Download Lecture 19: Induction

Magnetic Induction
Chapter 27
27.1 Induced currents
27.2 Induced EMF and Faraday’s Law
27.3 Lenz’s Law, Motional EMF
27.4 Inductance
27.5 Magnetic energy
27.6 Induced electric fields
Two symmetry situations
We found that:
current loop + magnetic field  torque
If there is no current and we turn loop by hand,
will the opposite occur?
torque + magnetic field  current ?
YES
Faraday’s Law
of induction
Experiment 1
1.
A current appears only of there is relative
motion between the loop and the magnet;
the current disappears when the relative
motion ceases.
2.
Faster motion produces greater current
3.
Current direction depends on which
magnetic pole is moving towards (or away)
Induced current and induced emf
Experiment 2
1.
If we close switch S, which turns on current in
right-hand loop, the meter registers a current
very briefly and suddenly in the left-hand loop.
2.
If we open the switch, another sudden and brief
current is measured, but in the opposite
direction.
i.e. there is an induced current (and thus an induced
emf) only when the current in the right-hand loop
is changing, and not when it is constant.
In both experiments an
induced current and
induced emf are
apparently caused by
something changing
What is this something that is changing?
Faraday discovered that:
An emf is induced in a loop when the
number of magnetic field lines that pass
through the loop is changing.
The values of the induced emf and
induced current are determined by the
rate at which the number of field lines is
changing (not the actual number).
Magnetic Flux
We need a way to calculate the amount of
magnetic field that passes through a loop.
We define a magnetic flux. Suppose a loop
enclosing area A is placed in a magnetic
field B. As for electric fields we define a
vector area, A.
When B makes an angle  with the normal
to the area, the flux through the loop is
B  A  B nˆA  BAcosθ
If dA is an element of area on surface S,
the magnetic flux through S is
ΦB   B  dA
Magnetic Flux
For the special case when the
magnetic field is perpendicular to
the loop, and the magnetic field is
uniform
Φ  BA
B
SI unit for magnetic flux is the
Weber:
We can now state Faraday’s
law as:
“ The magnitude of the emf
induced in a conducting loop is
equal to the rate at which the
magnetic flux through the loop
changes with time ”
1 Wb = 1 T.m2
EXERCISE: show that 1 Wb/s = 1 V
dΦB
ε
dt
Faraday’s Law of Induction
If we change the magnetic flux through
a coil of N turns, an induced emf
appears in every turn and the total emf
induced in the coil is the sum of these
individual emfs. It is
dΦB
ε  N
dt
ΦB   B  dA
To change the magnetic flux we can
change:
1.
the magnitude B of the magnetic
field within the coil
2.
the area of the coil, or the portion
of that area that lies within the
magnetic field (eg expanding the
coil or moving it in or out of the
field)
3.
the angle between the direction of
the field B and the area of the coil
(eg by rotation of the coil)
CHECKPOINT: The graph gives the
magnitude B(t) of a uniform magnetic
field that exists throughout a
conducting loop, perpendicular to the
plane of the loop. Rank the five regions
of the graph according to the magnitude
of the emf induced in the loop, greatest
first.
Clue: It is the
special case where
and A is constant
ΦB  BA
t
Answer:
b first
d and e tie
a and c tie (zero)
CHECKPOINT:
To change the magnetic flux we can
change:
If the circular conductor undergoes
thermal expansion while it is in a
uniform magnetic field, a current
will be induced clockwise around it.
Is the magnetic field directed
A.
into the page or
B.
out of the page?
1.
the magnitude B of the magnetic
field within the coil
2.
the area of the coil, or the portion
of that area that lies within the
magnetic field (eg expanding the
coil or moving it in or out of the
field)
3.
the angle between the direction of
the field B and the area of the coil
(eg by rotation of the coil)
Answer: out of the page
(The induced magnetic field is into
the page, opposing the increase in
flux outwards through the loop.)
To change the magnetic flux we can
change:
EMF is E = E0sin(2πft)
This is the principle of an
alternating-current generator
See Wolfson page 471
1.
the magnitude B of the magnetic
field within the coil
2.
the area of the coil, or the portion
of that area that lies within the
magnetic field (eg expanding the
coil or moving it in or out of the
field)
3.
the angle between the direction of
the field B and the area of the coil
(eg by rotation of the coil)
EXAMPLE: A uniform magnetic field makes an angle of 60° with the
plane of a circular coil of 300 turns and a radius of 4 cm. The magnitude
of the magnetic field increases at a rate of 85 T/s while its direction
remains fixed. Find the magnitude of the induced emf in the coil.
PICTURE THE PROBLEM: The induced emf equals N times the
rate of change of the flux through a single turn. Since B is
uniform, the flux through each turn is simply B = BAcos,
where A=r2.
NB what is the angle ?
EXERCISE: if the resistance of the coil is 200 , what is
the induced current?
27.3 Induction and energy: Lenz’s Law
If we move a bar magnet towards a wire
loop, an induced current flows and
energy is dissipated as heat in the wire.
Where did the energy come from?
Conservation of energy gives us the
direction of the induced current –
loop acts as a magnet – it is hard to
move a N pole towards another.
If we move a bar magnet away from the wire loop, an induced
current flows and energy is dissipated as heat in the wire.
This time work is needed to pull the magnet away
NB The induced flux of Bi always opposes the change in the flux of B, but
this does not mean that it necessarily points in the opposite direction to B.
When flux through loop is decreasing [(b) and (d)], the flux of Bi must
oppose this change and therefore Bi and B are in the same direction.
Two adjacent circuits
Just after switch is closed, I1 increases in
direction shown, inducing I2. The flux through
circuit 2 due to I2 opposes the the change in
flux due to I1.
As switch is opened, I1
decreases and flux through
circuit 2 changes. Induced
current I2 then tends to
maintain the flux through
circuit 2.
CHECKPOINT: The figure shows
three situations in which identical
circular conducting loops are in
uniform magnetic fields that are
either increasing or decreasing in
magnitude at identical rates.
Rank them according to the
magnitude of the current induced
in the loops, greatest first.
Answer: a and b tie, then c (zero)
True
True or False?
False
Induction
1.
The induced emf in a circuit is proportional
to the magnetic flux through the circuit.
False
2.
There can be an induced emf at an instant
when the flux through the circuit is zero.
True
3.
Lenz’s law is related to the conservation of
energy.
True