Download Jonti`s sixth lecture (Generators...energy in a magnetic field)

PHYSICS 1B
Today’s lecture:
Generators
Eddy Currents
Self Inductance
Energy Stored in a Magnetic Field
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Lenz's Law
Generators
Electric generators take in energy by work and transfer it out by electrical
transmission.
The AC generator consists of a loop of wire rotated by some external means in a
magnetic field.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
A Rotating Loop
Assume a loop with N turns, all of the same area,
rotating in a magnetic field.
The flux through one loop at any time, t, (using the
fact that 𝜃 = 𝜔𝑡 ):
𝛷𝐵 = 𝐵𝐴 cos 𝜃 = 𝐵𝐴 cos 𝜔𝑡
Therefore, the induced emf of the loop is:
𝜀 = −𝑁
i.e.
𝑑𝛷𝐵
𝑑𝑡
= −𝑁𝐵𝐴
𝑑
(cos 𝜔𝑡)
𝑑𝑡
𝜀 = 𝑁𝐴𝐵𝜔 sin 𝜔𝑡
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
A Rotating Loop
𝜀 = 𝑁𝐴𝐵𝜔 sin 𝜔𝑡
The relationship is sinusoidal, with a maximum value,
εmax given by:
𝜀max = 𝑁𝐴𝐵𝜔
εmax occurs when 𝜔𝑡 = 90⁰ or 2700
i.e. it occurs when the magnetic field is in the plane of
the coil, and the rate of change of the flux is at a
maximum.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
DC Generators
The DC (direct current) generator has essentially the same components as the
AC generator.
The main difference is that the contacts to the rotating loop are made using a
split ring called a commutator.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
DC Generators
In this configuration, the output voltage always has the same polarity.
It pulsates with time, as was the case with the AC generator.
To obtain a steady DC current, commercial generators use many coils and
commutators distributed so the pulses are out of phase.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Motors
Motors are devices into which energy is transferred by electrical transmission
while energy is transferred out by work.
In other words, a motor is a generator operating in reverse.
A current is supplied to the coil by a battery, and the torque acting on the
current-carrying coil causes it to rotate.
Useful mechanical work can be done by attaching the rotating coil to some
external device.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Motors
However…
As the coil rotates in a magnetic field, an emf is induced.
This induced emf always acts to reduce the current in the coil.
This is called a “back emf”
The back emf increases in magnitude as the rotational speed of the coil
increases.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Motors
The current in the rotating coil is therefore limited by the back emf.
The term back emf is commonly used to indicate an emf that tends to reduce the
supplied current.
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.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Hybrid Drive Systems
In a car with a hybrid drive system, a petrol engine and an electric motor are
combined to increase the fuel economy of the vehicle and reduce its emissions.
Power to the wheels can come from the petrol engine or the electric motor.
In normal driving, the electric motor accelerates the vehicle from rest to about
15 mph.
During these acceleration periods, the engine isn’t running, so petrol is not used,
and there is no emission.
At higher speeds, the motor and engine work together so that the engine is
always operating at, or near, its most efficient speed.
This results in a greatly improved fuel efficiency than that of a traditional car.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Quick Quiz
In an AC generator, a coil with N turns of wire spins in a magnetic field.
Of the following choices, which will not cause an increase in the emf generated
in the coil?
a) Replacing the coil wire with one of lower resistance
b) Spinning the coil faster
c) Increasing the magnetic field
d) Increasing the number of turns of wire on the coil
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Quick Answer
In an AC generator, a coil with N turns of wire spins in a magnetic field.
Of the following choices, which will not cause an increase in the emf generated
in the coil?
a) Replacing the coil wire with one of lower resistance
While reducing the resistance may increase the current that the generator
provides to a load, it does not alter the emf.
Since 𝜀 = 𝑁𝐴𝐵𝜔 sin 𝜔𝑡 , the emf depends on ω, B, and N, and so all the other
choices increase the emf.
Electricity & Magnetism – Lenz's Law & Generators
27/08/2013
PHYSICS 1B – Lenz's Law
Eddy Currents
Circulating currents called eddy currents are induced in
bulk pieces of metal moving through a magnetic field.
From Lenz’s law, their direction is to oppose the change
that causes them.
The eddy currents are in opposite directions as the plate
enters or leaves the field.
Eddy currents are often undesirable because they
represent a transformation of mechanical energy into
internal energy.
Electricity & Magnetism – Lenz's Law & Generators
28/08/2013
PHYSICS 1B – Lenz's Law
Eddy Currents - Example
The magnetic field here is directed into the screen.
The induced eddy current is counter-clockwise as the
plate enters the field.
It is opposite when the plate leaves the field (clockwise).
The induced eddy currents produce a magnetic retarding
force, and the swinging plate eventually comes to rest.
Electricity & Magnetism – Lenz's Law & Generators
28/08/2013
PHYSICS 1B – Lenz's Law
Eddy Currents - Final
There are a couple of ways to reduce the energy loses
resulting from eddy currents.
The conducting parts can:
• Be built up in thin layers separated by a nonconducting material
• Have slots cut in the conducting plate
Both these tricks prevent large current loops from
forming, and therefore increase the efficiency of the
device.
Electricity & Magnetism – Lenz's Law & Generators
28/08/2013
PHYSICS 1B – Lenz's Law
Quick Quiz
In equal-arm balances from the early twentieth century, it is
sometimes observed that an aluminium sheet hangs from one
of the arms and passes between the poles of a magnet.
This causes the oscillations of the equal-arm balance to decay
rapidly, when otherwise (without this magnetic braking), the
oscillation might continue for a long time.
The oscillations decay because:
a) The aluminium sheet is attracted to the magnet
b) Currents in the aluminium sheet set up a
magnetic field that opposes the oscillations
a) Aluminium is paramagnetic
Electricity & Magnetism – Lenz's Law & Generators
28/08/2013
PHYSICS 1B – Lenz's Law
Quick Answer
The oscillations decay because:
b) Currents in the aluminium sheet set up a magnetic field that
opposes the oscillations
When the aluminium sheet moves between the poles of the
magnet, eddy currents are established in the aluminium.
According to Lenz’s law, these currents are in a direction
so as to oppose the original change, which is the
movement of the aluminium sheet in the magnetic
field.
Electricity & Magnetism – Lenz's Law & Generators
28/08/2013
PHYSICS 1B – Lenz's Law
Applied Eddy Currents
Eddy currents aren’t always a bad thing.
In fact, they’re commonly used to look for unseen cracks in structures, such as
the wings of aircraft.
Eddy currents are induced in the structure. When the eddy current falls off, then
a crack is probably present, stopping the current!
Electricity & Magnetism – Lenz's Law & Generators
28/08/2013
PHYSICS 1B – Lenz's Law
Charging by induction
Induced emf is used to wirelessly recharge things.
An alternating current in the base induces an emf in
the device being charged.
The alternating current thus created is used to charge
the device’s battery.
The big advantage is – there are no exposed wires! So
no corrosion of the parts, and no chance of an electric
shock in the wet.
But – this is slower and less efficient…
This is very handy from a health and safety point of view.
Electricity & Magnetism – Lenz's Law & Generators
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Self-Inductance
When the switch is closed, the current does not
𝜀
immediately reach its maximum value of due
𝑅
to self-inductance.
There is also a self-induced emf, 𝜀𝐿
We use Faraday’s law to describe the effect.
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Self-Inductance
As the current increases with time, the magnetic
flux induced through the circuit also increases
with time.
This induces an emf in the circuit.
The direction of the induced emf is to cause an
induced current which establishes a magnetic
field opposing the change in the field.
i.e. opposing the direction of emf in the battery
The result is a gradual increase in the current to its final equilibrium value.
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Self-Inductance
So – a time-varying current in a circuit produces an induced emf opposing the
emf that initially set up the time-varying current.
This is the basis of the electrical circuit element called an inductor.
Energy is stored in the magnetic field of an inductor.
There is an energy density associated with the magnetic field.
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Self-Inductance in a Coil
a) A current in the coil produces a magnetic field directed towards the left
b) If the current increases, the increasing flux causes an induced emf
c) The polarity of the induced emf reverses if the current decreases
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Equations for Self-Inductance
The induced emf is proportional to the rate of change of the current:
𝑑𝐼
𝜀𝐿 = −𝐿
𝑑𝑡
Here, L is a constant of proportionality called the inductance of the coil.
The inductance of a coil depends on the geometry of the coil and its physical
characteristics.
The unit of inductance is the henry, H. 1H = 1
Electricity & Magnetism
V s
A
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Joseph Henry
1797 – 1878
American physicist.
First director of the Smithsonian
First president of the Academy of Natural Science
Improved the design of the electromagnet
Constructed one of the first motors
Discovered self-inductance, but didn’t publish his results!
Despite this, the unit of inductance is named in his honour…
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Terminology
We use emf and current when they are caused by batteries or other sources.
We use induced emf and induced current when they are caused by changing
magnetic fields.
When dealing with problems in electromagnetism, it is important to distinguish
between the two situations…
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Inductance of a Coil
Say we have a closely spaced coil, with N turns, through which a current I is
flowing. From Faraday’s law, we know:
𝑑𝛷𝐵
𝜀𝐿 = −𝑁
𝑑𝑡
We also know that 𝜀𝐿 = −𝐿
𝑑𝐼
𝑑𝑡
, so subbing this in…
𝑑𝐼
𝑑𝛷𝐵
𝐿 =𝑁
𝑑𝑡
𝑑𝑡
∴ 𝐿𝑑𝐼 = 𝑁𝑑𝛷𝐵
𝑁𝛷𝐵
𝜀𝐿
∴𝐿=
=−
𝑑𝐼
𝐼
𝑑𝑡
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Inductance of a Coil
𝑁𝛷𝐵
𝜀𝐿
𝐿=
=−
𝑑𝐼
𝐼
𝑑𝑡
The inductance is therefore a measure of the opposition to a change in the
current.
This can be compared to the resistance, which is a measure of the opposition to
the current itself…
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Inductance of a Solenoid
Now consider a uniformly wound solenoid which has N turns and length 𝑙. We
then have:
𝑁
𝐵 = 𝜇0 𝑛𝐼 = 𝜇0 𝐼
𝑙
We know 𝛷𝐵 = 𝐵𝐴, so therefore:
𝑁𝐴
𝛷𝐵 = 𝐵𝐴 = 𝜇0
𝐼
𝑙
Therefore:
𝑁𝛷𝐵 𝜇0 𝑁 2 𝐴
𝐿=
=
𝐼
𝑙
Note how the inductance just depends on the geometry – just like capacitance!
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Quick Quiz
The circuit shown in the figure consists of a resistor, an inductor, and an ideal
battery that has no internal resistance.
At the instant just after the switch is closed, across which circuit element is the
voltage equal to the emf of the battery?
a) The resistor
b) The inductor
c) Both the inductor and resistor
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Quick Answer
The circuit shown in the figure consists of a resistor, an inductor, and an ideal
battery that has no internal resistance.
At the instant just after the switch is closed, across which circuit element is the
voltage equal to the emf of the battery?
b) The inductor
As the switch is closed, there is no current, so
there is no voltage across the resistor
(remember, 𝑉 = 𝐼𝑅 )
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Quick Quiz
After a very long time, across which circuit element is the voltage equal to the
emf of the battery?
a) The resistor
b) The inductor
c) Both the inductor and resistor
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Quick Answer
After a very long time, across which circuit element is the voltage equal to the
emf of the battery?
a) The resistor
After a long time, the current has reached its
final value, and the inductor has no further
effect on the circuit…
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Quick Quiz
A coil with zero resistance has its ends labelled a and b.
The potential of the induced emf at a is higher than at b.
Which of the following could be consistent with this situation?
a) The current is constant and directed from a to b
b) The current is constant and directed from b to a
c) The current is increasing and is directed from a to b
d) The current is decreasing and is directed from a to b
e) The current is increasing and is directed from b to a
f)
The current is decreasing and is directed from b to a
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Quick Answer
A coil with zero resistance has its ends labelled a and b. The potential of the
induced emf at a is higher than at b. Which of the following could be consistent
with this situation?
d) The current is decreasing and is directed from a to b
e) The current is increasing and is directed from b to a
For the constant current in a) and b), there is no potential difference across the
resistance-less inductor.
In e), if the current increases, the emf induced in the inductor is in the opposite
direction, therefore from a to b, making a higher in potential than b.
Similarly, in d), the decreasing current induces an emf in the same direction as
the current, from a to b, again making the potential higher at a than b
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Energy in a Magnetic Field
In a circuit with an inductor, the battery must supply more energy than in a
circuit without an inductor.
Part of the energy supplied by the battery appears as internal energy in the
resistor.
The remaining energy is stored in the magnetic field of the inductor.
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Energy in a Magnetic Field
The power supplied by the battery is equal to the dissipation across the resistor,
of resistance R, plus the rate at which energy is stored in the inductor, of
inductance L. Therefore:
𝜀𝐼 =
𝐼2𝑅
𝑑𝐼
+ 𝐿𝐼
𝑑𝑡
In this equation:
• 𝜀𝐼 is the rate at which energy is supplied by the battery
• 𝐼 2 𝑅 is the rate at which the energy is being delivered to the resistor,
𝑑𝐼
• So, therefore, 𝐿𝐼
must be the rate at which energy is being stored in the
𝑑𝑡
magnetic field
Electricity & Magnetism
28/08/2013
PHYSICS 1B – Self Inductance and Energy Stored in a Magnetic Field
Energy in a Magnetic Field
If we let U denote the energy stored in the inductor at any time, then the rate at
which the energy is stored is:
𝑑𝑈
𝑑𝐼
= 𝐿𝐼
𝑑𝑡
𝑑𝑡
To find the total energy, we therefore integrate:
𝐼
1 2
𝑈 = 𝐿 𝐼𝑑𝐼 = 𝐿𝐼
2
0
(compare this result to that for electric fields in a capacitor: 𝑈𝐸 =
Electricity & Magnetism
1 𝑄2
2 𝐶
)
28/08/2013