FARADAY’S AND LENZ’
LAW
BOOK PG. 436 - 438
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MOTIONAL EMF AND MAGNETIC FLUX
(DERIVIATION)
• Motional emf = vBL
• Let a conducting rod being moved through a
magnetic field B
• During time 𝑡0 the rod has
t =0s
been moved to the right a
distance 𝑥0
• At a later time t, the rod
has moved an even
greater distance x
The speed of the rod is given by
∆𝑠
𝑣=
∆𝑡
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𝑡0 = 0.02𝑠
𝐴0
A
𝑥0
x
∆𝑠
𝑣=
∆𝑡
• 𝑣=
𝑥−𝑥0
𝑡−𝑡0
• Emf
𝑥−𝑥0
=
𝑡−𝑡0
substitute into emf = vBL
× 𝐵𝐿 =
𝑥𝐿−𝑥0 𝐿
𝑡−𝑡0
×𝐵
• From the sketch we can see that 𝑥0 𝐿 = 𝐴0
and 𝑥𝐿 = 𝐴
• Emf =
𝐴−𝐴0
𝑡−𝑡0
• Emf =
Φ−Φ0
𝑡−𝑡0
×𝐵 =
=
∆Φ
∆𝑡
𝐵𝐴−𝐵𝐴0
𝑡−𝑡0
This is almost
always written
𝑁∆Φ
as emf = - ∆𝑡
BA=Φ
• The induced emf equals the time rate of change of
the magnetic flux
• If there are N number of coils, emf =
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𝑁∆Φ
∆𝑡
EMF = -
𝑁∆Φ
∆𝑡
• The direction of the induced current is such that the
magnitude of force F acts on the rod to oppose its
motion, slowing the rod down
• The minus sign is a reminder that the induced
current creates a magnetic field that opposes the
force
• This is Faradays Law of electromagnetic induction
• ∆Φ = change in magnetic flux through 1 loop
• ∆𝑡 = time interval during which change occurs
•
∆Φ
∆𝑡
= average time rate of change of flux that
passes though one loop
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EXAMPLE
• A coil of wire consists of 20 turns or loops, each of which
has an area of 1.5× 10−3 𝑚2 . A magnetic field is
perpendicular to the surface of each loop at all times, so
that 𝜃 = 𝜃0 = 00 . At time 𝑡0 = 0𝑠, the magnitude of the
field at a location of the coil is 𝐵0 = 0.050𝑇. At a later
time t = 0.10s, the magnitude of the field at the coil has
increased to B = 0.060T.
• Find the average emf induced in the coil during this
time.
• 𝑒𝑚𝑓 =
Φ−Φ0
−𝑁
𝑡−𝑡0
= −𝑁
𝐵𝐴𝑐𝑜𝑠 𝜃−𝐵𝐴𝑜 cos 𝜃
𝑡−𝑡0
• 𝑒𝑚𝑓 = −20 × 1.5 × 10−3 cos 0
= −𝑁𝐴 cos 𝜃
0.060−0.050
0.10−0
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𝐵−𝐵0
𝑡−𝑡0
= −3.0 × 10−3 𝑉
DIRECTION OF THE INDUCED CURRENT
Lenz law:
• The direction of the induced current creates an
induced magnetic field that opposes the motion of
the inducing field
• Move N – pole towards coil
• Oppose – wants to repel
– creates N – pole
• Find direction of current using RHR2
• Move N – pole away from coil
• Oppose – wants to attract
– creates a S – pole
• Find direction of current using RHR2
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Lenz’s Law
• Changing the area of a coil in a constant magnetic
field induces a current
Magnetic flux will change if the area of the loop
changes:
• If the area is decreased, the current tries to maintain
the original flux by producing its own magnetic field
into the page, so increasing flux
LENZ LAW CONTINUED
• An induced emf drives a current around a circuit just as
a battery
• Conventional current from + to – terminal
• The net magnetic field penetrating a coil of wire results
from 2 combinations:
• Original magnetic field that • Induced current creates
its own magnetic field,
produces changing flux,
which is called induced
leads to an induced emf
magnetic field
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LENZ LAW
• The induced emf resulting from a changing
magnetic flux has a polarity that leads to an
induced current whose direction is such that the
induced magnetic field opposes the original flux
change
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REASONING STRATEGY
1. Determine whether the magnetic flux that is penetrating a
coil is increasing or decreasing
2. Find what direction of induced magnetic field must be so
that it can oppose the change in magnetic flux by adding
or subtracting from the original field
3. Finding direction of induced magnetic field then use RHR2 to
find direction of induced current
A current will be induced
In such a way that its
Magnetic flux will oppose
The incoming magnet
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SOLENOIDS AND STRAIGHT
CONDUCTORS
• The induction of polarity is in accordance with the
Law of energy conservation
If this solenoid would have an induced S – pole:
- Magnet would accelerate towards it because of attraction
- More induced current produced creating more acceleration
KE would increase indefinitely
- Energy would be created
- Impossible
- Induced current must
oppose the change
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producing it
PROBLEM SOLVING STEPS
Is the magnetic flux inside a loop increasing,
decreasing or unchanged?
v
S
N
• Flux is increasing
• Loop wants to oppose change
• Induces magnetic field that is opposing
increase
• By RHR2: current is up or ccw
Flux from magnet
Flux from induced current
Current ccw
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CONTINUED
v
• Flux inside loop is decreasing
• Loop wants to oppose change
• Creates magnetic field is same direction
as bar magnetic field, so increasing flux
• By RHR current is down or cw
Current is cw
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SUMMARY
• The magnetic field of induced current in a loop:
• If flux is increasing:
induced magnetic field points opposite direction as
external field to decrease total flux
• If flux is decreasing
induced magnetic field points in same direction as
external field to increase total flux
• Use the right hand rule to find the current that
produces the required flux density
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FALLING THROUGH A MAGNETIC FIELD
• The induced field is not always opposite the
external field because Lenz’s law requires only that
it must oppose the change in the flux that
generates the emf
• A copper ring passes through a rectangular region
with a constant magnetic field into the page.
Describe the flux changes at each stage
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COPPER RING
A. no current because no change in flux
B. Flux increases
- induced field opposes change
- wants to decrease field
- field must point out of page
- current must be ccw
C. No change in flux – no current
D. Flux decreases
- induced field wants to increase flux
- field must point in same direction as
external field or into the page
- current must be cw
E. No change in flux – no current
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