21.2 Faraday’s Law of Induction and Lenz’s Law
Changing magnetic flux and induced Emf
Faraday’s Law of Induction
• Magnetic Flux
• Changing Magnetic flux induces an EMF in a coil of wire
• Lenz’s Law
• Induced EMF in a Moving Conductor; Eddy Currents
• Faraday generalized:
Changing Magnetic Field induces an Electric Field
Imagine a coil of wire (area,
A) in a magnetic field
Magnetic Flux
- similar to electric flux.
θ = 90o
θ= 45o,
ΦΒ = BAcos45
• Electric Generators
• Transformers
B
θ
ΦΒ = 0
θ
θ= 0o,
ΦΒ = BA
(If B ⊥ A, flux is maximum)
(21-1)
• Self Inductance and Inductors
• Energy Stored in a Magnetic Field
• LR Circuit
Φ B through the coil ∝ number of
lines passing through the coil.
Units: weber (Wb),
1 Wb = 1 T·m2
21.2 Faraday’s Law of Induction and Lenz’s Law
21.2 Faraday’s Law of Induction; Lenz’s Law
Experiments by Faraday and others showed that…
if ΦB changes though a coil of wire, an emf is induced
Minus sign in Faraday’s Law tells us that the induced emf
opposes the original change, ie.
Moreover,
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]
the induced emf is proportional to the rate of
change of magnetic flux, φB through the coil.
[Faraday’s Law]
Number of loops
BIND
Examples
IIND
IIND
IIND
S
N
No
IIND
N
Induced Emf
Rate of change of
magnetic flux through coil
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
The magnetic flux
will also change if
the area of the
loop changes.
1. Magnetic flux, ΦB:
Is it 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.
If ΦB is decreasing, the induced magnetic field, BIND,
points in the same direction.
Similarly, flux will
change if the angle
between the loop and
the field changes.
Induced
current
3. Direction of the induced current can be determined
using the right-hand rule.
4. Remember that the external field and the field due to the
induced current are different.
Question...
A very long straight wire carries a
steady current down. A loop of wire
is moved towards the current.
v
What is the direction of the
induced current in the wire loop ?
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.
l
I
1. Counter clockwise
2. Clockwise
3. There is no induced current
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)
Lets 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
Question
21.6 Eddy Currents
Induced currents can flow in any
shaped conductors.
They are called eddy currents, and
the drag forces associated with them
can dramatically slow a conductor
moving into or out of a magnetic
field.
Rotating
metal
wheel
A plastic loop, a copper loop and an imaginary loop – all
of equal area - are placed in a changing magnetic field.
Across which loop is an induced emf (voltage) generated?
plastic
B changing
copper
“Giant Drop”
at Six Flags
Great America
flickr.com/photos/
mfullererie/2669019099/
Eddy
currents
1.
2.
3.
4.
The plastic loop
The copper loop
The imaginary loop
All loops
No loop
Drag force resists
motion of wheel
3
21.4 Changing Magnetic Flux Produces an Electric Field
21.5 Electric Generators
A generator transforms mechanical energy into electrical energy.
This surprising fact and other results suggest that
Faraday’s law can be generalized to the following:
A changing magnetic field induces an electric field.
The axle is rotated by an
external force e.g. falling
water or steam.
Axle
As it turns at constant
speed v, a sinusoidal emf,
is induced
B
- regardless of whether there are conductors around or not.
Generator eqn.
area of loop
(21-5)
number of turns in loop
Angular frequency (radians/s)
ω = 2πf, f = frequency
Please make your selection...
21.5 Electric Generators
Generator eqn.
A generator has a coil of wire
rotating in a magnetic field.
Max value:
The rotation rate INCREASES.
0
If the generator is connected to a circuit, an ac current flows.
Counter torque: As before, there is a drag force (torque)
that resists the motion when the generator is connected
to a circuit and current flows in the loops.
What happens to the maximum
output voltage of the generator?
1.
2.
3.
4.
It increases
It decreases
It varies sinusoidally
It stays the same
4
21.5 Electric Generators
Mechanical Energy
Electrical Energy
If we now pass I through
Up till now we have been changing
a coil and allow I to
φB and inducing a I in a coil.
change, then φB through
the coil also changes.
An electric generator can be used as a motor and vice versa.
If that changing φB passes
through a second coil an
emf can be induced in the
second coil.
This is the basis of a transformer
21.7 Transformers and Transmission of Power
A Transformer is a device for increasing or decreasing ac voltage.
• primary and secondary coil, either
interwoven or linked by an iron core.
21.7 Transformers and Transmission of Power
Transformers work only if the current is changing.
• 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.
Can show:
(21-6)
(must be rms or peak
values - transformer
doesn’t work for DC)
STEP-UP transformer increases
the voltage (NS > NP)
STEP-DOWN transformer
decreases the voltage (NP > NS)
Electricity is transmitted at high voltage (because less power is lost).
Transformers are then used to step down to a more useful voltage for the home.
5
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.
21.9 Self Inductance and Inductors
21.11 LR Circuit
Magnitude of L depends on the size and shape of the coil
and the presence of an iron core (which enhances L).
L can be calculated for an empty coil:
2
L = µ0N A
l
(21-9)
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)
What happens when a DC source is connected to a pure
inductor and resistor (or an inductor with resistance)?
Switch at 1: I changes rapidly at first
A large back emf develops across L which opposes the increasing current.
Initially most of the voltage drop is across the inductor
With time, I increases less rapidly. Eventually all the voltage drop is
across R.
-
+
2
-
+
+
-
1
Current in circuit
At time, t:
Switch on: Induced emf prevents current rising immediately to max value.
6
21.11 LR Circuit
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.
2
The energy density of the magnetic field is given by:
If the battery is removed from the circuit (switch → 2)
The current gradually decays away.
B
Switch off: Induced emf across inductor prevents I dropping immediately to zero
LR circuit similar to RC circuit but time
constant now is inversely proportional to R.
This is why there is a ‘reaction time’ when an
electromagnet is switched on.
where
(21-10)
Units: J/m3
(Energy per unit volume)
Summary of Chapter 21
• 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:
•Transformer changes
the magnitude of an ac
voltage:
• Energy density stored
in magnetic field:
B
7