21 Electromagnetic Induction Faradays
Law ac Circuits
We found in chapter 20 that an electric current
can give rise to a magnetic field, and that a
magnetic field can exert a force on a moving
charge.
I wonder if a magnetic field can somehow give
rise to an electric current
2
21.1 Induced emf
It is observed experimentally that changes in
magnetic fields induce an emf in a conductor.
An electric current is induced if there is a
closed circuit (e.g., loop of wire) in the
changing magnetic field.
A constant magnetic field does not induce an
emfit takes a changing magnetic field.
Passing the coil through the magnet would induce
an emf in the coil.
They need to calibrate their meter!
3
Note that change does not require observable
(to you) motion.
? A magnet may move through a loop of wire, or a
loop of wire may be moved through a magnetic
field (as suggested in the previous slide).
These involve observable motion.
? A changing current in a loop of wire gives rise
to a changing magnetic field (predicted by
Amperes law) which can induce a current in
another nearby loop of wire.
In the latter case, nothing observable (to your
eye) is moving, although, of course
microscopically, electrons are in motion.
As your text puts it induced emf is produced by
a changing magnetic field.
4
21.2 Faradays Law
To quantify the ideas of section 21.1, we define
magnetic flux. In an earlier chapter we briefly
touched on electric flux. This is the magnetic
analog.
Because we cant see magnetic fields directly,
we draw magnetic field lines to help us visualize
the magnetic field. Remember that magnetic field
lines start at N poles and end at S poles.
A strong magnetic field is represented by many
magnetic field lines, close together. A weak
magnetic field is represented by few magnetic
field lines, far apart.
We could, if we wished, actually calibrate by
specifying the number of magnetic field lines
passing through some surface that corresponded to
a given magnetic field strength.
5
Magnetic flux ?B is proportional to the number of
magnetic field lines passing through a surface.
http//hyperphysics.phy-astr.gsu.edu/hbase/magneti
c/fluxmg.html
6
Mathematically, magnetic flux ?B through a
surface of area A is defined by ?B B A cos
? OSE ?B B?A where B? is the component
of field perpendicular to the surface, A is the
area of the surface, and ? is the angle between B
and the normal to the surface.
7
When B is parallel to the surface, ?90 and ?B
0.
When B is perpendicular to the surface, ?0 and
?B BA.
The unit of magnetic flux is the T?m2, called a
weber 1 Wb 1 T ? m2 .
8
In the following discussion, we switch from
talking about surfaces in a magnetic field
to talking about loops of wire in a magnetic
field.
9
Now we can quantify the induced emf described
qualitatively in the previous section
Experimentally, if the flux through N loops of
wire changes by an amount ??B in a time ?t, the
induced emf is
Not an OSEnot quite yet.
This is called Faradays law of induction. It is
one of the fundamental laws of electricity and
magnetism, and an important component of the
theory that explains electricity and magnetism.
I wonder why the sign
10
Experimentally
an induced emf always gives rise to a current
whose magnetic field opposes the change in
fluxLenzs law.
Think of the current resulting from the induced
emf as trying to maintain the status quoto
prevent change.
If Lenzs law were not trueif there were a
sign in faradays lawthen a changing magnetic
field would produce a current, which would
further increased the magnetic field, further
increasing the current, making the magnetic field
still bigger
Among other things, conservation of energy would
be violated.
Well practice with this in a bit.
11
Ways to induce an emf
? change B
? change the area of the loop in the field
12
Ways to induce an emf (continued)
? change the orientation of the loop in the
field
13
Conceptual example 21-1 Induction Stove
An ac current in a coil in the stove top produces
a changing magnetic field at the bottom of a
metal pan.
The changing magnetic field gives rise to a
current in the bottom of the pan.
Because the pan has resistance, the current heats
the pan. If the coil in the stove has low
resistance it doesnt get hot but the pan does.
An insulator wont heat up on an induction stove.
Remember the controversy about cancer from power
lines a few years back? Careful studies showed
no harmful effect. Nevertheless, some believe
induction stoves are hazardous.
14
Conceptual example 21-2 Practice with Lenzs Law
In which direction is the current induced in the
coil for each situation shown?
(counterclockwise)
(no current)
15
(counterclockwise)
(clockwise)
16
Rotating the coil about the vertical diameter by
pulling the left side toward the reader and
pushing the right side away from the reader in a
magnetic field that points from right to left in
the plane of the page.
(counterclockwise)
Remember
Now that we are experts on the application of
Lenzs law, lets make our induced emf equation
official
This means it is up to you to use Lenzs law to
figure out the direction of the induced current
(or the direction of whatever the problem wants.
17
Example 21-3 Pulling a Coil from a Magnetic Field
A square coil of side 5 cm contains 100 loops and
is positioned perpendicular to a uniform 0.6 T
magnetic field. It is quickly and uniformly
pulled from the field (moving ? to B) to a region
where the field drops abruptly to zero. It takes
0.10 s to remove the coil, whose resistance is
100 ?.
B 0.6 T
5 cm
18
(a) Find the change in flux through the coil.
Initial ?Bi BA .
Final ?Bf 0 .
??B ?Bf - ?Bi 0 - BA - (0.6 T) (0.05 m)2
- 1.5x10-3 Wb .
19
(b) Find the current and emf induced.
?
The current must flow counterclockwise to induce
a downward magnetic field (which replaces the
removed magnetic field).
20
The induced emf is
The induced current is
21
(c) How much energy is dissipated in the coil?
Current flows only during the time flux
changes.
E Pt I2Rt (1.5x10-2 A) (100 ?) (0.1 s)
2.3x10-3 J .
(d) What was the average force required?
The loop had to be pulled out of the magnetic
field, so the pulling force did work. It is
tempting to try and set up a free body diagram
and use Newtons laws. Instead, energy
conservation gets the answer with less work.
If there no resistance in the loop, the current
would flow indefinitely. However, the resistance
quickly halts the flow of current once the
magnetic flux stops changing.
22
The flux change occurs only when the coil is in
the process of leaving the region of magnetic
field.
No flux change. No emf. No current. No work
(why?).
23
F
D
Flux changes. emf induced. Current flows. Work
done.
24
No flux change. No emf. No current. (No work.)
25
The energy calculated in part (c) is the energy
dissipated in the coils while the current is
flowing. The amount calculated in part (c) is
also the mechanical energy put into the system by
the force.
Ef Ei WotherI?f
0
See 2 slides back for F and D.
Ef Ei F D
F Ef / D
F (2.3x10-3 J) / (0.05 m) F 0.046 N
26
21.3 emf Induced in a Moving Conductor
Recall that one of the ways to induce an emf is
to change the area of the loop in the magnetic
field. Lets see how this works.
v
A U-shaped conductor and a moveable conducting
rod are placed in a magnetic field, as shown.
l
The rod moves to the right with a speed v for a
time ?t.
?A
v?t
The rod moves a distance v?t and the area of the
loop inside the magnetic field increases by an
amount ?A l v ?t .
27
The loop is perpendicular to the magnetic field,
so the magnetic flux through the loop is ?B
BA. The emf induced in the conductor can be
calculated using Amperes law
B and v are vector magnitudes, so they are always
. Wire length is always . You use Lenzs law
to get the direction of the current.
28
This kind of emf is called motional emf
because it took motion to induce it.
The induced emf causes current to flow in the
loop.
v
Magnetic flux inside the loop increases (more
area).
l
System wants to make the flux stay the same, so
the current gives rise to a field inside the loop
into the plane of the paper (to counteract the
extra flux).
?A
I
v?t
Clockwise current!
29
The induced emf causes current to flow in the
loop. Giancoli shows an alternate method for
getting ?, by calculating the work done moving
the charges in the wire.
Electrons in the moving rod (only the rod moves)
experience a force F q v B. Using the right
hand rule, you find the the force is up the
rod, so electrons move up.
Up here refers only to the orientation on the
page, and has nothing to do with gravity.
Because the rod is part of a loop, electrons flow
counterclockwise, and the current is clockwise
(whew, we got that part right!).
Remember, find the force direction, then reverse
it if the charge is an electron!
30
The work to move an electron from the bottom of
the rod to the top of the rod is W (force)
(distance) (q v B) (l).
Going way back to the beginning of the semester,
Wi?f q ?Vi?f .
F q v B
But ?Vi?f is just the change in potential along
the length l of the loop, which is the induced
emf.
Going way back to the beginning of the semester,
W (q v B) (l) (q ?).
Solving (q v B) (l) (q ?) for ? gives ? B l
v, as before.
31
I wont ask you to reproduce the derivation on an
exam, but a problem could (intentionally or not)
ask you to calculate the work done in moving a
charge (or a wire) through a magnetic field, so
be sure to study your text.
But you havent given us the OSE yet!
Good point! The derivation assumed B, l, and v
are all mutually perpendicular, so we really
derived this
where B? is the component of the magnetic field
perpendicular to l and v, and v? is the component
of the velocity perpendicular to B and l.
32
Example 21-4 An airplane travels 1000 km/h in a
region where the earths field is 5x10-5 T and is
nearly vertical. What is the potential
difference induced between the wing tips that are
70 m apart?
The derivation of ? B? l v? on slides 15 and 16
assumed the area through which the magnetic field
passes increased.
My first reaction is that the magnetic flux
through the wing is not changing because neither
the field nor the area of the wing is changing.
True, but wrong reaction! The alternate
derivation shows that the electrons in the moving
rod (airplane wing in this case) experience a
force, which moves the electrons.
33
The electrons pile up on the left hand wing of
the plane, leaving an excess of charge on the
right hand wing.
Our equation for ? gives the potential
difference.
No danger to passengers! (But I would want my
airplane designers to be aware of this.)
34
21.4 Changing Magnetic Flux Produces an Electric
Field
From chapter 16, section 6
OSE E F / q
From chapter 20, section 4
OSE F q v B sin?
For v ? B, and in magnitude only,
F q E q v B
E v B.
We conclude that a changing magnetic flux
produces an electric field. This is true not
just in conductors, but any-where in space where
there is a changing magnetic field.
35
Example 21-5 Blood contains charged ions, so
blood flow can be measured by applying a magnetic
field and measuring the induced emf. If a blood
vessel is 2 mm in diameter and a 0.08 T magnetic
field causes an induced emf of 0.1 mv, what is
the flow velocity of the blood?
OSE ? B? l v?
v ? / (B? l)
In Figure 21-11 (the figure for this example), B
is applied ? to the blood vessel, so B is ? to v.
The ions flow along the blood vessel, but the
emf is induced across the blood vessel, so l is
the diameter of the blood vessel.
v (0.1x10-3 V) / (0.08 T 0.2x10-3 m)
v 0.63 m/s
36
21.5 Electric Generators
Lets begin by looking at a simple animation of a
generator. http//www.wvic.com/how-gen-works.htm
Heres a freeze-frame.
Normally, many coils of wire are wrapped around
an armature. The picture shows only one.
Brushes pressed against a slip ring make
continual contact.
The shaft on which the armature is mounted is
turned by some mechanical means.
37
Lets look at the current direction in this
particular freeze-frame.
B is down. Coil rotates counter-clockwise.
Put your fingers along the direction of movement.
Stick out your thumb.
Bend your fingers 90. Rotate your hand until
the fingers point in the direction of B. Your
thumb points in the direction of conventional
current.
38
Alternative right-hand rule for current direction.
B is down. Coil rotates counter-clockwise.
Make an xyz axes out of your thumb and first two
fingers.
Thumb along component of wire velocity ? to B.
1st finger along B.
2nd finger then points in direction of
conventional current.
Hey! The picture got it right!
39
I know we need to work on that more. Lets zoom
in on the armature.
v
v??B
v?B
I
B
40
Forces on the charges in these parts of the wire
are perpendicular to the length of the wire, so
they dont contribute to the net current.
For future use, call the length of wire shown in
green h and the other lengths (where the two
red arrows are) l.
41
One more thing
This wire
connects to this ring
so the current flows this way.
42
Later in the cycle, the current still flows
clockwise in the loop
but now this wire
connects to this ring
so the current flows this way.
Alternating current! ac!
Again http//www.wvic.com/how-gen-works.htm
43
Dang! That was complicated. Are you going to
ask me to do that on the exam?
No. Not anything that complicated. But you
still need to understand each step, because each
step is test material.
Click here and scroll down to electrodynamics
to see some visualizations that might help you!
Understanding how a generator works is good,
but we need to quantify our knowledge.
We begin with our OSE ? B? l v?. (l was
defined on slide 16.) In our sample generator on
the last 7 slides, we had only one loop, but two
sides of the loop in the magnetic field. If the
generator has N loops, then ? 2 N B? l v?.
44
Back to this picture
This picture is oriented differently than Figure
21-13 in your text. In your text, ? is the angle
between the perpendicular to the magnetic field
and the plane of the loop.
45
?
The angle ? is the text is the same as the angle
between v??B and the vector v.
Thus, v? v sin ?.
46
B is ? to the wire, so
But the coil is rotating, so ? ? t, and v ? r
? (h/2). The diameter of the circle of
rotation, h, was defined on slide 16.
where A is the area of the loop, f is the
frequency of rotation of the loop, and ? 2 ? f.

47
Example 21-6 The armature of a 60 Hz ac
generator rotates in a 0.15 T magnetic field. If
the area of the coil is 2x10-2 m2, how many loops
must the coil contain if the peak output is to be
?0 170 V?
48
18-8 Alternating Current
In chapter 21 we learned how a coil of wire
rotating in a magnetic field creates ac current.
A large magnet rotating inside coils of wire
would also produce ac current. (ac current is
redundant, isnt it?)
49
The voltage produced by the generator is
sinusoidal
The generator web page uses U instead of V as the
symbol for voltage.
50
In chapter 21, we wrote
Using ? 2?f, V instead of ?, and grouping
constants together, in chapter 18 Giancoli
decrees the formula
V0 is the peak voltage and in the US, the
frequency of ac power in your home is f 60 Hz.
Using Ohms law
Where I0 is the peak current.
51
Even though the average current is zero, power is
still lost due to resistance
It is easy to show (a bit of calculus) that the
average value of sin2(2? f t) is ½, so the
average power developed in a resistance is
The bar above the P denotes average. I will
write it Pavg when using text.
Note that the equations for Pavg contain
sorry, eqn. editor wont let me put bar over 2
characters
52
The rms (root mean square) value of a quantity is
obtained by taking the square root of the average
of the square of that quantity. The equations
for Pavg contain the mean square values of
current and voltage.
We write
and
Then
Not sure why I didnt make this official last
year. I think Ill make it official this year
53
Un the US, we talk about rms voltage when we
refer to household line voltage. The peak
voltage is thus
Example 18-11 (a) Calculate the resistance and
the peak current in a 1000 W hair dryer connected
to a 120 V line. (b) what happens if it is
connected to a 240 V line in Britain?
(a) Our OSE P IV I2R V2/R works if we
replace P by Pavg and I and V by Irms and Vrms.
54
(No Transcript)
55
(b) Assume the hair dryers resistance does not
change with temperature (in reality, it probably
increases).
You just melted your hair dryer!
56
Example 18-12 Each channel of a stereo receiver
is capable of an average power output of 100 W
into an 8 ? loudspeaker. What is the rms voltage
and rms current fed to the speaker (a) at the
maximum power of 100 W, and (b) at 1 W?
well use these for both parts
57
(a) at 100 W and 8 ?
(b) at 1 W and 8 ?
58
21-7 Transformers
59
No, no, no
A transformer is a device for increasing or
decreasing an ac voltage.
Pole-mounted transformer
ac-dc converter
Power Substation
60
A transformer is basically two coils of wire
wrapped around each other, or wrapped around an
iron core.
When an ac voltage is applied to the primary
coil, it induces an ac voltage in the secondary
coil.
A step up transformer increases the output
voltage in the secondary coil a step down
transformer reduces it.
61
The ac voltage in the primary coil causes a
magnetic flux change given by
The changing flux (which is efficiently carried
in the transformer core) induces an ac voltage in
the secondary coil given by
Dividing the two equations gives the transformer
equation
For a step-up transformer, NS gt NP and VS gt VP
(the voltage is stepped up).
62
For a step-down transformer, NS lt NP and VS lt VP
(the voltage is stepped down).
Transformers only work with ac voltages a dc
voltage does not produce the necessary changing
flux.
A step-up transformer increases the voltage. Is
this an example of getting something for
nothing?
No, because even though transformers are
extremely efficient, some power (and therefore
energy) is lost.
If no power is lost, we can use P IV to get
flipped!
63
If transformers only work on ac, how come you
showed a picture of an ac-dc converter a few
slides back?
Ever wanted to cut open one of those ac-dc
converters and see what they look like inside?
An ac-dc converter first steps down the 120 volt
line voltage, and then converts the voltage to dc
fewer turns in the secondary coil
a diode is a device that lets current flow one
way only (dc)
64
Pictures of ac-dc converter came from
http//www.howstuffworks.com/inside-transformer.ht
m
Example 21-9 A transformer for home use of a
portable radio reduces 120 V ac to 9 V dc. The
secondary contains 30 turns and the radio draws
400 mA. Calculate (a) the number of turns in the
primary (b) the current in the primary and (c)
the power transformed.
65
(No Transcript)
66
The power output to the secondary coil is
This is the same as the power input to the
primary coil because our transformer equation
derivation assumed 100 efficient transformation
of power.
67
Example 21-10 An average of 120 kW of electrical
power is sent to a small town from a power plant
10 km away. The transmission lines have a total
resistance of 0.40 ?. Calculate the power loss
if power is transmitted at (a) 240 V and (b)
24,000 V.
This problem does not use the transformer
equation, but it shows why transformers are
useful.
68
(a) at 240 V
(a) at 24000 V
More than 80 of the power would be wasted if it
were transmitted at 240 V, but less than 0.01
is wasted if the power is transmitted at 24000 V.
69
http//www.howstuffworks.com/power.htm
We skipped section 21.6 on counter emf. Read it
to see why motors burn out when they cant turn,
and why your house lights might dim when the
fridge comes on.
70
Chapter 22 Electromagnetic Waves
Every student of EM should be exposed to
electromagnetic waves. Here is your exposure.
22.1 Maxwells Equations
Maxwells equations involve calculus. They
represent the fundamental laws of electricity and
magnetism. We have seen simpler forms of some of
them.
71
In words, Maxwells equations are 1a
generalized form of Coulombs law, relating
electric fields to their sources (charges) 2a
law relating magnetic fields to magnetic
poles 3an equation describing how an electric
field is produced by a changing magnetic field
(Faradays Law) 4an equation describing how a
magnetic field is produced by an electric current
or changing electric field (Amperes Law)
That a changing electric field can produce a
magnetic field is not one of the predictions of
Amperes law it was hypothesized by Maxwell and
verified after his death.
72
Every student should be exposed to Maxwells
equations, so here they are in their integral
form
73
Maxwells equations are to EM as Newtons laws
are to mechanics, except Maxwells equations are
relativistically correct, and Newtons laws are
not.
22.3 Production of Electromagnetic Waves
Your text shows how electromagnetic waves can be
produced by oscillating charges on conductors.
These waves travel through space even long after
they are far away from the charges that produced
them.
E and B in the radiation fields drop off as 1/r,
and the intensity of the waves drops off as 1/r2.
74
The electric and magnetic fields are
perpendicular to each other and to the direction
of propagation of the wave. They are also in
phase. See figure 22.7, page 666.
From Maxwells equations you can show that
electromagnetic waves travel with a speed v
1/(?0?0)1/2 which is equal to 3x108 m/s, the
speed of light.
75
22.5 Light as an EM Wave The Electromagnetic
Spectrum
Light was known to behave like a wave long before
Maxwell showed that the speed of EM waves is the
same as the speed of light. Eventually, it came
to be recognized that light is just an example of
an EM wave.
The frequencies of visible light lie between
about 4x10-7 and 7.5x10-7 m, or 400 to 750 nm.
The frequency, wavelength, and speed of a wave
are related by v f?, so for EM waves, and for
light, c f?.
Visible light represents only a minute portion of
the electromagnetic spectrum, see figure 22.10.
EM waves are typically produced by acceleration
of charged particles.
76
The sun emits large amounts of IR, visible, and
UV radiation. We detect the IR as heat, the
visible as light, and the UV through skin damage.
Note from figure 22.10 that x-rays, gamma rays,
and radio waves are just EM waves, like light,
only of different frequencies.
(sorry, rushed scan)
77
Symbols for cut and paste ? ? ? l ?
? ? ? ? ? ? ? ? ?
? ? ?
http//www.howstuffworks.com/power.htm http//www.
howstuffworks.com/inside-transformer.htm http//ww
w.sweethaven.com/acee/forms/frm0502.htm