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Transcript of Hitt Two long straight wires pierce the plane of the paper at vertices of an equilateral triangle as...
hittTwo long straight wires pierce the plane of the paper at vertices of an equilateral triangle as shown. They each carry 3A in the same direction. The magnetic field at the third vertex (P) has the magnitude and direction (North is up):
(1) 20 μT, west (2) 17 μT, east (3) 15 μT, north (4) 26 μT, south (5) none of these
4 cm
P
4 cm
Long, straight, parallel wires carry equal currents into or out of page. Rank according to the magnitude of the force on the central wire, largest to smallest.
1. d, c, a, b 2. a, b, c, d 3. b, c, d, a 4. c, a, b, d5. b, d, c, a
A horizontal power line carries a current of 5000 A from south to north. Earth's magnetic eld (60 μT) is directed towards north and inclined down-ward 50 degrees to the horizontal. Find the magnitude and direction of the magnetic force on 100 m of the line due to the Earth's field.
(1)23 N, west (2) 23 N, east (3) 30N, west
(4) 30N, east (5) none of these
50
B
N
I
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)
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 acurately predicted by what is
known as Lenz's
Lenz's Rule
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 haces the corresponding north pole of the b
B
ar magnet
approaching the loop. The loop the approaching magnet and thus opposes
the change in which generated the induced current.repels
(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 :
B
As the bar magnet approaches the loop the magnet field points towards the left
and its magnitude increases with time at the location of the loop. Thus the magnitude
of the loop magnetic flux also
B
increases. The induced current flows in the
(CCW) direction so that the induced magnetic field opposes
the magnet field . The net field . The induced current is t
i
net i
B
B B B B
counterclockwise
B B
hus trying
to from increasing. Remember it was the increase in that generated
the induced current in the first place.
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 :
B
As the bar magnet moves away from the loop the magnet field points towards the left
and its magnitude decreases with time at the location of the loop. Thus the magnitude
of the loop magnetic flux
B
also decreases. The induced current flows in the
(CW) direction so that the induced magnetic field adds to
the magnet field . The net field . The induced current is thus
i
net i
B
B B B B
clockwise
B B
trying
to from decreasing. Remember it was the decrease in that generated
the induced current in the first place.
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 :
B
As the bar magnet approaches the loop the magnet field points towards the right
and its magnitude increases with time at the location of the loop. Thus the magnitude
of the loop magnetic flux als
B
o increases. The induced current flows in the
(CW) direction so that the induced magnetic field opposes
the magnet field . The net field . The induced current is thus try
i
net i
B
B B B B
clockwise
B B
ing
to from increasing. Remember it was the increase in that generated
the induced current in the first place.
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 magnet field points towards the
right and its magnitude decreases with time at the location of the loop. Thus
the magnitude of the loop magnetic flux
B
B also decreases. The induced current
flows in the (CCW) direction so that the induced magnetic field
adds to the magnet field . The net field . The induced cur
i
net i
B
B B B B
counterclockwise
B B
rent is thus
trying to from decreasing. Remember it was the decrease in that
generated the induced current in the first place.
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-magnetic fie
Induction and energy transfers
do work
ld 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
Consid
R
er 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 constant
sp
L
B
eed . The magnetic flux through the loop
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
2 3
The rate at which thermal energy is dissipated on
( )
The magnetic forces on the wire sides are shown
in the figure. Forces and cancel each other.
Force
th
R
BLv B L vP i R R
R R
F F
F
eqs.1
1 1
2 2
1
2 2 2
1
sin 90
The rate at which the external agent is
producing mechanical work ( )
If we compare equations 1 and 2 we see that indeed the
me
ext
BLviL B F iLB iLB LB
R
B L vF
R
B L vP F v
R
eqs.2
chanical 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 mechanical energy that moves the plate is
transformed into thermal energy that heats up the pla
eddy currents
te.
(30 –13)
Consider the copper ring of radius shown in the
figure. It is paced in a uniform magnetic field
pointing into the page, that icreases as function
of time. The resulting chan
r
B
Induced electric fields
ge in magnetic flux
induces a current i in the counterclock-wise
(CCW) direction.
The presence of the current in the conducting ring implies that an induced
electric field must be present in order to set the electrons on 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 absense 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
in 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 eqs.1 with eqs.2
we get:
B
E ds
d
dtd
E ds
eqs.1
eqs.2
E
E
2 2
2
cos 0 2
22
B
BB
E ds Eds E ds rE
d dBr B r
dt dtdB
rE rd
dt
r dBE
dtt
(30 –15)
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 unifrom
magnetic field ino
N
NA n
i
B ni
Inductance
side the solenoid.
The solenoid magnetic flux B NBA
B
2The 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 solenoid
B o
B
N n n A i
Li L
inductance
22
. The inductance depends
on the geometry of the particular inductor.
For the solenoid oBo
n AiL n A
i i
Inductance of the solenoid
2 oL 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
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 determined the re
B
B
Li
d diL
dt dt
the Henry (symbol: H)
An inductor has inductance
sulting emf known as
em
1 H if a current
change of 1 A/s results in a self-induced emf of
f
1 V
.
.
B
L
d diL
dt dt
SI unit for L :
Eself induced
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 a and the equivelent circuit is shown in
the lower figure. It cont
t
RL circuits
ains 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 function of time . We
write Kirchhoff's
R L
RL
i t
E
loop rule starting at point x and
moving around the loop in the clockwise direction.
0 di
L iRd
dt
iiR L
dt 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 th( 1 e) "t Li t e
R
i
R
E
time cons " of
the RL circuit.
tant
/( ) 1 ti t eR
E L
R
/
/
/
( ) 1 Here
The voltage across the resistor 1 .
The voltage across the inductor
The solution gives 0 at 0 as required by the
initial conditio
t
tR
tL
Li t e
R R
V iR e
diV L e
dti t
E
E
E
n. 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 c
E
E
E
E
the current, for all
practical purposes has reached its terminal value / . R
onstants
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. Consider t
Energy stored in a magnetic field
he circuit
shown in the figure. Kirchhoff's loop rules gives:
2
2
If we multiply both sides of the equation we get:
The term describes the rate at which the batter delivers energy to the circuit
The term is the rate at which
di diL iR i Li i R
dt dti
i R
E E
E
thermal 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
diLi
dtdU di
Li dU Lididt 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 soleniod of length and loop area
that has windings per unit length. The solenoid
carries a current i that generates a uniform magnetic
field o
A
n
B ni
Energy density of a magnetic field
inside the solenoid. The magnetic field
outside the solenoid is approximately zero.
2 221
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 in
oB
BB
n A iU Li
Uu V
V
22 2 2 2 2 2 2
side
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.
o o oB
o o
n A i n i n i Bu
A
2
2B
o
Bu
(30 –21)
N2
N1 Consider two inductors
which are placed close enough
so that the magnetic field of one
can influence the other.
Mutual Induction
1 1
2 21 1
1
In fig.a 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 curent varies wit
i B
M i
i
2 12 21 21
h 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 their relative position. (30 –22)
12 21
diM
dtE
N2
N1
2 2
1 12 2
2
In fig.b 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 curent varies wit
i B
M i
i
1 21 12 12
h 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 we
d diM M
dt dt
E
ll as their relative position.
21 12
diM
dtE
(30 –23)
N2
N1
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 geometr
M M M M M
M
mutual inductance
the He
y of th
nry (H)
e two
inductors as well as their relative position.
The expressions for the induced emfs across the two inductors become:
The SI unit for M :
21
diM
dtE 1
2
diM
dtE
(30 –24)
1 2Mi 2 1Mi
Electrons are going around a circle in a counterclockwise direction as shown. At the center of the circle they produce a magnetic field that is:
A. into the pageB. out of the pageC. to the leftD. to the rightE. zero
e
Long parallel wires carry equal currents into or out of the page. Rank according to the magnitude of the net magnetic field at the center of the square.
1. C,D, (A,B) 2. A, B, (C,D) 3. B, A, C, D4. D, (A, B), C
Partial Loops (cont.)
• Note on problems when you have to evaluate a B field at a point from several partial loops
– Only loop parts contribute, proportional to angle (previous slide)
– Straight sections aimed at point contribute exactly nothing
– Be careful about signs, e.g.in (b) fields partially cancel, whereas in (a) and (c) they add
In a series of experiments Michael Faraday in England
and Joseph Henry in the US were able to generate
electric currents without the use of batteries
Below we describe some of the
Faraday's experiments
se experiments that
helped formulate whats 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 we
1.
2.
3. 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 appears induced current
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 as
" "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
of what induction is. It is one of the four " of electromagnetism"
all of which are statements of experim
Maxwell's equations
ental 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:
BMagnetic Flux Φ
1 we divide the surface that has the loop as its border
into area 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 a
Bd BdA
n B
2.
3.
2: T m known as the Weber (symbol
ll the terms. cos
We can express Faraday's law of induction in the folowin
W
g
b)
form:
B BdA B dA
SI magnetic flux unit
B
The magnitude of the emf induced in a conductive loop is equal to rate
at which the magnetic flux Φ through the loop changes with time
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 angle
BBMethods for changing Φ through a loop
1.
2.
3.
ˆ between and
Problem 30-11
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)