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PHYS 219
Spring semester 2015
Lecture 14: Faraday’s Law (1831),
Lenz’s Law (1835)
and
Magnetic Induction
Ron Reifenberger
Birck Nanotechnology Center
Purdue University
Lecture 14
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Overview
In 1831, Faraday discovered that a changing
magnetic field creates an electric field
 This effect is called magnetic induction
 Faraday’s discovery couples electricity and
magnetism in a fundamental way
Magnetic induction is the key to MANY
technologically relevant inventions.
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Faradays’ Law –
Electromagnetic Induction
Magnetic
flux
move magnet thru loop
loop with
area A
 
 B  magnetic flux  B  A
a b
Eab  
 B
t
Origin of minus
sign is Lenz’s Law
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Implications
Faraday showed that an electric current is produced in a wire loop only
when the magnetic field thru the loop is changing  A changing magnetic
field produces an electric field.
1.
An electric field produced in this way is called an
induced electric field
2. The phenomena is called electromagnetic induction
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Defining Change in Magnetic Flux with Time
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Wire
loop
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Time =t
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Time =t+Δt
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Time =t
Time =t+Δt
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Faraday’s Law relies on the concept of magnetic flux
Magnetic Flux is the product of an area (m2)
with a component of a magnetic field (T)
passing perpendicularly through it.
Convenient to
visualize an
“area” vector
A
perpendicular
to plane of
loop
Magnetic flux
is similar to
the concept of
electric flux
B
ΦB = BA cos(Θ)
Θ measures the angle between B and A
Units: ΦB = T•m2 = 1 Weber = 1 Wb
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Magnetic Flux is easy to calculate for
uniform magnetic fields
=3.0
m2
What is the magnetic flux
through the surface?
=2.0 T
=60o
ΦB = BA cos(Θ)
=2.0 T × 3.0 m2 × cos(60o)
=6.0 × 0.5 Tm2= 3 Wb
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Why cos ?
B is a vector!
A is  to green plane
B
B//

B =Bcos 
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Faraday’s Law: An induced emf is produced in a coil whenever
the magnetic flux changes with time. Here we focus on an
increasing flux through a loop.
2. E is the induced
ΔΦB
emf that develops
E =Δt
when the magnetic
flux thru the loop
1. B is an external
increases with time.
applied magnetic
This E drives the
field that changes
current I
with time. You
control B!
3. I is the
induced
current that
flows due to E
Subtle Point: The negative
sign indicates that the
induced emf is opposite to
change in magnetic flux
that causes it.
Key Idea: An emf is produced by the changing magnetic flux.
The emf in turn produces an induced current I.
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IMPORTANT
E
 B
t
Minus sign is due
to Lenz’s Law
Four ways the flux can change:
magnitude of
B changes
with time
loop rotates so
the angle
changes with
time
w
ℓ
v
B constant
but area A
changes with
time:
A=ℓ w
where ℓ = v ∆t
loop moves
from one
region to
another and
magnitude
of B field is
different
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Example: What is the induced emf if the magnetic field through a
six turn coil increases at a rate of 0.17 T/s?
E =-
The negative sign
indicates that the
induced emf acts to
“oppose” the change in
magnetic flux that
causes it
E=-
 
ΔΦB Δ(B  A)
ΔB
=
=A
Δt
Δt
Δt
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A = πR = 0.88 m
E =
ΔB
= +0.17 T / s
Δt
E =(0.88 m2 )(0.17 T / s) = 0.15 V
ΔΦB
Δt
It is often easier to take the absolute
value of Farady’s Law to find the
magnitude of the induced emf and then
use Lenz’s Law to find the direction of
the induced current that results.
Since coil has six turns, E = 6 × (0.15 V) = 0.90 V
a
b
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Review: You need to know the direction of the
magnetic field produced by current in a wire
Long straight wire
Wire loop
I
B
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Lenz’s Law
There are many ways to state Lenz’s Law. Here
is one that makes sense to me:
An induced electric current is produced by a
changing magnetic field. The induced current
will flow in a direction such that it will create
its own induced magnetic field that opposes
the changing magnetic field that created it.
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Lenz’s Law – Example 1
I
A. Assume a metal loop in which the applied magnetic field (solid
arrows) passes upward through it
B. Assume the magnetic flux increases with time
C. The induced magnetic field produced by the induced current
must oppose the change in flux
D. Therefore, the induced magnetic field (dotted arrows) must be
downward
E. The induced current must therefore be clockwise (CW) when
viewed from the top of coil
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Lenz’s Law – Example 2 – the dropped magnet
The situation
when the North
pole just enters
the loop
Be able to
distinguish
the applied
field from
the induced
field
loop
An induced current must flow in the loop to produce a magnetic
field (inside the loop) that opposes the change in flux.
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Lenz’s Law – Example 3 - Motion to and from a loop
(Focus on the Change in Flux w.Time)
a)
Iind
b)
N
Steady motion
away from loop
Iind
N
Steady motion
toward loop
loop
loop


increasing time
increasing time
The induced current always produces a magnetic
field that opposes (counteracts) the change in flux
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Lenz’s Law – Example 4
Jumping Rings
Induced
magnetic
field
“PCA – ACR”
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Up Next – Induced emf
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Appendix
Right Hand Rules:
 Force on moving charge q in B-field
 Current Loop – direction of B-field at center of loop
 Current in long straight wire - direction of B-field that curls
around the wire
 Induced current (Lenz’s Law) – direction of current induced
by changing magnetic flux
Make sure you can distinguish between the
different situations
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Why is Lenz’s Law Confusing?
There are four cases to consider for one loop!
Bapplied
conducting
loop
Bapplied
conducting
loop
If Bapplied
INCREASES
with time
If Bapplied
DECREASES
with time
If Bapplied
INCREASES
with time
Iinduced
Iinduced
Iinduced
Iinduced
CCW
CW
CW
CCW
Don’t memorize it – think it through
If Bapplied
DECREASES
with time
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If you like computer simulations, check out
Important:
Changing
magnetic
field could
also be
produced by a
2nd coil
https://phet.colorado.edu/sims/faradays-law/faradays-law_en.html
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