22.3 Magnetic Flux
Normal
vector
MAGNETIC FLUX
The magnetic flux is proportional to the number
of field lines that passthrough a surface.
Directional boundary path.
Circulation sense related to normal vector by RHR-2
Φ B = BA cos φ
Unit: T·m2 = “weber” (Wb)
FARADAY’S LAW OF ELECTROMAGNETIC INDUCTION
The average emf induced in a coil of N loops is
 Φ − Φo 
∆Φ
 = − N
E = − N 
∆t
 t − to 
Review:
SI Unit of Induced
Emf: volt (V)
1
22.5 Lenz’s Law
LENZ’S LAW (gives you the direction of the induced current)
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.
Change in flux  induced emf  induced current
 induced B field opposes change in flux
Reasoning Strategy
1. Determine whether the magnetic flux that penetrates the coil is
increasing or decreasing.
2. Find what the direction of the induced magnetic field must be so
that it can oppose the change in flux by adding or subtracting from
the original field.
3. Use RHR-2 to determine the direction of the induced current from the
induced field in step 2.
2
22.5 Lenz’s Law
Case 2: The Emf Produced by a Moving Magnet
A permanent magnet is approaching a
loop of wire. The external circuit consists
of a resistance. Find the direction of the
induced current and the polarity of
the induced emf.
Answer:
1. As the magnet moves to the right,
the field lines at the loop get closer
together  magnetic flux to the
right is increasing
2. Induced magnetic field must point
to the left to oppose the increase.
3. The induced current is then given
by RHR-2 as shown in figure (b)
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22.5 Lenz’s Law
Case 2: The Emf Produced by a Moving Coil.
There is a constant magnetic field directed into the
page in the shaded region. The field is zero outside
the shaded region.
A Coil slides through the region. For each of the five
positions, determine whether an induced current
exists or not. If it does, find its direction.
1. No field  no flux change  no current
2. Flux is increassing into the page  induced
magnetic field points out of page  induced
current in the CCW direction
3. Flux is negative, but not changing  no
induced current
4. Flux is is decreasing into the page induced
magnetic field points into the page 
induced current in the CW
5. Same as 1: no induced current.
4
HOW AN AC GENERATOR PRODUCES EMF
Simple external
contacts:
Unlike in a DC
motor (Ch. 21), the
brushes DO NOT
switch the contacts
back-and-forth
Device being
powered
22.7 The Electric Generator
ω=
Zooming in
on the loop
v
r
The magnetic
force on all parts
of the loop are
vertical by RHR-1
Vertical force
produces NO emf
on the horizontal
segments
EMF by
induction
On each vertical segment, the
magnetic forces generate
motional emf given by
E = (vB sin θ )L = BLv sin θ
Induced current flows up
on the segment in front
and down on the back
r
Mechanical
power input
But v = rω = ω ⋅ 12 W
E ↑,↓ = 12 BLWω sin θ = 12 BAω sin θ
5
22.7 The Electric Generator
ω=
θ = ωt
NOTE: ω ≠ W
W = wdith of the coil
v
r
ω = angular speed of
≡
rotation of the loop/coil
+
(is equivalent to) −
−
E ↑,↓ = 12 BAω sin θ
+
= 12 BAω sin ω t
For each loop in a coil we
have two of these terms
ELOOP = BAω sin θ
Emf induced in a rotating planar coil of N turns
E = NABω sin ω t
E (t )
or
V (t )
E (t ) = E0 sin ω t
or
ω = 2π f
V (t ) = V0 sin ω t
Peak emf / source voltage
E0 = V0 = NBAω
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