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Chapter 27: Electromagnetic Induction

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Title: Chapter 21: Electric Charge and Electric Field Author: Chiaki Yanagisawa Created Date: 5/27/2004 8:57:33 PM Document presentation format: On-screen Show – PowerPoint PPT presentation

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Title: Chapter 27: Electromagnetic Induction

1
Chapter 27 Electromagnetic Induction
• Discovery of Faradys law of induction

2

An emf in volts is induced in a circuit that is
equal to the time rate of change of the total
circuit
• The flux through the circuit may be changed in
several different ways
• B may be made more intense.
• The coil may be enlarged.
• The coil may be moved into a region of stronger
field.
• The angle between the plane of the coil and B may
change.

3
• Faradys law of induction (contd)

4
• Induced electric field

Consider work done in moving a test charge
around the loop in one revolution of induced emf.
Work done by emf
Work done by electric field
emf 2prE for a circular loop
For a circular current loop
In general
5
• Lenzs Law
• Direction of induced emf and Lenzs law

Extend Faradys law to solenoids with N turns
Why the minus sign and what does it mean?
Number of turns
Lenzs Law
The sign of the induced emf is such that it tries
to produce a current that would create a magnetic
flux to cancel (oppose) the original flux change.
or the induced emf and induced current are in
such a direction as to oppose the change that
produces them!
6
• Lenzs Law
• Example 1

7
• Lenzs Law
• Example 2
• The bar magnet moves towards loop.
• The flux through loop increases, and an emf
induced in the loop produces current in the
direction shown.
• B field due to induced current in the loop
(indicated by the dashed lines) produces a flux
opposing the increasing flux through the loop due
to the motion of the magnet.

8
• Motional Electromotive Force
• Origin of motional electromotive force I

FE
FB
9
• Motional Electromotive Force
• Origin of motional electromotive force I (contd)

10
• Motional Electromotive Force
• Origin of motional electromotive force II

v constant
B
.
Bind
11
• Motional Electromotive Force
• Origin of motional electromotive force II
(contd)

12
• Motional Electromotive Force
• Origin of motional electromotive force II
(contd)

13
• Motional Electromotive Force
• Origin of motional electromotive force II
(contd)

14
• Motional Electromotive Force
• Origin of motional electromotive force II
(contd)

15
• Motional Electromotive Force
• Origin of motional electromotive force III

16
• Motional Electromotive Force
• Origin of motional electromotive force III
(contd)

17
• Motional Electromotive Force
• Origin of motional electromotive force III
(contd)

18
• Motional Electromotive Force
• A bar magnet and a loop (again)

In this example, a magnet is being pushed towards
a closed loop. The number of field lines linking
the loop is evidently increasing. There is
relative motion between the loop and the field
lines and an observer at any point in the metal
of the loop, or the charges in the loop, will see
an E field
Also we have
19
• Motional Electromotive Force
• Example A bar magnet and a loop (contd)

loop
For the example just considered, let us see what
happens in a small interval dt. The relative
displacement ?loopdt causes a small area of B
field to enter the loop. For a length dL of the
loop the ddFB passing inside is d(dA)B dL
?loopdt sin? B. We can see this as
Integrating this expression right round the
circuit ( i.e. over dL) shows that this ?B
interpretation recovers Faradays law. You will
also see that the sign of E is consistent with
Lenzs Law.
20
• Motional Electromotive Force

v
v
• Example A generator (alternator)

top
The armature of the generator is rotating in a
uniform B field with angular velocity ? this can
be treated as a simple case of the E ?B field.
On the ends of the loop ?B is perpendicular to
the conductor so does not contribute to the emf.
On the top ?B is parallel to the conductor and
has the value E ?B cos ? ?RB cos ?t. The
bottom conductor has the same value of E in the
opposite direction but the same sense of
circulation.
B
bottom
21
• Eddy Current
• Eddy current Examples

22
• Eddy Current
• Eddy current Examples (contd)

23
• Eddy Current
• Eddy current prevention

24

The orange represents a magnetic field pointing
into the screen and let say it is increasing at
a steady rate like 100 gauss per sec. Then we put
a copper ring In the field as shown below. What
does Faradays Law say will happen?
Current will flow in the ring. What will happen
If there is no ring present?
Now consider a hypothetical path Without any
copper ring.There will be an induced Emf
with electric field lines as shown above.
In fact there will be many concentric circles
everywhere in space.
The red circuits have equal areas. Emf is the
same in 1 and 2, less in 3 and 0 in 4. Note no
current flows. Therefore, no thermal energy is
dissipated
25
Example
• A magnetic field is ? to the board (screen) and
uniform inside a radius R. The magnetic field is
increasing at a steady rate. What is the
magnitude of the induced field at a distance r
from the center?

E is parallel to dl
Notice that there is no wire or loop of wire. To
26
Example with numbers
Suppose dB/dt - 1300 Gauss per sec and R 8.5 cm
Find E at r 5.2 cm
Find E at 12.5 cm
27
• Self Inductance
• Self induction

When a current flows in a circuit, it creates a
magnetic flux which links its own circuit. This
is called self-induction. (Induction was the
old word for the flux linkage FB). The strength
of B is everywhere proportional to the I in the
circuit so we can write
L is called the self-inductance of the circuit
L depends on shape and size of the circuit. It
may also be thought as being equal to the flux
linkage FB when I 1 amp.
The unit of inductance is the henry
28
Self Inductance
• Calculation of self inductance A solenoid

Accurate calculations of L are generally
difficult. Often the answer depends even on the
thickness of the wire, since B becomes strong
close to a wire.
In the important case of the solenoid, the first
approximation result for L is quite easy to
Hence
Then,
So L is proportional to n2 and the volume of the
solenoid
29
Self Inductance
• Calculation of self inductance A solenoid
(contd)

Example the L of a solenoid of length 10 cm,
area 5 cm2, with a total of 100 turns is
L 6.2810-5 H 0.5 mm
diameter wire would achieve 100 turns in a single
layer. Going to 10 layers would increase L by a
factor of 100. Adding an iron or ferrite core
would also increase L by about a factor of 100.
The expression for L shows that µ0 has units H/m,
c.f, Tm/A obtained earlier
30
Self Inductance
• Calculation of self inductance A toroidal
solenoid

The magnetic flux inside the solenoid
Then the self-inductance of the solenoid
If N 200 turns, A 5.0 cm2 , and r 0.10 m
Then when the current increases uniformly from
0.0 to 6.0 A in 3.0 ms, the self-induced emf E
will be
31
• Self Inductance
• Stored energy in magnetic field

Why is L an interesting and very important
quantity? This stems from its relationship to
the total energy stored in the B field of the
circuit which we shall prove below.
When I is first established, we have a finite
(self-induced emf)
The source of I does work against the
self-induced emf in order to raise I to its final
value.
power work done per unit time
32
• Self Inductance
• Stored energy in magnetic field Example

Returning to our expression for the energy stored
in an inductance we can use it for the case of a
solenoid. Using formulae we have already obtained
for the solenoid
and
Hence
Energy per unit volume in the field
33
• Self Inductance
• Inductor

A circuit device that is designed to have a
particular inductance is called an inductor or a
choke. The usual symbol is
a
I
variable source of emf
L
b
34
• Mutual Inductance
• Transformer and mutual inductance

The classic examples of mutual inductance are
transformers for power conversion and for making
high voltages as in gasoline engine ignition.
A current I1 is flowing in the primary coil 1 of
N1 turns and this creates flux B which then
links coil 2 of N2 turns. The mutual inductance
M2 1 is defined such that the induction F2 is
given by
M2 1Mutual Inductance of the coils
Generally, M 1 2 M 2 1
Also
35
• Mutual Inductance
• Changing current and induced emf

Consider two fixed coils with a varying current
I1 in coil 1 producing magnetic field B1. The
induced emf in coil 2 due to B1 is
proportional to the magnetic flux through coil 2
f2 is the flux through a single loop in coil 2
and N2 is the number of loops in coil 2. But we
know that B1 is proportional to I1 which means
that F2 is proportional to I1. The mutual
inductance M is defined to be the constant
of proportionality between F2 and I1 and depends
on the geometry of the situation.
The induced emf is proportional to M and to the
rate of change of the current .
36
• Mutual Inductance
• Example

Now consider a tightly wound concentric solenoids.
Assume that the inner solenoid carries current
I1 and the magnetic flux on the outer solenoid
FB2 is created due to this current. Now the flux
produced by the inner solenoid is
The flux through the outer solenoid due to this
magnetic field is
37
• Mutual Inductance
• Example of inductor Car ignition coil

Two ignition coils, N116,000 turns, N2400 turns
wound over each other. l10 cm, r3 cm. A current
through the primary coil I13 A is broken in 10-4
sec. What is the induced emf ?
Spark jumps across gap in a spark plug and
ignites a gasoline-air mixture
38
• The R-L Circuit
• Current growth in an R-L circuit

Consider the circuit shown. At t lt 0 the switch
is open and I 0. The resistance R can include
the resistance of the inductor coil.
The switch closes at t 0 and I begins to
increase, Without the inductor the full current
would be established in nanoseconds. Not so with
the inductor.
Kirchhoffs Loop Rule
Power balance
Multiply by I
39
• The R-L Circuit
• Current growth in an R-L circuit (contd)

Rate at which energy is stored up in the inductor.
Power supplied by the battery
Power dissipated as heat in the resistor
If energy in inductor is
then
or
Integrate from t 0 (I 0) to t ? (I If)
So, the energy stored in an inductor carrying
current I is
40
• The R-L Circuit
• Current growth in an R-L circuit (contd)

Kirchhoffs Loop Rule
I then increases until finally dI/dt 0
At t 0, I 0
Compare with
Current in an LR circuit as function of time
41
• The R-L Circuit
• Current growth in an R-L circuit (contd)

Integrating between (I 0, t 0) and (I I, t
t)
42
• The R-L Circuit
• Current growth in an R-L circuit (contd)

Now we raise e to the power of each side
43
• The R-L Circuit
• Discharging an R-L circuit

Add switch S2 to be able to remove the battery.
And add R1 to protect the battery so that it is
protected when both switches are closed.
First S1 has been closed for a long enough time
so that the current is steady at its final value
I0.
At t0, close S2 and open S1 to effectively
remove the battery. Now the circuit abcd carries
the current I0.
Kirchhoffs loop rule
44
• The R-L Circuit
• Discharging an R-L circuit (contd)

Now lets calculate the total heat produced in
resistance R when the current decreases from I0
to 0.
Rate of heat production
Energy dissipated as heat in the resistor
The current as a function of time
The total energy
The total heat produced equals the energy
originally stored in the inductor
45
• The L-C Circuit
• Complex number and plane

real part Re(z)x, imaginary part Im(z)y
Complex number z x iy
46
• The L-C Circuit
• Simple harmonic oscillation

47
• The L-C Circuit
• Simple harmonic oscillation (contd)

48
• The L-C Circuit
• Simple harmonic oscillation (contd)

49
• The L-C Circuit
• An L-C circuit and electrical oscillation

S
Consider a circuit with an inductor and
a capacitor as shown in Fig. Initially
the capacitor C carries charge Q0
At t0 the switch closes and charge flows through
inductor producing self-induced emf.
The current I is by definition
Acceleration equation for a mass on a spring
Kirchhoffs loop rule
50
• The L-C Circuit
• An L-C circuit and electrical oscillation
(contd)

The solution of this equation is simple harmonic
motion.
Now lets figure out what A and f are. For that
choose initial condition as I(0)0 and Q(0)Q0.
Then AQ0 and f0.
The charge and current are 90o out of phase with
the same angular frequency w I is at maximum when
Q0, and Q is at maximum when I0.
51
• The L-C Circuit
• An L-C circuit and electrical oscillation
(contd)

The charge and current are 90o out of phase with
the same angular frequency w I is at maximum when
Q0, and Q is at maximum when I0.
-I(t)
52
• The L-C Circuit
• An L-C circuit and electrical oscillation
(contd)

The electric energy in the capacitor
The electric energy oscillates between its
maximum Q02 and 0.
The magnetic energy in the inductor
The magnetic energy oscillates between its
maximum Q02 /(2C) and 0.
UtotUeUm constant
Ue(t)
Um(t)
53
• The L-R-C Circuit
• Another differential equation

54
• The L-R-C Circuit
• Another differential equation (contd)

55
• The L-R-C Circuit
• Another differential equation (contd)

56
• The L-R-C Circuit
• An L-R-C circuit and electrical damped
oscillation

At t0 the switch is closed and a capacitor with
initial charge Q0 is connected in series across
an inductor.
Initial condition
A loop around the circuit in the direction of
the current flow yields
Since the current is flowing out of the capacitor,
57
• The L-R-C Circuit
• An L-R-C circuit and electrical damped
oscillation (contd)

If R2lt 4LC, the solution is
Note that if R0,no damping occurs.