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Lecture 9
Magnetic field
Magnetic fields are everywhere. Since ancient times the magnetic field
of the Earth was used for orientation and magnetic stones
(magnetite) were known.
The magnetic field of a magnet is apparently similar to that of an
electric dipole. However, inside the magnet the lines point in
opposite way as for the dipole. The lines of magnetic field are
closed, there are no elementary magnetic charges.
Theoretically there are reasons why one might expect that magnetic
monopoles exist but they have not been seen experimentally (and
would be very exotic object anyway).
The elementary magnet is the electron. Not only has electric charge
but it also acts as a tiny magnet. A proton is also a tiny magnet but
its strength is 2000 times smaller. A macroscopic magnet has a
significant fraction of its electrons aligned in the same direction.
N
S
The lines of magnetic field are closed. There are no magnetic
charges where the lines can start or end.
No magnetic monopoles!!
We learn about magnetic fields by the force it
produces on electric charges. However, for a
charge to feel a force in magnetic field, the
charge has to be moving.
The force (called Lorentz force) is given by:
Where
product):
is a vector given by (vector
Right hand rule
Clicker question
The magnetic field is measured in T (Tesla). What
should that unit correspond to?
A.
B.
C.
D.
(N C) / m
m / (C s)
(N s) / (C m)
(N m) / (C s)
Some comments
• In the charge is not moving (v=0) there is no
magnetic force
• If the velocity is parallel to the magnetic field
then there is no force (q=0).
• The magnetic force is always perpendicular to
the velocity.
• Unit of magnetic field are T (Tesla) given by:
Earth: 30 mT
Lab: 10 T
• A charge moving in a magnetic field and in the
absence of other forces, describes a circle.
Cyclotron frequency
Lecture 10
Magnetic force on a current
A current is charges in motion. Therefore a wire through which a
current circulates will experience a force if inside a magnetic
field. To compute it we need to know the velocity.
Force per unit length
Again we use right hand rule to find the direction of
the force.
Clicker question
A cable carrying a current I=10A is located inside a
perpendicular magnetic field B=0.1T.
If the cable is 10cm long, compute the magnetic force
acting on the cable.
A.
B.
C.
D.
10 N
1N
0.1 N
0.01 N
F = I L B = 10A 0.1T 0.1m=0.1 N
• Interesting observation: Hall effect
• It turns out that the magnetic force can be used
to distinguish if the carriers of the current have
positive or negative charge. It turns out that
they have negative, they are electrons.
Sources of magnetic field.
• We discussed permanent magnets produce magnetic fields,
the electron itself being a tiny magnet.
• However it turns out that there is another way to produce a
magnetic field and that is by charges in motion. A charge in
motion not only feels a magnetic force, it produces a
magnetic field of its own!
• Again the most common case is that of wires with currents.
• The intensity of the magnetic field is given by Ampere’s law.
Ampere’s law
• The lines of magnetic field are closed so the flux through a surface is zero.
We first need to introduce the concept of circulation instead. It is similar
to the concept of work done by a force.
• Ampere’s law:
• Where I is the current that pierces the surface surrounded by the loop.
• Example: magnetic field of a straight wire
• Example II: Magnetic field of a solenoid
Displacement current
• There is an ambiguity when considering which surface is surrounded
by a loop. It seems trivial, but lead Maxwell to the amazing
discovery of the fact that a changing electric field also produces a
magnetic field. This effect is called the displacement current.
Lecture 11
Force between currents
• A current creates a magnetic field, another current feels a force in
such magnetic field therefore there are magnetic forces between
currents.
Clicker question
Two parallel cables carry each a current I=10A. If
they are 1m long and are separated by a distance
of 1cm, compute the force between them?
A.
B.
C.
D.
2x10-1 N
2x10-3 N
2x10-5 N
2x10-7 N
Magnetic induction
• If a cable moves in a magnetic field, the charges inside it will feel a
force and then a current will circulate. A cable moving in a
magnetic field acts as a battery!!
L
• One can increase the effect by using a coil, that
is many conductors:
If we move the coil we have a emf in the
circuit (as before).
However if we move the magnet the same
effect appears!
Both effects can be summarized in the Faraday and Lenz law:
The emf (electro motive force) generated by a magnetic field is
equal, in magnitude to the rate of variation of the magnetic flux
through the circuit. Its polarity is such that the current generated
in the loop opposes the change in flux
• Example
+
Same as before!!
Inductors
• A solenoid (or coil) when part of a circuit is also called an inductor.
It stores energy in a magnetic field (similarly as a capacitor stores
energy in an electric field).
l
• What happens if the current changes?
Inductance:
Units: 1H (Henry) = 1 V s/A
A: Cross section,
l : length,
N: number of turns.
Comparison with capacitor
• Remember that for a capacitor
Summary:
Capacitor:
Inductor:
Transformers
• Transformers are used to increase or decrease the voltage for
different applications.
• Consider a load with power P. If the power source of voltage V
is connected to the load with a cable of small resistance R, the
power dissipated in the cable is Pc=I2 R. If R is small then
Pload=I V and we have Pc = Pload2 R / V2.
So the larger the V, the smaller the power lost in the cable!!.
Clicker question
We need a transformer to reduce the voltage from
120V to 12V. The primary has 1000 turns.
The secondary should have
A. 10 turns
B. 100 turns
C. 1000 turns
D. 10000 turns
Lecture 12
RL circuits
In the same way that we charge a capacitor using a battery we
can “charge” (store energy) an inductor. The basic circuit
contains simply a battery, a resistor and an inductor. It is
called RL circuit.
The main difference with an RC circuit is that the inductor
opposes any sudden change in current, so the current slowly
rises to its final value.
Understanding the RL circuit.
The end result is a current circulating through the circuit and
equal to:
Initially, the inductor opposes any change in current, so the full
voltage of the battery shows across the inductor and we have:
We can approximate the time it takes to charge as:
Comparison between L and C:
Initially the capacitor behaves as a short circuit. The current
appear to flow unimpeded through the capacitor (although in
reality charge is accumulating and not going through).
Initially the inductor behaves as an open circuit. Current does
not flow through it (although current could flow a voltage is
generated that opposes it).
In the final stage, the capacitor behaves as an open circuit. The
only way to have current flowing is to change the external
voltage.
In the final stage, the inductor behaves as an short circuit. The
inductor behaves as a cable with negligible resistance.
Application: ElectroMagnetic braking
• Braking by definition is to slow down a vehicle.
• Since energy is conserved, the kinetic energy is
usually converted into heat. In a regular car this
happens at the brake pads by friction.
• Magnetic braking works by attaching a magnet to
the wheels and moving it in the presence of a
conductor (coil). The current can be converted
into heat by Joule effect.
• Alternatively, the current can be used to charge a
battery and the energy can be recovered
increasing the efficiency of the vehicle.
Clicker question
An RL circuit has an inductor of L=10H and a
resistor of 100W.
What it the time constant associated with this
circuit?.
A.
B.
C.
D.
0.01 sec
0.1 sec
1 sec
10 sec
Lecture 13
AC generator
• Moving a cable in a magnetic field acts as a battery. If it moves
in circle the motion can be continuous:
• Looking more carefully, however, the polarity of
the battery flips at each half turn since the flux
increases and decreases.
Analogy:
• Let us look at how the polarity changes:
Compute by analogy using circular motion.
• Going back to the original calculation
• This is called an AC generator. The polarity of
the current flips every half period. For example
household current alternates 60 times a
second. This called frequency and is measured
in Hz (Hertz).
We usually write:
It repeats itself if we do
Therefore:
AC resistor circuit.
To average the power we use
V0: peak
Lecture 14
AC circuits: Capacitors and Inductors
• One can understand what happens by the
following reasoning: If the frequency is very large,
the capacitor has no time to charge and behaves
as an open circuit. If the frequency is very slow,
the capacitor will always be fully charged and no
current will go through:
• Two circuits:
Another example: AC coupling
• An inductor works in the opposite way. It opposes
any sudden changes in current, so if the frequency
is high, it will generate an opposing voltage and
behave as an open circuit. If the changes are very
slow, the magnetic flux changes very slowly, no
voltage is generated and therefore behaves as a
cable.
Clicker question
The following circuit acts as :
A. High pass filter
B. Low pass filter
• Two circuits:
Another example: Rectifier
Capacitor to ground helps to eliminate line frequency.