Download 21.2 Faraday`s Law of Induction and Lenz`s Law

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
Zero flux
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
B
θ
Φ Β = BAcos45
θ
Φ Β = BA
Max flux
Units: weber (Wb),
1 Wb = 1 T·m2
(21-1)
number of lines passing through
the coil ∝ Φ B through the coil
What is the total magnetic flux through any closed surface?
21.2 Faraday’s Law of Induction; Lenz’s Law
Experiments by Faraday and others showed that…
Minus sign in Faraday’s Law tells us that the
induced emf opposes the original change, ie.
if ΦB changes though a coil of wire, an emf is induced and
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.
BIND
Number of loops
Examples
IIND
[Faraday’s Law]
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
the plane of the page
points out of the page
into the page
into the page
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.
.. or if the angle
between the loop
and
the
field
changes.
Induced
current
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.
1
Question...
Question...
A very long straight wire carries a
steady current down. A loop of
wire is moved towards the current.
What is the magnetic flux
through the wire loop ?
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?
IIND
BIND
21.3 (Motional) EMF in a Moving Conductor (21-3 and Ex 21-8)
l
I
v
Area, A1
A) Counter clockwise
B) Clockwise
C) There is no induced current
21.3 A “Slide generator"
Wire loop
What happens?
An emf is induced in the rod of magnitude:
A) BA1
B) BA2
C) B(A1 - A2)
ε = Blv
Fe
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.
A
F
Now there is a continuous
path for the electrons and
the induced emf causes a
current to flow.
Because the conductor is
moving this is sometimes
referred to as motional emf
Question...
(21-3 and Ex 21-8)
Rest the moving rod on a
U-shaped conductor…
(21-3)
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.
F
F
Note,
is different from the upward force
e,
on the electrons that produced the initial current.
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
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
21.5 Electric Generators
Rotating
metal
wheel
Eddy
currents
The axle is rotated by an
external force e.g. falling water
or steam. As it turns a
sinusoidal emf, is induced:
Generator eqn.
area of each turn
B points into
page here
Note, Faraday’s law can be generalized to:
A changing magnetic field induces an electric field.
- regardless of whether there are conductors around or not.
Axle
Mechanical
A generator transforms MECHANICAL
energy into ELECTRICAL energy.
(21-5)
number of
turns on coil
Peak voltage,
B
Electrical
Output voltage of generator
Angular frequency (radians/s)
ω = 2πf,
f = frequency (Hz)
= When sine
function = +1
When the generator is connected to a circuit, an ac current flows. This creates
a drag force (counter torque) which resists the motion of the motor.
2
21.5 Electric Generators
21.7 Transformers and Transmission of Power
A generator has a coil of wire rotating in a
magnetic field. The rotation rate INCREASES.
Up till now we have been changing the magnetic
flux through a coil and inducing a current.
What happens to the maximum output
voltage of the generator? It..
1. increases
2. decreases
3. varies sinusoidally
4. stays the same
If instead we pass a changing I through a coil,
the magnetic flux it produces also changes.
Note,
Mechanical
Energy
- pass that changing flux through a 2nd
coil, an emf can be induced in the 2nd coil.
Electrical
Energy
This is the basis of a transformer
An electric generator can be used as a motor and vice versa.
21.7 Transformers and Transmission of Power
(changing)
21.9 Self Inductance and Inductors
A Transformer is a device for increasing or decreasing ac voltage.
What is an inductor?
• primary and secondary coil:
interwoven or linked by an iron core.
- its just a coil of wire
• 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:
[rms or peak
values]
(21-6)
transformer requires ac
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
21.9 Self Inductance
(21-9)
-
+
Induced emf
-
+
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
21.9 Self Inductance
The induced emf is proportional to the rate of change of
the current and it opposes the change (Lenz’s Law):
L = self-inductance
Units: henry, H.
1 H = 1 V·s/A = 1 Ω·s.
When this coil of wire is put in a circuit it has
interesting effects because of Faraday’s Law
Induced emf tries to prevent
the current from increasing as
it enters the inductor at A
Self inductance, L depends on the size and shape of the
coil and the presence of an iron core (which increases L).
L can be calculated for an empty coil using this formula:
N2A
L = µ0
l
A
N loops
l
Induced emf tries to prevent
the current from decreasing.
An Inductor resists any change in the current.
3
* a pure inductor + resistor in series could represent a real coil of wire or an electromagnet
Question…
21.11 LR Circuit
The current through a 220 mH inductor increases
from 0.4 to 1.6 Amperes in 640 ms.
What happens when a pure inductor and resistor* are
connected to a battery and switch?
Switch ON (position 1):
I increasing
220 mH
What is the induced emf across the inductor?
a)
b)
c)
d)
e)
-0.13 V
-0.41 V
0V
+2.7V
+8.4 V
-
+
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
-
+
+
-
1
With time: I increases less rapidly.
Eventually: All the voltage drop is across R.
I
Current flowing
in the circuit
Induced emf prevents current rising immediately to max value.
21.11 LR Circuit
21.10 Energy Stored in a Magnetic Field
Switching OFF (position 2):
Current in the circuit
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:
When the battery is removed from the circuit the current gradually dies away
PE per unit
volume
Units: J/m3
=
1 2 (21-10)
LR circuit similar to RC circuit but time
constant is inversely proportional to R.
where
What is the energy density of the Earth’s magnetic field?
Induced emf across inductor prevents I dropping immediately to zero
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:
=
1 2 4