Chapter 30 Induction and Inductance
In this chapter we will study the following topics:
-Faraday’s law of induction -Lenz’s rule -Electric field induced by a changing magnetic field -Inductance and mutual inductance - RL circuits -Energy stored in a magnetic field
(30–1)
In a series of experiments, Michael Faraday in England
and Joseph Henry in the U.S. were able to generate
electric currents without the use of batteries.
Below
Faraday's
3
0.2 Two exp
Experiment
e
s
riments
we describe some of these experiments that
helped formulate what is known as "Faraday's law
of induction."The circuit shown in the figure consists of a wire loop connected to a sensitive
ammeter (known as a "galvanometer"). If we approach the loop with a permanent
magnet we see a current being registered by the galvanometer. The results can be
summarized as follows:
A current appears only if there is relative motion between the magnet and the loop.
Faster motion results in a larger current.
If
1.
2.
3. we reverse the direction of motion or the polarity of the magnet, the current
reverses sign and flows in the opposite direction.
The current generated is known as " "; the emf that appearinduced current s
is known as " "; the whole effect is called " "induced emf induction. (30–2)
In the figure we show a second type of experiment
in which current is induced in loop 2 when the
switch S in loop 1 is either closed or opened. When
the current in loop 1 is constant no induced current
is observed in loop 2. The conclusion is that the
magnetic field in an induction experiment can be
generated either by a permanent magnet or by an
electric current in a coil.
loop 1loop 2
Faraday summarized the results of his experiments in what is known a
30-3 Faraday's Law of Induction
s
" "Faraday's law of induction.An emf is induced in a loop when the number of magnetic field lines that
pass through the loop is changing.
Faraday's law is not an explanation of induction but merely a description of
what induction is. It is one of the four " of electromagnetism,"
all of which are statements of experimen
Maxwell's equations
tal results. We have already encountered
Gauss' law for the electric field, and Ampere's law (in its incomplete form).(30–3)
B
n̂
dA
The magnetic flux through a surface that borders
a loop is determined as follows:
Magnetic Flux Φ
1
B
We divide the surface that has the loop as its border
into elements of area . dA
.
For each element we calculate the magnetic flux through it: cos .
ˆHere is the angle between the normal and the magnetic field vectors
at the position of the element.
We integrate
Bd BdA
n B
2.
3.
2: T m known as the Weber (symbol
all the terms. cos
We can express Faraday's law of induction in the followin
Wb).
g form:
B BdA B dA
SI magnetic flux unit
The magnitude of the emf induced in a conductive loop is equal to the
rate at which the magnetic flux Φ through the loop changes with time.B
E
B B dA
Bd
dt
E
(30–4)
B
n̂
dA
cosB BdA B dA
Change the magnitude of within the loop.
Change either the total area of the coil or
the portion of the area within the magnetic field.
Change the angl
B
Methods for Changing Φ Through a Loop
1.
2.
3.
B
ˆe between and .
Problem 30-13
cos cos
sin
2
2 sin 2
B
B
B n
NAB NabB t
dNabB t
dtf
fNabB t
An Example.
E
E
B
n̂
loop(30–5)
Bd
dt
E
We now concentrate on the negative sign
in the equation that expresses Faraday's law.
The direction of the flow of induced current
in a loop is accurately predicted by what is
known as L
30 - 4 Lenz's Rule
enz's rule.
An induced current has a direction such that the magnetic field due to the
induced current opposes the change in the magnetic flux that induces the current.
Lenz's rule can be implemented using one of two methods:
In the figure we show a bar magnet approaching a loop. The induced current flows
in the direction indicated becaus
1. Opposition to pole movement
e this current generates an induced magnetic field
that has the field lines pointing from left to right. The loop is equivalent to a
magnet whose north pole faces the corresponding north pole of the bar magnet
approaching the loop. The loop the approaching magnet and thus opposes
the change in that generated the induced current.Brepels
(30–6)
N S
magnet motion
Bar magnet approaches the loop
with the north pole facing the loop.
2. Opposition to flux change
Example :a
As the bar magnet approaches the loop, the magnetic field points toward the left
and its magnitude increases with time at the location of the loop. Thus the magnitude
of the loop magnetic flux alB
B
net
so increases. The induced current flows in the
(CCW) direction so that the induced magnetic field opposes
the magnetic field . The net field . The induced current
i
i
B
B B B B
counterclockwise
is thus trying
to from increasing. Remember that it was the increase in that
generated the induced current in the first place. B B prevent
(30–7)
N S
magnet motion
Bar magnet moves away from the loop
with north pole facing the loop.
2. Opposition to flux change
Example :b
As the bar magnet moves away from the loop, the magnetic field points
toward the left and its magnitude decreases with time at the location of the
loop. Thus the magnitude of the loop magnetic flu
B
net
x also decreases. The
induced current flows in the (CW) direction so that the induced
magnetic field adds to the magnetic field . The net field .
The induced current
B
i iB B B B B
clockwise
is thus trying to from decreasing. Remember
that it was the decrease in that generated the induced current in the first
place.
B
B
prevent
(30–8)
S N
magnet motion
Bar magnet approaches the loop
with south pole facing the loop.
2. Opposition to flux change
Example :c
As the bar magnet approaches the loop, the magnetic field points toward the
right and its magnitude increases with time at the location of the loop. Thus
the magnitude of the loop magnetic flux B
B
net
also increases. The induced current
flows in the (CW) direction so that the induced magnetic field
opposes the magnetic field . The net field . The induced current is
t
i
i
B
B B B B
clockwise
hus trying to from increasing. Remember that it was the increase in
that generated the induced current in the first place. B
B
prevent
(30–9)
S N
magnet motion
Bar magnet moves away from the loop
with south pole facing the loop.
2. Opposition to flux change
Example :d
As the bar magnet moves away from the loop, the magnetic field points toward
the right and its magnitude decreases with time at the location of the loop. Thus
the magnitude of the loop magnetic fl
B
net
ux also decreases. The induced current
flows in the (CCW) direction so that the induced magnetic
field adds to the magnetic field . The net field . The induced
B
i iB B B B B
counterclockwise
B
current is thus trying to from decreasing. Remember that it was the
decrease in that generated the induced current in the first place. B
prevent
(30–10)
By Lenz's rule, the induced current always opposes
the external agent that produced the induced current.
Thus the external agent must always on the
loop-magneti
30 - 5 Induction and Energy Transfers
do work
c field system. This work appears as
thermal energy that gets dissipated on the resistance
of the loop wire.
Lenz's rule is actually a different formulation of
the principle of energy conservation.
R
Consider the loop of width shown in the figure.
Part of the loop is located in a region where a
uniform magnetic field exists. The loop is being
pulled outside the magnetic field region with cons
L
B
tant
speed . The magnetic flux through the loop is
. The flux decreases with time:
B
B
v
BA BLx
d dx BLvBL BLv i
dt dt R R
EE
(30–11)
2 2 2 22
th
2 3
The rate at which thermal energy is dissipated on is
( )
The magnetic forces on the wire sides are shown
in the figure. Forces and cancel each other:
Force
R
BLv B L vP i R R
R R
F F
eq. 1
1 1
2 2
1
2 2 2
ext 1
, sin 90 ,
. The rate at which the external agent is
producing mechanical work is ( )
If we compare equations 1 and 2 we see that indeed the
BLvF iL B F iLB iLB LB
R
B L vF
R
B L vP Fv
R
eq. 2
mechanical work done by the external agent that moves
the loop is converted into thermal energy that appears
on the loop wires.
(30–12)
We replace the wire loop in the previous example
with a solid conducting plate and move the plate
out of the magnetic field as shown in the figure.
The motion between the plate and indB
Eddy Currents
uces a
current in the conductor and we encounter an opposing
force. With the plate, the free electrons do not follow
one path as in the case of the loop. Instead, the electrons
swirl around the plate. These currents are known as
" " As in the case of the wire loop, the net
result is that the mechanical energy that moves the plate is
transformed into thermal energy that heats up
eddy currents.
the plate.
(30–13)
Consider the copper ring of radius shown in the
figure. It is placed in a uniform magnetic field
pointing into the page, which increases as a function
of time. The resu
r
B
30 - 6 Induced Electric Fields
lting change in magnetic flux
induces a current in the counterclockwise
(CCW) direction.
i
The presence of the current in the conducting ring implies that an induced
electric field must be present in order to set the electrons in motion.
Using the argument above we can reformulate Farada
i
E
y's law as follows:
A changing magnetic field produces an electric field.
The induced electric field is generated even in the absence of the
copper ring.
Note :
(30–14)
Consider the circular closed path of radius shown in
the figure to the left. The picture is the same as that
on the previous page except that the copper ring has
been removed. The path is now an a
r
bstract line.
The emf along the path is given by the equation
( )
The emf is also given by Faraday's law:
( ). If we compare eq. 1 with eq. 2
we get:
B
E d
dE d
s
d
dt
s
eq. 1
eq. 2
E
E
2 2
2
cos 0
22
.
2
BB
B
E ds Eds E ds rE
d dBr B r
dt dtdB
rE rd
dt
r dBE
dtt
(30–15)
0
Consider a solenoid of length that has loops of
area each, and windings per unit length. A current
flows through the solenoid and generates a uniform
magnetic field
N
NA n
i
B
30 - 7 Inductance
inside the solenoid.
The solenoid magnetic flux is .B
ni
NBA
B
20The total number of turns . The result we got for the
special case of the solenoid is true for any inductor: . Here is a
constant known as the of the solenoi
B
B
N n n A i
Li L
inductance
220
0
d. The inductance depends
on the geometry of the particular inductor.
For the solenoid, .B n AiL n A
i i
Inductance of the Solenoid
20 L n A
(30–16)
loop 1loop 2In the picture to the right we
already have seen how a change
in the current of loop 1 results
in a change in the flux through
loop 2, and thus creates an
induced emf in loop 2.
30 - 8 Self - Induction
If we change the current through an inductor this causes
a change in the magnetic flux through the inductor
according to the equation . Using Faraday's
law we can determine the resu
B
B
Li
d diL
dt dt
the henry (symbol: H)
An inductor has inductance 1 H if a current
change of 1 A/s results in a self
lting emf known as
-induced emf o
f
emf:
1 V
.
.
Bd diL
dt dt
L
SI unit for :
self - induc
L
ed E
di
Ldt
E
(30–17)
Consider the circuit in the upper figure with the switch
S in the middle position. At 0 the switch is thrown
in position and the equivalent circuit is shown in
the lower figure.
t
a
30 - 9 RL Circuits
It contains a battery with emf ,
connected in series to a resistor and an inductor
(thus the name " circuit"). Our objective is to
calculate the current as a function of time . We
write Kir
R L
RL
i t
E
chhoff's loop rule starting at point and
moving around the loop in the clockwise direction:
0 di
L iR
x
diiR L
t dd t E E
/
The initial condition for this problem is (0) 0. The solution of the differential
equation that satisfies the initial condition is
The constant is known as the "( ) 1 .t Li t e
R R
i
time constantE
" of
the circuit. RL
/( ) 1 ti t eR
E L
R
(30–18)
/
/
/
( ) 1 Here .
The voltage across the resistor 1 .
The voltage across the inductor .
The solution gives 0 at 0 as required by the
initial conditi
t
tR
tL
Li t e
R R
V iR e
diV L e
dti t
E
E
E
on. The solution gives ( ) / .
The circuit time constant / tells us how fast
the current approaches its terminal value:
( ) 0.632 /
( 3 ) 0.950 /
( 5 ) 0.993 /
If we wait only
i R
L R
i t R
i t R
i t R
a few time
E
E
E
E
the current, for all
practical purposes, has reached its terminal value / . R
constants
E .
(30–19)
We have seen that energy can be stored in the electric field
of a capacitor. In a similar fashion, energy can be stored in
the magnetic field of an inductor. Co
30 - 10 Energy Stored in a Magnetic Field
nsider the circuit
shown in the figure. Kirchhoff's loop rule gives
2
2
. If we multiply both sides of the equation we get: .
The term describes the rate at which the battery delivers energy to the circuit.
The term is the rate at which t
di diL iR i Li i Rdt dt
i
i R
E E
E
hermal energy is produced on the resistor.
Using energy conservation we conclude that the term is the rate at which
energy is stored in the inductor: . We integrate
both
BB
diLidt
dU diLi dU Lidi
dt dt
2 2
sides of this equation: .2 2
ii
oB
o
L i LiU Li di
2
2B
LiU
(30–20)
B Consider the solenoid of length and loop area
that has windings per unit length. The solenoid
carries a current that generates a uniform magnetic
field
A
n
i
30 - 11 Energy Density of a Magnetic Field
0 inside the solenoid. The magnetic field
outside the solenoid is approximately zero.
B ni
2 22 01
The energy stored by the inductor is equal to .2 2
This energy is stored in the empty space where the magnetic field is present.
We define as energy density where is the volume
B
BB
n A iU Li
Uu V
V
2 2 2 2 2 2 2 20 0 0
0 0
inside
the solenoid. The density .2 2 2 2
This result, even though it was derived for the special case of a uniform
magnetic field, holds true in general.
B
n A i n i n i Bu
A
2
0
2B
Bu
(30–21)
N2N1 Consider two inductors
that are placed close enough
so that the magnetic field of one
can influence the other.
30 - 12 Mutual Induction
1 1
2 21 1
1
In fig. we have a current in inductor 1. That creates a magnetic field
in the vicinity of inductor 2. As a result, we have a magnetic flux
through inductor 2. If current varies w
a i B
M i
i
2 12 21 21
ith time, then we have a time-varying
flux through inductor 2 and therefore an induced emf across it:
. is a constant that depends on the geometry
of the two inductors as we
d diM M
dt dt
E
ll as on their relative position. (30–22)
12 21
diM
dtE
N2N1
2 2
1 12 2
2
In fig. we have a current in inductor 2. That creates a magnetic field
in the vicinity of inductor 1. As a result, we have a magnetic flux
through inductor 1. If current varies w
b i B
M i
i
1 21 12 12
ith time, then we have a time-varying
flux through inductor 1 and therefore an induced emf across it:
. is a constant that depends on the geometry
of the two inductors as w
d diM M
dt dt
E
ell as on their relative position. 21 12
diM
dtE
(30–23)
N2N1
12 21
diM
dtE
21 12
diM
dtE
12 21 12 21It can be shown that the constants and are equal: .
The constant is known as the " " between the two coils.
Mutual inductance is a constant that depends on the geomet
M M M M M
M
mutual inductance
th
ry
e h
of the
enry (
two
inductors as well as on their relative position.
The expressions for the induced emfs across the two inductors be
)
ome:
H
c
The SI unit for :M
21
diMdt
E 12
diMdt
E
(30–24)
1 2Mi 2 1Mi
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