21.2 Faraday’s Law of Induction and Lenz’s Law
Magnetic Flux
- similar to
electric flux
Imagine a coil of wire (area, A)
in a magnetic field
θ = 90o
Faraday’s Law of Induction
• Changing Magnetic flux induces an EMF
• Lenz’s Law
• Induced EMF in a Moving Conductor; Eddy Currents
• Faraday generalized:
Changing Magnetic Field induces an Electric Field
• Electric Generators
• Transformers
• Self Inductance and Inductors
• Energy Stored in a Magnetic Field
• LR Circuit
21.2 Faraday’s Law of Induction and Lenz’s Law
θ= 45o,
ΦΒ = BAcos45
Max flux
Units: weber (Wb),
1 Wb = 1 T·m2
Rate of change of
magnetic flux through coil
number of lines passing through
the coil ∝ Φ B through the coil
21.2 Faraday’s Law of Induction; Lenz’s Law
Minus sign in Faraday’s Law tells us that the
induced emf opposes the original change, ie.
The current produced by an induced emf moves
in a direction such that its magnetic field
opposes the original change in flux. [Lenz’s Law]
[Faraday’s Law]
BIND
Examples
IIND
Induced Emf
(21-1)
What is the total magnetic flux through any closed surface?
if ΦB changes though a coil of wire, an emf is induced and
Number of loops
θ
θ= 0o,
ΦΒ = BA
Experiments by Faraday and others showed that…
the induced emf is proportional to the rate of
change of magnetic flux, φB through the coil.
B
θ
Zero flux
IIND
IIND
S
N
No
IIND
N
Pull the loop out of the South magnetic pole North magnetic pole
Magnetic field increases magnetic field which moving toward loop moving toward loop in
points out of the page
the plane of the page
into the page
into the page
1
21.2 Faraday’s Law of Induction; Lenz’s Law
21.2 Faraday’s Law of Induction; Lenz’s Law
Problem Solving using Lenz’s Law
1. Magnetic flux, ΦB:
The magnetic flux
will also change if
the area of the
loop changes.
What direction?
Is Φ B increasing, decreasing, or constant?
2. Induced magnetic field tries to keep the flux constant.
If ΦB is increasing, the induced magnetic field,
BIND, points in the opposite direction.
Similarly, flux will
change if the angle
between the loop and
the field changes.
Induced
current
Question...
A very long straight wire carries a
steady current down. A loop of
wire is moved towards the current.
Question...
What is the magnetic flux
through the wire loop ?
IIND
v
21.3 EMF Induced in a Moving Conductor (21-3 and Ex 21-8)
Here is another way to induce an emf in a conductor…
A conducting rod moves
to the right with velocity,
v, perpendicular to a
magnetic field, B.
What is the direction of the
induced current in the wire loop?
BIND
If ΦB is decreasing, the induced magnetic field,
BIND, points in the same direction.
3. Direction of the induced current can be determined
using RHR-1.
4. Remember that the external field and the field due
to the induced current are different.
l
I
Area, A1
A) Counter clockwise
B) Clockwise
C) There is no induced current
Wire loop
A) BA1
B) BA2
C) B(A1 - A2)
What happens?
An emf is induced in the rod of magnitude:
ε = Blv
(21-3)
2
Question...
21.3 EMF Induced in a Moving Conductor (21-3 and Ex 21-8)
Rest the moving rod on a
U-shaped conductor…
Fe
A
F
Now there is a continuous
path for the electrons and
the induced emf causes a
current to flow.
A 737 is flying at 200 m/s through a
region where the Earth’s magnetic
field is 5 x 10-5 T and pointing DOWN.
How much potential difference is
created across the 35 m wingspan ?
I
v
B
But …
the induced current interacts with the magnetic field, producing
a drag force (F=ILB) that resists the motion of the rod.
Note, F is different from the upward force Fe,
on the electrons that produced the initial current.
1.
2.
3.
Zero because there is no closed circuit
for a current to flow.
0.35 V with wing A positively charged
0.35 V with wing A negatively charged
21.6 Eddy Currents
Induced currents (Eddy currents)
can flow in any shaped conductor.
Drag forces associated with eddy
currents can dramatically slow a
conductor moving into or out of a
magnetic field.
Drag force resists
motion of wheel
21.5 Electric Generators
Rotating
metal
wheel
Eddy
currents
B points into
page here
Note, Faraday’s law can be generalized to:
A generator transforms MECHANICAL energy into ELECTRICAL energy.
The axle is rotated by an
external force e.g. falling
water or steam.
As it turns at constant
speed v, a sinusoidal emf,
is induced
B
Generator eqn.
A changing magnetic field induces an electric field.
- regardless of whether there are conductors around or not.
Axle
area of loop
(21-5)
number of turns in loop
Angular frequency (radians/s)
ω = 2πf, f = frequency
3
21.5 Electric Generators
Generator eqn.
Max value:
0
If the generator is connected to a circuit, an ac current flows.
Again, there is a drag force (known as counter torque) that resists the
motion when the generator is connected to a circuit and current flows.
Mechanical Energy
Electrical Energy
Please make your selection...
A generator has a coil of wire rotating in a magnetic field.
The rotation rate INCREASES.
What happens to the
maximum output voltage
of the generator?
1.
2.
3.
4.
An electric generator can be used as a motor and vice versa.
It increases
It decreases
It varies sinusoidally
It stays the same
21.7 Transformers and Transmission of Power
Up till now we have been changing φB and
inducing a I in a coil.
If we now pass a changing I through a coil,
then the magnetic flux through the coil also
changes.
21.7 Transformers and Transmission of Power
A Transformer is a device for increasing or decreasing ac voltage.
• primary and secondary coil:
interwoven or linked by an iron core.
• Nearly all magnetic flux produced by primary
coil passes through secondary coil.
When an ac voltage is applied to the primary
coil an ac voltage of the same frequency is
induced in the secondary coil.
(21-6)
Can show:
When that changing flux passes through
a 2nd coil, an emf can be induced in the
2nd coil.
This is the basis of a transformer
[rms or peak values]
STEP-UP transformer increases
the voltage (NS > NP)
STEP-DOWN transformer
decreases the voltage (NP > NS)
Transformers play an important role in the transmission of electricity
4
21.9 Self Inductance and Inductors
21.9 Self Inductance and Inductors
Self inductance
What is an inductor?
The induced emf is proportional to the rate of change of
the current and it opposes the change (Lenz’s Law):
Basically its just a coil of wire
L = self-inductance
Units: henry, H.
1 H = 1 V·s/A = 1 Ω·s.
But, when this coil of wire is put in a circuit it has interesting
effects because of Faraday’s Law and induced emf.
If I changes in a single coil, then φB changes and an emf is
induced in that same coil. This is known as self inductance
+
-
-
+
Induced emf tries to prevent
the current from increasing as
it enters the inductor at A
Induced emf tries to prevent
the current from decreasing.
An Inductor resists any change in the current.
Question…
21.9 Self Inductance and Inductors
Self inductance, L depends on the size and shape of the
coil and the presence of an iron core (which increases L).
(21-9)
The current through a 220 mH inductor increases
from 0.4 to 1.6 Amperes in 640 ms.
I increasing
220 mH
L can be calculated for an empty coil:
2
L = µ0N A
l
What is the induced emf across the inductor?
A
N loops
l
Example: Calculate L for a tightly wrapped solenoid, 7 cm
long with 150 loops and cross-sectional area, A = 0.20 cm2.
2
-7
2
-4
L = µ0N A = (4π x 10 )(150) (0.2 x 10 ) = 8.1 µH
l
(0.07)
a)
b)
c)
d)
e)
-0.13 V
-0.41 V
0V
+2.7V
+8.4 V
5
* a pure inductor + resistor in series could represent a real coil of wire or an electromagnet
21.11 LR Circuit
21.11 LR Circuit
What happens when a DC source is
connected to a pure inductor and resistor*?
Switch at position 1:
Initially I increases rapidly
A large emf develops across L to
oppose the increasing current.
Most of the voltage drop
is across the inductor
-
+
-
+
2
2
+
-
1
With time I increases less rapidly.
If the battery is removed from the circuit (switch → 2)
the current gradually decays away.
Eventually All the voltage drop is across R.
Switch off: Induced emf across inductor prevents I dropping immediately to zero
Current in
circuit
at time, t:
LR circuit similar to RC circuit but time
constant is inversely proportional to R.
where
Switch on: Induced emf prevents current rising immediately to max value.
Summary of Chapter 21
21.10 Energy Stored in a Magnetic Field
We saw in section 17-9, that energy can be stored in
an electric field ( uE = 12 ε0 E2 ).
Energy can also be stored in a magnetic field,
for example in an inductor or solenoid.
The energy density of the magnetic field is given by:
Energy per
unit volume
Units: J/m3
‫ݑ‬஻ =
1 ‫ܤ‬ଶ
2 ߤ଴
• Magnetic flux:
• Changing magnetic flux induces
an emf:
• Induced emf opposes the original flux change.
• Changing magnetic field induces an electric field
• Electric generator converts mechanical energy to electrical energy.
Changing magnetic flux in the coils induce an emf, which drives an
alternating current through an external circuit.
• Self inductance:
(21-10)
•Transformer changes
the magnitude of an ac
voltage:
• Energy density stored
in magnetic field:
‫ݑ‬஻ =
1 ‫ܤ‬ଶ
2 ߤ଴
6