1. 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.

2. N S
The lines of magnetic field are closed. There are no magnetic charges where the lines can start or end.

3. No magnetic monopoles!!

4. 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 is a vector given by (vector product):

5. Right hand rule

6. Clicker question
The magnetic field is measured in T (Tesla). What should that unit correspond to?
(N C) / m
m / (C s)
(N s) / (C m)
(N m) / (C s)

7. 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: T

8. A charge moving in a magnetic field and in the absence of other forces, describes a circle.
Cyclotron frequency

9. 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

10. Again we use right hand rule to find the direction of the force.

11. Clicker question F = I L B = 10A 0.1T 0.1m=0.1 N
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.
10 N
1 N
0.1 N
0.01 N
F = I L B = 10A 0.1T 0.1m=0.1 N

12. 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.

13. 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.

14. 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.

15. Example: magnetic field of a straight wire

16. Example II: Magnetic field of a solenoid

17. 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.

18. 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.

19. 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?
2x10-1 N
2x10-3 N
2x10-5 N
2x10-7 N

20. 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

21. 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!

22. 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

23. Example
Same as before!! - +

24. 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

25. What happens if the current changes?
Inductance:
Units: 1H (Henry) = 1 V s/A
A: Cross section,
l : length,
N: number of turns.

26. Comparison with capacitor
Remember that for a capacitor
Summary:
Capacitor:
Inductor:

27. Transformers

28. 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!!.

29. Clicker question
We need a transformer to reduce the voltage from 120V to 12V. The primary has 1000 turns.
The secondary should have
10 turns
100 turns
1000 turns
10000 turns

30. 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.

32. Understanding the RL circuit
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:

33. 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.

34. 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.

35. 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?.
0.01 sec
0.1 sec
1 sec
10 sec

36. 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:

37. Looking more carefully, however, the polarity of the battery flips at each half turn since the flux increases and decreases.
Analogy:

38. Let us look at how the polarity changes:

39. Compute by analogy using circular motion.

40. 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).

41. We usually write:
It repeats itself if we do
Therefore:

42. AC resistor circuit.
To average the power we use
V0: peak

43. Lecture 14 AC circuits: Capacitors and Inductors

44. 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:

45. Two circuits:

46. Another example: AC coupling

47. An inductor works in the opposite way
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.

48. Clicker question The following circuit acts as : High pass filter
Low pass filter

49. Two circuits:

50. Another example: Rectifier
Capacitor to ground helps to eliminate line frequency.