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Physics 212
Lecture 17
Faraday’s Law
 
d B
emf   E  d   
dt
Physics 212 Lecture 17, Slide 1
Music
Will I get points for this question?
A)
B)
C)
D)
E)
Yes and no
Maybe
Who is “I”? You? You won’t get points.
I get The point
It’s a pointless question
Faraday’s Law: emf  
dB
f d  
dt
and f is force / unit charge:
where
 
 B   B  dA
f  EvB
A changing magnetic flux can drive current around a
loop  electromagnetic battery!
When B-dot-A changes, what “f” causes the EMF?
•
“A” or “dot” changes (moving or rotating loops)
 f = v x B = familiar magnetic force
• B changes
 f = E = induced electric field
NEW TYPE OF ELECTRIC FIELD!
Caused by dB/dt not charges; its field lines make loops!
Physics 212 Lecture 17, Slide 6
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the
loop.
B
A
Think of B as the number of field lines passing through the surface
There are many ways to change this…
Physics 212 Lecture 17, Slide 7
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the
loop.
B
Move loop to a place where
the B field is different
A
EMF caused by
magnetic force: f = v x B
Physics 212 Lecture 17, Slide 8
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the
loop.
B
Rotate the loop
A
EMF caused by
magnetic force: f = v x B
Physics 212 Lecture 17, Slide 9
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the
loop.
B
Rotate the loop
A
EMF caused by
magnetic force: f = v x B
Physics 212 Lecture 17, Slide 10
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the
loop.
B
Rotate the loop
A
EMF caused by
magnetic force: f = v x B
Physics 212 Lecture 17, Slide 11
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the
loop.
Nothing’s
moving ...
B
So no
magnetic
force …
Change the B field
A
dB/dt creates an electric field !
often called an “induced E-field”
EMF caused by
electric force: f = E
Physics 212 Lecture 17, Slide 12
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the
loop.
2) The emf will make a current flow if it can (like a battery).
I
Physics 212 Lecture 17, Slide 13
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the loop.
2) The emf will make a current flow if it can (like a battery).
3) The current that flows generates a new magnetic field.
I
Physics 212 Lecture 17, Slide 14
Checkpoint 1
Suppose a current flows in a horizontal conducting loop in such a way that the magnetic flux
produced by this current points upward.
As viewed from above, in which direction is this current flowing?
Physics 212 Lecture 17, Slide 15
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the loop.
2) The emf will make a current flow if it can (like a battery).
3) The current that flows induces a new magnetic field.
4) The new magnetic field opposes the change in the original
magnetic field that created it  Lenz’s Law
B
dB/dt
Physics 212 Lecture 17, Slide 16
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
In Practical Words:
1) When the flux B through a loop changes, an emf is induced in the loop.
2) The emf will make a current flow if it can (like a battery).
3) The current that flows induces a new magnetic field.
4) The new magnetic field opposes the change in the original
magnetic field that created it  Lenz’s Law
B
Demo
dB/dt
Physics 212 Lecture 17, Slide 17
Checkpoint 2
A magnet makes the vertical magnetic field shown by the red arrows. A horizontal conducting
loop is entering the field as shown.
At the instant shown, what is the direction of the additional flux produced by the current
induced in the loop?
Physics 212 Lecture 17, Slide 18
Checkpoint 3
A magnet makes the vertical magnetic field shown by the red arrows. A horizontal conducting
loop passes through the field from left to right as shown.
The upward flux through the loop as a function of time is shown by the blue trace.
Which of the red traces best represents the current induced in the loop as a function of time
as it passes over the magnet? (Positive means counter-clockwise as viewed from above.)
Physics 212 Lecture 17, Slide 19
Faraday’s Law: emf  
dB
f d  
dt
where
 
 B   B  dA
Executive Summary:
emf→current→field a) induced only when flux is changing
b) opposes the change
Physics 212 Lecture 17, Slide 20
Old Checkpoint 2
A horizontal copper ring is dropped from rest directly above the north pole of a permanent magnet
(copper is not
ferromagnetic)
Will the acceleration a of the falling ring in the presence of the magnet
be any different than it would have been under the influence of just
gravity (i.e. g)?
A. a > g
B. a = g
C. a < g
“Please do not display
this in lecture but
that picture on this
checkpoint with the
falling conducting loop
looked a LOT
McDonalds like french
fries.”
Physics 212 Lecture 17, Slide 21
Old Checkpoint 2
A horizontal copper ring is dropped from rest directly above the north pole of a permanent magnet
F
O
X
B
B
Like poles repel
(copper is not
ferromagnetic)
Ftotal < mg
Will the acceleration a of the falling ring in the presence of the magnet
be any different than it would have been under the influence of just
gravity (i.e. g)?
A. a > g
B. a = g
C. a < g
a<g
This one is hard !
B field increases upward as loop falls
Clockwise current (viewed from top) is induced
Physics 212 Lecture 17, Slide 22
Old Checkpoint 2
A horizontal copper ring is dropped from rest directly above the north pole of a permanent magnet
HOW
IT
WORKS
Looking down
B
(copper is not
ferromagnetic)
Will the acceleration a of the falling ring in the presence of the magnet
be any different than it would have been under the influence of just
gravity (i.e. g)?
A. a > g
B. a = g
C. a < g
This one is hard !
B field increases upward as loop falls
Clockwise current (viewed from top) is induced
Main Field produces horizontal forces
“Fringe” Field produces vertical force
I
I
B
IL X B points UP
Ftotal < mg
a<g
Demo !
dropping magnets
e-m cannon
Physics 212 Lecture 17, Slide 23
Calculation
A rectangular loop (height = a, length = b,
resistance = R, mass = m) coasts with a constant
velocity v0 in + x direction as shown. At t =0, the
loop enters a region of constant magnetic field B
directed in the –z direction.
y
a
v0
B
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x
What is the direction and the magnitude of the
force on the loop when half of it is in the field?
• Conceptual Analysis
–
–
Once loop enters B field region, flux will be changing in time
Faraday’s Law then says emf will be induced
• Strategic Analysis
–
–
–
Find the emf
Find the current in the loop
Find the force on the current
Physics 212 Lecture 17, Slide 24
Calculation
A rectangular loop (height = a, length = b,
resistance = R, mass = m) coasts with a constant
velocity v0 in + x direction as shown. At t =0, the
loop enters a region of constant magnetic field B
directed in the –z direction.
emf  
d B
dt
y
a
v0
B
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x
What is the magnitude of the emf induced
in the loop just after it enters the field?
(A) e = Babv02
(B) e = ½ Bav0 (C) e = ½ Bbv0 (D) e = Bav0 (E) e = Bbv0
y
a
v0
B
x
x
x
x x x x
b
x x x x x x x
a
x x x x x x x
x x x x x x x
Change in Flux = dB = BdA = Bav0dt
x
In a time dt
it moves by v0dt
The area in field
changes by dA = v0dt a
d B
 Bav o
dt
Physics 212 Lecture 17, Slide 25
Calculation
emf  
A rectangular loop (height = a, length = b,
resistance = R, mass = m) coasts with a constant
velocity v0 in + x direction as shown. At t =0, the
loop enters a region of constant magnetic field B
directed in the –z direction.
y
a
v0
d B
dt
B
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x
What is the direction of the current induced
in the loop just after it enters the field?
(A) clockwise
(B) counterclockwise
(C) no current is induced
emf is induced in direction to oppose the change in flux that produced it
y
a
v0
B
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
Flux is increasing into the screen
Induced emf produces flux out of screen
x
Physics 212 Lecture 17, Slide 26
Calculation
A rectangular loop (height = a, length = b,
resistance = R, mass = m) coasts with a constant
velocity v0 in + x direction as shown. At t =0, the
loop enters a region of constant magnetic field B
directed in the –z direction.
emf  
y
a
v0
d B
dt
B
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x
What is the direction of the net force
on the loop just after it enters the field?
(A) +y
(B) -y
(C) +x
(D) -x

 
Force on a current in a magnetic field: F  IL  B
y
b
a
B
x x x x x x x
v0
I
x x x x x x x
• Force on top and bottom segments cancel (red arrows)
• Force on right segment is directed in –x direction.
x
Physics 212 Lecture 17, Slide 27
Calculation
A rectangular loop (height = a, length = b,
resistance = R, mass = m) coasts with a constant
velocity v0 in + x direction as shown. At t =0, the
loop enters a region of constant magnetic field B
directed in the –z direction.
What is the magnitude of the net force on
the loop just after it enters the field?
(A) F  4aBvo R (B) F  a 2 Bvo R
(C) F  a 2 B 2vo2 / R

 
F  IL  B
emf  
y
a
v0
d B
dt
B
b x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x
F  IL  B
e = Bav0
(D) F  a 2 B 2vo / R
 
F  ILB since L  B
y
b
a
F
B
x x x x x x x
v0
I
e
Bavo
I 
R
R
x x x x x x x
B 2 a 2vo
 Bav o 
F 
aB 
R
R


ILB
x
Physics 212 Lecture 17, Slide 28
Follow-Up
A rectangular loop (sides = a,b, resistance = R,
mass = m) coasts with a constant velocity v0 in + x
direction as shown. At t =0, the loop enters a
region of constant magnetic field B directed in a
the –z direction.
t = dt: e = Bav0
y
b
v0
B
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
x
What is the velocity of the loop when half of it is
in the field?
Which of these plots best represents the velocity as a function of
time as the loop moves form entering the field to halfway through ?
(A)
(B)
(C)
D)
(E)
X
This is not obvious,
but we know v must
decrease
Why?
X
b
a
Fright
B
x x x x x x x
v0
I
x x x x x x x
X
Fright points to left
Acceleration negative
Speed must decrease
Physics 212 Lecture 17, Slide 29
Follow-Up
A rectangular loop (sides = a,b, resistance = R,
mass = m) coasts with a constant velocity v0 in + x
direction as shown. At t =0, the loop enters a
region of constant magnetic field B directed in
the –z direction.
y
b
a
What is the velocity of the loop when half of it is
in the field?
v0
B
x x x x x x x
x x x x x x x
x x x x x x x
x x x x x x x
e = Bav0
x
Which of these plots best represents the velocity as a function of
time as the loop moves form entering the field to halfway through ?
dv
(A)
F  a B v / R (D)
m
• Why (D), not (A)?
2
2
dt
–
F is not constant, depends on v
a 2 B 2v
dv
F 
m
R
dt
Challenge: Look at energy
v  vo e t
2 2
a
where   B
mR
Claim: The decrease in kinetic energy of
loop is equal to the energy dissipated as
heat in the resistor. Can you verify??
Physics 212 Lecture 17, Slide 30