Transcript Chapter 18

Chapter 31
Faraday’s Law
Faraday’s Experiment
• A primary coil is connected to a battery and a
secondary coil is connected to an ammeter
• The purpose of the secondary circuit is to detect
current that might be produced by a (changing)
magnetic field
• When there is a steady current in the primary circuit, the
ammeter reads zero
Faraday’s Experiment
• When the switch is opened, the ammeter reads a
current and then returns to zero
• When the switch is closed, the ammeter reads a current
in the opposite direction and then returns to zero
• An induced emf is produced in the secondary circuit by
the changing magnetic field
Electromagnetic Induction
• When a magnet moves toward a
loop of wire, the ammeter shows the
presence of a current
• When the magnet moves away from
the loop, the ammeter shows a
current in the opposite direction
• When the magnet is held stationary,
there is no current
• If the loop is moved instead of the
magnet, a current is also detected
Electromagnetic Induction
• A current is set up in the circuit as
long as there is relative motion
between the magnet and the loop
• The current is called an induced
current because is it produced by
an induced emf
Faraday’s Law and Electromagnetic
Induction
• Faraday’s law of induction: the instantaneous emf
induced in a circuit is directly proportional to the
time rate of change of the magnetic flux through the
circuit
• If the circuit consists of N loops, all of the same area,
and if FB is the flux through one loop, an emf is
induced in every loop and Faraday’s law becomes
dF B
  N
dt
Faraday’s Law and Lenz’ Law
• The negative sign in Faraday’s Law is included to
indicate the polarity of the induced emf, which is found
by Lenz’ Law:
• The current caused by the induced emf travels in the
direction that creates a magnetic field with flux
opposing the change in the original flux through the
circuit
dF B
  N
dt
Heinrich Friedrich
Emil Lenz
1804 – 1865
Faraday’s Law and Lenz’ Law
• Example:
• The magnetic field, B, becomes smaller with time and
this reduces the flux
• The induced current will produce an induced field,
Bind, in the same direction as the original field
dF B
  N
dt
Faraday’s Law and Lenz’ Law
• Example:
• Assume a loop enclosing an area A lies in a uniform
magnetic field
• Since ΦB = B A cos θ, the change in the flux, ΔΦB, can
be produced by a change in B, A or θ
dF B
  N
dt
Chapter 31
Problem 17
A coil formed by wrapping 50 turns of wire in the shape of a square is
positioned in a magnetic field so that the normal to the plane of the
coil makes an angle of 30.0° with the direction of the field. When the
magnetic field is increased uniformly from 200 μT to 600 μT in 0.400 s,
an emf of magnitude 80.0 mV is induced in the coil. What is the total
length of the wire?
Motional emf
• A straight conductor of length ℓ
moves perpendicularly with constant
velocity through a uniform field
• The electrons in the conductor
experience a magnetic force
FB = q v B
• The electrons tend to move to the
lower end of the conductor
• As the negative charges accumulate
at the base, a net positive charge
exists at the upper end of the
conductor
Motional emf
• As a result of this charge separation,
an electric field is produced in the
conductor
• Charges build up at the ends of the
conductor until the downward
magnetic force is balanced by the
upward electric force
FE = q E = q v B;
E = v B;
• There is a potential difference
between the upper and lower ends of
the conductor
Motional emf
• The potential difference between the
ends of the conductor (the upper end
is at a higher potential than the lower
end):
ΔV = E ℓ = B ℓ v
• A potential difference is maintained
across the conductor as long as
there is motion through the field
• If the motion is reversed, the polarity
of the potential difference is also
reversed
Motional emf in a Circuit
• As the bar (with zero resistance) is
pulled to the right with a constant
velocity under the influence of an
applied force, the free charges
experience a magnetic force along the
length of the bar
• This force sets up an induced current
because the charges are free to move
in the closed path
•
dF B
d ( Blx )
 

dt
dt
The changing magnetic flux through
dx
the loop and the corresponding
  Bl
 Blv
induced emf in the bar result from the
dt
change in area of the loop
Motional emf in a Circuit
• The induced, motional emf, acts like a
battery in the circuit
B v
  B v and I 
R
• As the bar moves to the right, the
magnetic flux through the circuit
increases with time because the area
of the loop increases
• The induced current must be in a
direction such that it opposes the
change in the external magnetic flux
(Lenz’ Law)
Motional emf in a Circuit
• The flux due to the external field is
increasing into the page
• The flux due to the induced current
must be out of the page
• Therefore the current must be
counterclockwise when the bar moves
to the right
• If the bar is moving toward the left, the
magnetic flux through the loop is
decreasing with time – the induced
current must be clockwise to produce
its own flux into the page
Motional emf in a Circuit
• The applied force does work on the
conducting bar, thus moving the
charges through a magnetic field and
establishing a current
• The change in energy of the system
during some time interval must be
equal to the transfer of energy into the
system by work
• The power input is equal to the rate at
which energy is delivered to the
2
resistor
ε
 Fappv   I B  v 
R
Chapter 31
Problem 24
A conducting rod of length ℓ moves on two horizontal frictionless rails. A
constant force of magnitude 1.00 N moves the bar at a uniform speed of 2.00
m/s through a magnetic field that is directed into the page. (a) What is the
current in an 8.00-Ω resistor R? (b) What is the rate of energy dissipation in
the resistor? (c) What is the mechanical power delivered by the constant
force?
Induced emf and Electric Fields
• An electric field is created in the conductor as a result
of the changing magnetic flux
• Even in the absence of a conducting loop, a changing
magnetic field will generate an electric field in empty
space (this induced electric field is nonconservative,
unlike the electric field produced by stationary charges)
• The emf for any closed path can be expressed as the
 
line integral
 E  ds
• Faraday’s law can be written in a general form:
 
dF B
 E  ds   dt
Lenz’ Law – Moving Magnet Example
• As the bar magnet is moved to the right toward a
stationary loop of wire, the magnetic flux increases
with time
• The induced current produces a flux to the left, so the
current is in the direction shown
• When applying Lenz’ Law, there are two magnetic
fields to consider: changing external and induced
Lenz’ Law – Rotating Loop Example
• Assume a loop with N turns, all of the
same area rotating in a magnetic field
• The flux through the loop at any time t
is FB = BAcosq = BAcoswt
• The induced emf in the loop is
dFB
ε  N
dt
 NABω sin ωt
• This is sinusoidal, with max = NABw
AC Generators
• Alternating Current (AC) generators convert
mechanical energy to electrical energy
• Consist of a wire loop rotated by some external means
(falling water, heat by burning coal to produce steam,
etc.)
• As the loop rotates, the magnetic flux through it
changes with time inducing an emf and a current in the
external circuit
AC Generators
• The ends of the loop are connected to slip rings that
rotate with the loop; connections to the external circuit
are made by stationary brushes in contact with the slip
rings
• The emf generated by the rotating loop:
dFB
ε  N
dt
 NABω sin ωt
DC Generators
• Components are essentially the same as that of an ac
generator
• The major difference is the contacts to the rotating
loop are made by a split ring, or commutator
• The output voltage always has the same polarity
• The current is a pulsing current
DC Generators
• To produce a steady current, many loops and
commutators around the axis of rotation are used
• The multiple outputs are superimposed and the output
is almost free of fluctuations
Motors
• Motors are devices that convert electrical energy into
mechanical energy (generators run in reverse)
• A motor can perform useful mechanical work when a
shaft connected to its rotating coil is attached to some
external device
• As the coil begins to rotate, the induced back emf
opposes the applied voltage and the current in the coil
is reduced
• The induced emf explains why the power requirements
for starting a motor and for running it are greater for
heavy loads than for light ones
Answers to Even Numbered Problems
Chapter 31:
Problem 28
(a) to the right
(b) out of the plane of the paper
(c) to the right
(d) into the paper