PHYSICS 1B
Today’s lecture:
Motional emf
and
Lenz’s Law
Electricity & Magnetism
27/08/2013
PHYSICS 1B – Faraday’s Law
Applications of Faraday’s Law - GFCI
A GFCI is a Ground Fault Circuit Interrupter.
It is designed to protect users of electrical
appliances against an electric shock.
When the currents in the wires run in opposite
directions, the flux is zero.
When the return current in wire 2 changes, then the flux is no longer zero.
The resulting induced emf can be used to trigger a circuit breaker.
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Applications of Faraday’s Law – Pickup Coil
The pickup coil of an electric guitar uses
Faraday’s law.
The coil is placed near the vibrating string and
causes a portion of the string to become
magnetised.
When the string vibrates, the magnetised
segment produces a changing flux through the
coil.
The induced emf is fed into an amplifier.
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Motional emf
A motional emf is the emf induced in a conductor
moving through a constant magnetic field.
The conductor is in motion – hence the name!
The electrons in the conductor experience a force:
𝑭𝑩 = 𝑞𝒗 × 𝑩
which is directed along 𝑙.
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Motional emf
Under the influence of the force, the electrons move to
the lower end of the conductor and accumulate there.
As a result of the charge separation, an electric field, E
is produced inside the conductor.
The charges accumulate at both ends of the conductor
until they are in equilibrium with regard to the electric
and magnetic forces.
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Motional emf
In equilibrium, then, the force due to the magnetic field
is balanced by the force due to the electric field.
i.e.
𝑞𝐸 = 𝑞𝑣𝐵
or
𝐸 = 𝑣𝐵
A potential difference is maintained between the ends
of the conductor as long as it continues to move
through the magnetic field.
If the direction of the motion is reversed, then the sign
of the potential difference will also be reversed.
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Motional emf – a loop rotating in a magnetic field
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Electrical Resistance
The electrical resistance of an element of an electric circuit is the opposition to
the passage of an electric current through that element.
An object of uniform cross section has a resistance proportional to its resistivity
and its length, and inversely proportional to its cross-sectional area.
Almost all materials have some resistance, except for superconductors.
The resistance, R, of an object is defined as the ratio of the voltage across (V) it
to the current through it (I). The unit of resistance is the Ohm (Ω).
i.e.
𝑅=
𝑉
𝐼
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
The Sliding Conducting Bar
A bar moving through a uniform field is
shown (to the left) and the equivalent
circuit diagram is shown (to the right).
The wiggly line is a resistor, which has
resistance R.
Assume that the bar has zero resistance.
The stationary part of the circuit has a resistance R.
The work done by the applied force appears as internal energy in the resistor.
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
The Sliding Conducting Bar
The magnetic flux is
𝛷𝐵 = 𝐵𝑙𝑥
The induced emf is therefore:
𝑑𝛷𝐵
𝑑
𝑑𝑥
𝜀=−
=−
𝐵𝑙𝑥 = −𝐵𝑙
= −𝐵𝑙𝑣
𝑑𝑡
𝑑𝑡
𝑑𝑡
Using our equation for resistance, the current is therefore:
𝜀
𝐵𝑙𝑣
𝐼=
=
𝑅
𝑅
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
The Sliding Conducting Bar: Forces
The applied force, Fapp, does work on the conducting bar.
This moves the charges through a magnetic field.
The magnetic force, 𝐹𝐵 = 𝐵𝐼𝑙 opposes the motion
Its direction is opposite to the applied force (check with the right hand rule!)
Since the bar is moving at constant speed (i.e. we have no acceleration), we
must have
𝐹𝑎𝑝𝑝 = 𝐹𝐵 = 𝐵𝐼𝑙
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
The Sliding Conducting Bar: Energy Considerations
The change in energy must be equal to the transfer of
energy into the system by the work done by the applied
force on the conducting bar.
So we’d have to model our circuit as a non-isolated system.
The power input is equal to the rate at which energy is
delivered to the resistor
Therefore:
(since we said 𝐼 =
𝐵𝑙𝑣
𝑅
)
𝐵2 𝑙2 𝑣 2 𝜀 2
𝑃 = 𝐹𝑎𝑝𝑝 𝑣 = 𝐵𝐼𝑙 𝑣 =
=
𝑅
𝑅
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Quick Quiz
As an aircraft flies from Sydney to Melbourne, it passes through the Earth’s
magnetic field, which is directed upwards.
As a result, a motional emf is developed between the wingtips.
Which wingtip is positively charged?
a) The left wing
b) The right wing
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Quick Answer
As an aircraft flies from Sydney to Melbourne, it passes through the Earth’s
magnetic field, which is directed upwards. As a result, a motional emf is
developed between the wingtips. Which wingtip is positively charged?
b) The right wing
The Earth’s magnetic field has an upward component in the southern
hemisphere. As the plane flies south, the right-hand rule indicates that positive
charges will experience a force directed to the west.
Thus, the right wingtip becomes positively charged, and the left wingtip
becomes negatively charged.
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Quick Quiz
In the figure, an applied force of magnitude Fapp results in
a constant speed v and a power input P.
Imagine that the force is increased so that the constant
speed of the bar is doubled to 2v.
Under these conditions, the new force and the new power
input are:
a) 2F and 2P
b) 4F and 2P
c) 2F and 4P
d) 4F and 4P
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Faraday’s Law
Quick Answer
Imagine that the force is increased so that the constant
speed of the bar is doubled to 2v. Under these conditions,
the new force and the new power input are:
c) 2F and 4P
The force on the wire is of magnitude 𝐹𝑎𝑝𝑝 = 𝐹𝐵 = 𝐼𝑙𝐵,
𝐵𝑙𝑣
with 𝐼 =
.
𝑅
Thus, the force is proportional to the speed and the force
doubles.
Because 𝑃 = 𝐹𝑎𝑝𝑝 𝑣, the doubling of the force and the
speed results in the power becoming four times as large.
Electricity & Magnetism – Induction & Inductance
27/08/2013
PHYSICS 1B – Lenz's Law
Lenz’s Law
Faraday’s law indicates that the induced emf and the change in flux have
opposite algebraic signs:
𝑑𝛷𝐵
𝜀=−
𝑑𝑡
This has a physical interpretation that has come to be known as Lenz’s law.
Developed by German physicist Heinrich Lenz (1804 – 1865).
Also carried out research into the physical properties of
seawater, as well as the climate around the world.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Lenz’s Law
Lenz’s law: the induced current in a loop is in the direction that creates a
magnetic field that opposes the change in magnetic flux through the area
enclosed by the loop.
In other words, the induced current tends to keep the
original magnetic flux through the circuit from changing.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Lenz’s Law, Example
The conducting bar slides on the two fixed conducting rails.
The magnetic flux due to the external magnetic field through the enclosed area
increases with time as the bar is slid to the right.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Lenz’s Law, Example
Since the flux through the enclosed area is increasing, the induced current must
produce a magnetic field out of the screen.
From the right hand rule, we therefore see that the induced current must be
counter-clockwise.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Lenz’s Law, Example
If we were to move the bar in the opposite direction (panel b), then the flux
through the loop decreases with time.
The direction of the induced current would therefore be clockwise, producing a
magnetic field into the screen.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Induced Current Directions: Example
A magnet is placed near a metal loop.
If the magnet is pushed towards the loop
(panel a), the induced current produces a
field directed to the left to counteract the
increasing external flux (panel b).
If the magnet is moved away from the loop
(panel c), then the induced current produces
a field directed to the right to counteract the
decreasing external flux (panel d)
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Quick Quiz
In the figure, a magnet is being moved in the vicinity of a solenoid that is
connected to a very sensitive ammeter.
The south pole of the magnet is nearest the
solenoid, and the ammeter indicates a
clockwise current (as viewed from above) in
the solenoid.
The person is:
a) Inserting the magnet
b) Pulling it out
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Quick Answer
The person is:
a) Inserting the magnet
Because the current induced in the solenoid
is clockwise, as viewed from above, the
magnetic field lines produced by the current
point downwards in the figure.
Thus, the upper end of the solenoid must be
acting as a magnetic south pole.
For this situation to be consistent with Lenz’s law, the south pole of the bar
magnet must be approaching the solenoid. Remember: like poles repel!
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Quick Quiz
In this figure, a circular loop of wire is being dropped towards a wire that carries
a current that is flowing to the left.
The direction of the induced current in the loop of wire is:
a) Clockwise
b) Counter-clockwise
c) Zero
d) Impossible to determine
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Quick Answer
The direction of the induced current in the loop of wire is:
b) Counter-clockwise
At the position of the loop, the magnetic field lines
point into the page. The loop is entering a region of
stronger magnetic field as it drops towards the wire,
so the flux is increasing.
The induced current must set up a magnetic field
that opposes this increase.
To do this, it creates a magnetic field directed
out of the page.
This requires a counter-clockwise current in the loop (from the right-hand rule).
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
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.
The induced field is non-conservative, unlike the electric field produced by
stationary charges.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Induced emf and Electric Fields
The emf, ε, for any closed path can be expressed as the line integral of E.ds over
that path - i.e.
𝜀=
𝑬. 𝑑𝒔
Therefore, we can write Faraday’s law in a general form:
𝑑𝛷𝐵
𝑬. 𝑑𝒔 = −
𝑑𝑡
The field cannot be an electrostatic field (i.e. one created by a charge), since
electrostatic fields are conservative, and therefore the integral around a closed
path would be zero. It isn’t!
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Quick Quiz
In a region of space, the magnetic field increases at a constant rate.
This changing magnetic field induces an electric field that:
a) Increases in time
b) Is conservative
c) Is in the direction of the magnetic field
d) Has a constant magnitude
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Quick Answer
In a region of space, the magnetic field increases at a constant rate. This
changing magnetic field induces an electric field that:
d) Has a constant magnitude
The constant rate of change of B will result in a constant rate of change of the
magnetic flux.
According to our general version of Faraday’s law,
constant, then E is constant in magnitude.
Electricity & Magnetism – Lenz's Law & Generators
𝑬. 𝑑𝒔 = −
𝑑𝛷𝐵
𝑑𝑡
, if
𝑑𝛷𝐵
𝑑𝑡
is
27/08/2013