Download March 16 Induction and Inductance Chapter 31

March 16
Induction and
Inductance
Chapter 31
Review/Demo (Fig. 31-1)
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A current can produce a B
field
Can a B field generate a
current?
Move a bar magnet in and out
of loop of wire
Moving magnet towards loop
causes current in loop
o Current disappears when magnet
stops
o Move magnet away from loop
current again appears but in
opposite direction
o Faster motion produces a
greater current
o
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Review: Faraday’s law
r r
Φ B = ∫ B •dA
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Magnetic flux through area A
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dA is vector of magnitude dA that is
⊥ to the differential area, dA
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If B is uniform and ⊥ to A then
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SI unit is the weber, Wb
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Φ B = BA
Wb = T⋅m
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Review: Faraday’s law
• If B is constant within coil
r r
ΦB = ∫ B •dA = BAcosθ
ε
• We can change the magnetic flux
through a loop (or coil) by
o Changing magnitude of B
field within coil
o Changing area of coil, or
portion of area within B field
o Changing angle between B
field and area of coil (e.g.
rotating the coil)
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dΦ B
= −N
dt
ε
dB
= − NA cosθ
dt
ε
dA
= − NB cosθ
dt
ε
d (cosθ )
= − NBA
dt 4
Checkpoint #2
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Three identical circular conductors in uniform B fields
that are either increasing or decreasing in magnitude at
identical rates. Rank according to magnitude of current
induced in loop, greatest first.
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Use Lenz’s law to find direction of Bi
Use right-hand rule to find direction of current
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INC=increase
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Situation (a):
o
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B increases out of page, so Bi is into page
From right-hand rule, induced current is clockwise
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INC=increase
DEC=decrease
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Situation (b) top:
B increases into the page, so Bi is out of the page
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Situation (b) bottom:
B decreases out of the page, so Bi is out of the page
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In both cases from the right-hand rule, induced current is
counter-clockwise
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INC=increase
DEC=decrease
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Situation (c) top:
o
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Situation (c) bottom:
o
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B decreases out of page, so Bi is out of the page
B increases out of the page, so Bi is into the page
Total Bi is zero and total current is zero
Rank magnitude of current induced in loops
a & b tie, then c
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Problem 31-3 (Fig.31-9)
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Loop has width W=3.0m and height H=2.0m
Loop in non-uniform and varying B field ⊥ to loop
and directed into the page
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B = 4t x
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2
Q : What is magnitude and
direction of induced emf
around loop at t=0.10s?
Since magnitude B is changing in time, flux through
the loop is changing so use Faraday’s law to
calculate induced emf
dΦ B
ε
March 16, 2004
=−
PHY 184
dt
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Problem 31-3 (Fig.31-9)
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2
B = 4t x
B is not uniform so need to
calculate magnetic flux using
2
r r
ΦB = ∫ B •dA
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B ⊥ to plane of loop and only
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changes
r rin x direction
B •dA = BdA = BHdx
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At time t:
3
x 
2
ΦB = ∫ BHdx = 4t H ∫ 0 x dx = 4t H  = 72t
 3 0
2
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3
2
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Problem 31-3 (Fig.31-9)
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Now use Faraday’s law to
find the magnitude of the
induced emf
ε
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B = 4t 2 x 2
2
dΦ B d (72t )
=
=
= 144t
dt
dt
At t=0.10s, emf = 14 V
Find direction of emf by Lenz’s law
o
o
B is increasing in time directed into the page, so Bi is in
opposite direction - out of the page
Right-hand rule – current (and emf) are counterclockwise
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Loop + magnet (Fig.31-10)
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If you pull a loop at a
constant velocity, v, through
a B field, you must apply a
constant force, F
As you move loop to right,
less area is in B field so
magnetic flux decreases and
current is induced in loop
Magnetic flux when B is ⊥
and constant to area is
Φ B = BA = BLx
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Loop + magnet (Fig.31-10)
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Using Faraday’s law
ε
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dΦ B d
dx
=
= BLx = BL
dt
dt
dt
Remember v = dx/dt so
ε = BLv
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where L is the length of the loop and v is
⊥ to B field
B is decreasing so Bi is in same direction
(into page), so the current is clockwise
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Loop + magnet (Fig.31-10)
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Since loop carries current
through a B field there is
a force given by
r
r r
FB = iL × B
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Use right-hand rule to find
direction of FB on segments
of loop in B field
Find forces, F2 and F3 , cancel
each other
Force, F1 = iLB , opposes your
force
March 16, 2004
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r
r
Fapp = − F1
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Loop + magnet (Fig.31-10)
ε = BLv
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The circuit diagram is
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With
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Then
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And
March 16, 2004
ε = iR
ε
BLv
i= =
R
R
2 2
B Lv
F1 = iLB =
R
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Loop + magnet (Fig.31-10)
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What happens if we
push the wire in?
i
B is increasing so Bi is
in the opposite direction
(out of page), so the
current is counterclockwise.
v
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Inductance (19)
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Checkpoint #3 – Four wire loops with edge lengths of
either L or 2L. All loops move through uniform B
field at same velocity. Rank the four loops according
to maximum magnitude of induced emf, greatest first.
L
2L
ε = BLv
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c & d tie, then a
& b tie
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Loop + magnet (Fig.31-10)
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Energy is conserved - so
where does the work you do
moving the loop in and out go?
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The current flowing through
the resistance produces heat
at the rate
2 2 2
B Lv
P=i R=
R
2
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since
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BLv
i=
R
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Eddy currents (Fig. 31-12)
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Instead of a loop of wire,
what happens when a bulk
piece of metal moves
through a B field?
Free electrons in metal move
in circles as if caught in a
whirlpool called eddy
currents
A metal plate swinging
through a B field will
generate eddy currents
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Eddy currents (Fig. 31-12)
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Eddy currents will oppose
the change that caused
them – Lenz’s law
Induced eddy currents will
always produce a retarding
force when plate enters or
leaves B field causing the
plate to come to rest
Cutting slots in metal plate
will greatly reduce the eddy
currents
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Eddy currents
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Induction and eddy currents are used for braking
systems on some subways and rapid transit cars
Moving vehicle has electromagnet (e.g. solenoid)
which is positioned near steel rails
Current in electromagnet generates B field
Relative motion of B field to rails induces eddy
currents in rails
Eddy currents produce a drag force on the moving
vehicle
Eddy currents decrease steadily as car slows giving
a smooth stop
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Eddy currents
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Eddy currents often undesirable since
they dissipate energy in form of heat
Moving conducting parts often laminated
o
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Build up several thin layers separated by
nonconducting material
Layered structure confines eddy currents to
individual layers
Used in transformers and motors to
minimize eddy currents and improve
efficiency
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Inductance (units)
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Inductor is a device used to produce
and store a desired B field (e.g.
solenoid)
A current, i, in an inductor with N turns
produces a magnetic flux, ΦB, in its
central region
N
Φ
B
Inductance, L is defined as L =
i
SI unit is henry, H
2
H = T ⋅m / A
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Inductance of a solenoid
NΦ B
L=
i
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What is inductance of a solenoid?
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First find flux of single loop in solenoid
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# of turns (N ) per unit length (l )
r r
ΦB = ∫ B •dA = BA = µ0 niA
2
L = lµ 0 n A
n = N /l
L
2
= µ0n A
or
l
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Thus
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Depends only on the physical properties of
the solenoid
March 16, 2004
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