Induced voltages and Inductance
Faraday’s Law
concept #1, 4, 5, 8, 13
Problem # 1, 3, 4, 5, 6, 9, 10, 13, 15, 24,
23, 25, 31, 32a, 34, 37, 41, 43, 51, 61
• Last chapter we saw that a current produces a
magnetic field.
• In 1831 experiments by Michael Faraday and
Joseph Henry showed that a changing
magnetic field could induce a current in a
circuit.
Faraday’s setup.
Switch
Ammeter
0.0 mA
+
_
Battery
• The coil with the switch is connected to a battery.
(Primary coil)
• When current goes through a coil, it produces a
magnetic field.
• The coils are wrapped around an iron ring to
intensify the magnetic field.
• The secondary coil is hooked up to an ammeter.
This coil is not hooked up to a battery.
When the switch in the primary coil is closed,
the ammeter reads a current in the secondary
coil for a short moment, then returns to zero.
When the switch is opened, the ammeter
momentarily measures a current in the
opposite direction before returning to zero.
When there is a steady current in the primary
coil, there is no current read by the ammeter.
• Conclusion: An electric current can be
produced from a changing magnetic field.
• The current produced in the secondary coil
occurs only for the instant the magnetic field
through the secondary coil is changing.
• The secondary circuit behaves as though a
source of emf (a battery) was connected to it
for a short time.
• An induced emf is produced in the secondary
circuit by the changing magnetic field.
emf – electric motive force, not really a force.
This is a source of electrical work/energy per
unit charge.
work/charge = volt
Devices that increase the potential energy of
circulating charges (batteries, generators) are
sources of emf.
Think of emf as a voltage increase.
While the magnetic field inside the secondary
coil was changing, the secondary coil acted as
is it was connected to a battery.
The changing magnetic field induced an
electric field in the secondary wire that caused
the current to flow.
Changing magnetic fields induce electric fields.
A constant magnetic field can also induce an
electric field. Example of this is an electric
generator.
If the magnetic field is constant then what is
changing?
The property that creates an electric field is the
changing of the magnetic flux.
Magnetic flux
Magnetic flux is defined in the same way electric
flux was defined earlier.
Magnetic flux through a loop, is proportional
to the strength of the B-field passing through
the plane of the loop and the area of the loop.
B = B A cos
B cos is the component of the B-field that is
perpendicular to the loop.
B
side view of loop
Magnetic flux
• The value of the magnetic flux is proportional
to the number of B-field lines passing through
the loop.
•
= B A cos is maximized when = 0. This is
when the B-field is perpendicular to the loop.
B
• see fig. 20.3 for maximizing/minimizing the flux
• work example 20.1
Faraday’s law of Induction
Consider a wire loop connected to an ammeter.
Moving a magnet towards the loop will induce a
current in one direction.
When the magnet is stationary, there is no
induced current.
Moving the magnet away from the loop induces
a current in the opposite direction.
This is similar to the Faraday experiment shown earlier.
Faraday’s Law of induction
• An emf is induced in a circuit when the
through the circuit changes with time.
B
• The instantaneous emf induced in a circuit
equals the negative rate of change of B with
respect to time.
• This is Faraday’s Law of magnetic induction.
N
B
t
Lenz’s Law: The induced current travels in the
direction that creates a magnetic field with
flux opposing the change in the original flux
through the circuit.
If the flux is increasing in one direction, the
induced current will be in the direction so that
its own magnetic flux will be in the direction
opposite of the original flux.
Nature wants to keep the flux constant.
The induced magnetic flux does not have to be
in the opposite direction of the original flux.
Fig. 20.5. The original flux is upwards. As the
B-field is reduced in strength, the flux is
reduced. Lenz’s law will show that the induced
current will be in the direction so that the
induced B-field is in the upward direction.
work out example 20.2
Application of Faraday’s law
• Ground fault interrupters, used to protect
against short circuits
• Pickups on electric guitars, converts the
vibrations of the strings to an electrical signal.
Motional emf
• Earlier we changed the B-field with time.
• Now we keep the field constant. Look at the
emf induced in a conductor moving through a
magnetic field.
fig 20.12
Look at straight conductor of length (L) moving with
constant velocity through a uniform B-field pointing into
the page.
In this example the velocity is normal to B-field.
Force on the electrons (-) is FB = qvB downward
Free electrons build up on the lower end, leaving a net
positive charge on the upper end. This produces an Efield in the conductor.
Electrons keep moving until the magnetic force is balanced
out by the electric force qE.
qE = qvB
or E = vB
The potential difference across the conductor is
given by V = E L
so V = EL = BLv
The upper end of the conductor will be at a
higher potential than the lower end.
Now the conductor is part of a closed loop.
See pictures on page 670.
Conducting bar of length L slides along two fixed
parallel conducting rails. Let the stationary part of
the loop have a resistance R. A uniform and constant
B-field is perpendicular to the plane of the loop.
As the bar is pulled to the right, a magnetic force
acts on the free charges in the bar. Since the bar is
part of a closed loop, an induced current circulates.
The change in B and the induced current are
produced from the change in area of the loop
as the bar moves.
If bar moved a distance x in time t, the flux
changes as
B.
( A = L x)
B =B A = BL x
using Faraday’s Law with 1 loop (N = 1) and
ignoring the direction for now
B
t
x
BL
t
BLv
We call this motional emf since it is produced
from the motion of a conductor through a
magnetic field.
if we want to find the induced current, we use
the resistance of the circuit R.
V = IR I = V/R
I
R
BLv
R
gives the magnitude of the induces current.
Use Lenz’s Law and right hand rule to get
direction.
Lenz’s Law revisited
See pictures in book to practice getting the
direction of the induced current. page 674
Look to see what is the direction of the change of
flux.
The induced current will produce flux in the
opposite direction of the change in flux.
http://micro.magnet.fsu.edu/electromag/java/le
nzlaw/
Generators
• generators and motors operate on the
principle of electromagnetic induction
• generator converts mechanical energy to
electrical energy
• motor converts electrical energy to
mechanical
Electrical generator
Consists of a wire loop(s) rotated in a magnetic
field.
In a hydroelectric plant, falling water turns the
blades of a turbine to rotate the loop.
As the loop rotates, the magnetic flux through
the loop changes with time.
Changing flux induces an emf and a current.
= NBA sin t
the varies with time
N = number of turns
A = area or coil
B = field strength
= angular speed
Maximum emf from a generator
occurs when t = = 900 or 2700
= NBA
This occurs when the plane of the loop is
parallel to the magnetic field.
f where f is the frequency of rotation
In U.S. and Canada f = 60 Hz.
Motor is a generator acting in reverse.
Current is supplied to a loop by some outside
source.
The magnetic torque on a carrying current
loop, forces the loop to rotate.
Remember from Ch. 19 that the torque on a
current loop is: = B I A sin
Self Inductance
Consider a circuit (current loop) with a switch, a
resistor, and a source of emf (battery).
When the switch is closed, the current does not
change from 0 to its maximum value of I = /R
immediately.
Faraday’s Law prevent this.
As the current increases from zero, a magnetic
field is produced and the flux through the loop
increases. The increasing flux induces an emf that
opposes the change in magnetic flux.
• The net voltage across the resistor is the emf
from the battery minus the induced emf.
fig 20.24
• The opposing emf results in a gradual increase
in current.
• When the switch is closed, the current does
not immediately drop to zero either.
• This effect is called self-inductance
also see fig 20.25
Self Inductance
Faraday’s Law
= -N
B/
t
The change in flux comes from the change in
the current.
= - L I/ t
L = the inductance of the device
If the current is increasing, the induced emf is
negative.
Inductance
Inductance has units of henry (H)
1 H = 1 V s/A
These units can be found from the previous eqn.
By setting equal the equations:
= -N
and
B/ t
N
We can find: L N
B
I
B
I
= - L I/ t
Inductance of a solenoid
B
0
B
L
nI
BA
N
B
I
N
N
)I
0(
l
N
AI
0
l
N 2 AI
0
l
I
2
N
A
0
l
nl
2
L
V
(nl )
A
0
l
volume Al
2
n
Al
0
2
n
0 V
example 20.7
Calculate inductance of solenoid of 300 turns and
length 25 cm and area 4x10-4m2.
2
4
2
300
(
4
x
10
m
)
7
2
L = 0N A/length = (4 10 Tm / A)
0.25 m
L = 0.181 mH
Calculate self induced emf in the solenoid if the
current decreases at rate of 50 A/s.
= -L I/ t = -0.181mH (-50 A/s)
=9.05 mV
(remember: H = V s/A)
RL circuits
Series circuit with a resistor and an inductor.
Resistor is used to limit the current flowing
through the circuit. Resistance has units of ohms
( ).
Inductor is a closely wrapped coil of many turns.
Has an inductance.
When current is turned on, it raises gradually to
the maximum value of I = V/R
V is the voltage provided by the battery.
Save this for next segment of class. Will be on
third exam.
Energy stored in a magnetic field
Inductors store energy in a magnetic field.
This is a potential energy.
PEL = ½ L I2
(L for inductance I for current)
Note that this is similar to the expression for the
energy stored in a capacitor (PEC = ½ CV2)
example 20.10 part b.
12 V battery is hooked up to a 25
inductor.
resistor and a 5 H
Find the max current:
Imax = V/R = (12V)/(25 ) = 0.48 A
Find energy stored in the inductor when the current is maxed:
energy = ½ (5 H)(0.48A)2 = 0.576 J
Worry about part c for later.