Physics 2113
Jonathan Dowling
Lecture 32: WED 01 APR
Ch30.1–4
Induction and Inductance I
Fender Stratocaster
Solenoid Pickup
Magnetic Circuit Breaker
As the normal operating or "rated" current flows through the sensing coil, a
magnetic field is created around that coil.
When the current flow increases, the strength of the magnetic field increases,
drawing the spring-biased, movable magnetic core toward the pole piece.
As the core moves inward, the efficiency of the magnetic circuit is increased,
creating an even greater electromagnetic force.
When the core is fully "in", maximum electromagnetic force is attained.
The armature is attracted to the pole piece, unlatching a trip mechanism thereby
opening the contacts.
Faraday's Experiments
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.
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.
1. A current appears only if there is relative motion between the magnet and the loop.
2. Faster motion results in a larger current.
3. If 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 "induced current"; the emf that appears
is known as "induced emf"; the whole effect is called "induction."
Changing B-Field Induces a Current in a Wire Loop
Note Current Changes Sign With Direction
No Current When Magnet Stops
loop 1
loop 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.
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.
Loop Two is Connected
To A Light Bulb.
Loop One
Has a 60 Hz
Alternating Current
The Current in Loop
One Produces a
Rapidly Changing
Magnetic Field in
Loop Two That
Induces a Current in
Loop Two — Lighting
the Bulb!
Induction Charges My Electric Toothbrush!
•
Faraday’s Law: What? The Flux!
!
A time varying magnetic FLUX
dA B
creates an induced EMF
•  Definition of magnetic flux is
similar to definition of electric
flux. Units of Magnetic Flux: =
Telsa • m2 = Weber!
! !
Φ B = ∫ B ⋅ dA
S
E EMF
dΦB
=−
dt
dA
•  Take note of the MINUS sign!!
•  The induced EMF acts in
such a way that it OPPOSES
the change in magnetic flux
(“Lenz’s Law”).
dΦ B
dB
=A
dt
dt
EMF is proportional to the slope dB/dt
EEMF =
Eb > Ed = Ee > Ea = Ec = 0
Lenz’s Law
•  The Loop Current Produces a B Field
that Opposes the CHANGE in the bar
magnet field.
•  Upper Drawing: B Field from Magnet
is INCREASING so Loop Current is
Clockwise and Produces an Opposing
B Field that Tries to CANCEL the
INCREASING Magnet Field
•  Lower Drawing: B Field from Magnet
is DECREASING so Loop Current is
Counterclockwise and Tries to
BOOST the Decreasing Magnet Field.
Ia = Ib > Ic = 0
30.4.1. Consider the situation shown. A triangular, aluminum loop
is slowly moving to the right. Eventually, it will enter and pass
through the uniform magnetic field region represented by the
tails of arrows directed away from you. Initially, there is no
current in the loop. When the loop is entering the magnetic
field, what will be the direction of any induced current present
in the loop?
a) clockwise
b) counterclockwise
c) No current is induced.
30.4.1. Consider the situation shown. A triangular, aluminum loop
is slowly moving to the right. Eventually, it will enter and pass
through the uniform magnetic field region represented by the
tails of arrows directed away from you. Initially, there is no
current in the loop. When the loop is entering the magnetic
field, what will be the direction of any induced current present
in the loop?
a) clockwise
b) counterclockwise
c) No current is induced.
30.4.2. Consider the situation shown. A triangular, aluminum loop
is slowly moving to the right. Eventually, it will enter and pass
through the uniform magnetic field region represented by the
tails of arrows directed away from you. Initially, there is no
current in the loop. When the loop is exiting the magnetic
field, what will be the direction of any induced current present
in the loop?
a) clockwise
b) counterclockwise
c) No current is induced.
30.4.2. Consider the situation shown. A triangular, aluminum loop
is slowly moving to the right. Eventually, it will enter and pass
through the uniform magnetic field region represented by the
tails of arrows directed away from you. Initially, there is no
current in the loop. When the loop is exiting the magnetic
field, what will be the direction of any induced current present
in the loop?
a) clockwise
b) counterclockwise
c) No current is induced.
30.4.3. A rigid, circular metal loop begins at rest in a uniform
magnetic field directed away from you as shown. The loop is
then pulled through the field toward the right, but does not
exit the field. What is the direction of any induced current
within the loop?
a) clockwise
b) counterclockwise
c) No current is induced.
30.4.3. A rigid, circular metal loop begins at rest in a uniform
magnetic field directed away from you as shown. The loop is
then pulled through the field toward the right, but does not
exit the field. What is the direction of any induced current
within the loop?
a) clockwise
b) counterclockwise
c) No current is induced.
Example
•  A closed loop of wire encloses
an area of A = 1 m2 in which in a
uniform magnetic field exists at
300 to the PLANE of the loop.
The magnetic field is
DECREASING at a rate of
!
dA
B
300
60°
! !
Φ B = ∫ B ⋅ dA
= BAcos(60 0 ) = BA /2
dB/dt = 1T/s. The resistance of the
dΦB A dB
wire is 10 Ω.
E =
=
dt
2 dt
•  What is the induced current?
Is it
…clockwise or
…counterclockwise?
E
A dB
i=
=
R 2R dt
(1m2 )
i=
(1T/s) = 0.05A
2(10Ω)
Example
•  3 loops are shown.
B
II
•  B = 0 everywhere except in
III
I
the circular region I where B
is uniform, pointing out of
the page and is increasing
at a steady rate.
• III encloses no flux so EMF=0
•  Rank the 3 loops in order of • I and II enclose same flux so
EMF same.
increasing induced EMF.
•  Are Currents in Loops I & II
–  (a) III < II < I ?
Clockwise or Counterclockwise?
–  (b) III < II = I ?
–  (c) III = II = I ?
30.4.4. A coil of wire that forms a complete loop is moving with a constant speed
v toward a very long, current carrying wire, only a portion of which is
shown. What affect, if any, does the current carrying wire have on the coil
of wire?
a) Since the magnetic field increases as the coil
approaches the wire, a current is induced in the coil.
B ∝1 / r
r!
b) The rectangle will be distorted as it is pulled in the
direction of the current in the wire.
c) Close to the wire, a magnetic force acts on the loop that accelerates the loop
away from the wire.
d) Since the magnetic field around the wire is not changing, there is no effect on
the coil.
e) Since the coil and the wire are not touching, there is no effect.
30.4.4. A coil of wire that forms a complete loop is moving with a constant speed
v toward a very long, current carrying wire, only a portion of which is
shown. What affect, if any, does the current carrying wire have on the coil
of wire?
a) Since the magnetic field increases as the coil
approaches the wire, a current is induced in the coil.
b) The rectangle will be distorted as it is pulled in the
direction of the current in the wire.
c) Close to the wire, a magnetic force acts on the loop that accelerates the loop
away from the wire.
d) Since the magnetic field around the wire is not changing, there is no effect on
the coil.
e) Since the coil and the wire are not touching, there is no effect.
Example
B=
µ 0i
2πr
i
•  An infinitely long wire carries a constant
R
current i as shown
r=R+x
•  A square loop of side L is moving
x
L
towards the wire with a constant velocity dR/dt=v
L
v.
•  What is the EMF induced in the loop
Choose a “strip” of width dx
when it is a distance R from the loop?
located as shown.
Flux thru this “strip”
ΦB =
L
∫
0
µ0iLdx
2π (R + x)
µ0iLdx
dΦ = BLdx =
2π (R + x)
L
⎡ µ 0iL
⎤
=⎢
ln( R + x)⎥
⎣ 2π
⎦0
µ 0iL ⎡ R + L ⎤
=
ln ⎢
2π
⎣ R ⎥⎦
dΦB
E =−
dt
µ0 Li d ⎧ ⎡
=−
⎨ln⎢1+
2π dt ⎩ ⎣
L ⎤⎫
⎬
⎥
R ⎦⎭
Example
dΦB
E =−
dt
µ 0 Li d ⎧ ⎡ L ⎤ ⎫
=−
⎨ln ⎢1 + ⎥ ⎬
2π dt ⎩ ⎣ R ⎦ ⎭
µ 0 Li dR ⎡ R ⎤ L
=
2π dt ⎢⎣ R + L ⎥⎦ R 2
⎤
µ 0i ⎡ L2
=
v⎢
⎥
2π ⎣ ( R + L) R ⎦
i
R
dR/dt=v
L
x
What is the DIRECTION of the
induced current?
•  Magnetic field due to wire points
INTO page and gets stronger as
you get closer to wire
•  So, flux into page is
INCREASING
•  Hence, current induced must be
counter clockwise to oppose this
increase in flux = CCW
B
Example : The Generator
•  A square loop of wire of side L
is rotated at a uniform
frequency f in the presence of
a uniform magnetic field B as
shown.
•  Describe the EMF induced in
the loop.
! !
Φ B = ∫ B ⋅ dA
S
θ = ωt
f = 2πω
L
B
B
!
= BL cos(θ )
dΦB
2 dθ
2
E =−
= BL
sin(θ ) = BL (2πf )sin( 2πft )
dt
dt
2
30.3.2. A circular ring is rotated
clockwise at a constant rate
for an extended period of
time using the apparatus
shown. Which of the graphs
below correctly shows the
magnetic flux through the
ring as a function of time?
Note: At time t = 0 s, the
plane of the ring is
perpendicular to the
direction of the magnetic
field.
30.3.2. A circular ring is
rotated clockwise at a
constant rate for an
extended period of time
using the apparatus
shown. Which of the
graphs below correctly
shows the magnetic flux
through the ring as a
function of time? Note:
At time t = 0 s, the plane
of the ring is
perpendicular to the
direction of the magnetic
field.