How does a vending machine choose the real coins from the slugs? Although it seems quite unrelated to the topic of our discussion in this chapter, we can answer the question as follows.

In coin operated
vending machines, eddy currents are used to detect counterfeit coins, or slugs. The coin rolls past a stationary magnet, and eddy currents slow its speed. The strength of the eddy currents, and thus the amount of slowing, depends on the conductivity of the coin's metal. Slugs are slowed to a different degree than genuine coins, thus they are directed to the rejection slot.”



Vending_machine.jpg


This chapter tells us about the conversion of energy from one to another. The example above shows us the conversion of mechanical energy to a different form.


Induction and Inductance


In the previous chapter, we determined that a current produces a magnetic field. This came as a surprise to the scientists who first discovered the effect. What was even more surprising was the discovery of the reverse effect: a magnetic field can produce an electric field. The link between a magnetic field and an electric field is now referred to as Faraday’s Law of Induction. Although it appears complicated, examples of this basic science are almost everywhere.

Experiment


This experiment shows a conducting loop connected to a sensitive ammeter. Because there is no battery or any other source of emf, we can conclude that there is no current in the circuit. Once a bar magnet is moved towards the loop, a current appears in the circuit. Once the magnet is moved away, a current suddenly appears, however it is now in the opposite direction. After a while, the following would be discovered:

1. A current appears only if there is a relative motion between the loop and the magnet; the current disappears when the relative motion between them ceases.
2. Faster motion produces a greater current.
3. If moving the magnet’s north pole toward the loop causes clockwise current, then moving the north pole away causes counterclockwise current.

30.1.bmp
Induced Current – the current produced in the loop.
Induced emf – the work done per unit charge to produce that current.
Induction – the process of producing the current and emf.

Faraday’s Law of Induction

Michael Faraday discovered that an emf and a current can be induced in a loop (as previously discussed) by just changing the amount of magnetic field flowing through the loop. However, Faraday’s law doesn’t explain why a current and an emf are induced; it only helps in visualizing the induction.

In order to calculate the amount of magnetic field that passes, the amount of electric field that passes through a surface must be calculated as well.

30.2.bmp

When calculating the magnetic flux through the loop enclosing an area A, the equation is:30.3.bmpWhere Da is a vector of magnitude dA that is perpendicular to a differential area dA

From the equation, it is visible that the SI unit for magnetic flux is the tesla-square meter, which is called the weber (abbreviated Wb):
1 weber = 1Wb = 1 T·m2
The magnitude of emf E induced in a conducting loop = the rate at which magnetic flux øB through that loop changes with time.
Faraday’s Law is formally written as:
faradaylaw.bmp

If we change the magnetic flux through a coil of N turns, the total emf induced in these coils is the sum of these individual induced emfs.
faradaylaw2.bmp

Lenz’s Law

After Faraday’s law of induction, Heinrich Lenz contrived a rule for determining the direction of an induced current in a loop. When a magnetic field travels through a ring, EMF appears around the ring. For example, if you are observing a magnetic field passing through a ring, and then remove the energy source, the magnetic field will still be present for a limited amount of time. Although the power source was removed, the emf causes the magnetic field to remain unchanged for a limited period of time.
_1.bmp

In this figure, the magnetic field on the ring is being increased. The induced emf will cause the situation to remain as is, causing an induced magnetic field to be created, which then counters the increasing magnetic field. The right-hand rule gives the direction of the induced current based on the direction of the induced field. For example, if you were to place your right hand around the ring, your right thumb would point in the direction of the induced magnetic field.



Electric Guitars

_2.bmpUnlike an acoustic guitar, the sound from an electric guitar does not come from the hollow body. The oscillations of the metal strings are sensed by electric pickups. The pickups send signals to an amplifier and a set of speakers. Wire connecting the instrument to the amplifier is coiled around a small magnet. The magnetic field produces a north and south pole in the string just above the magnet. This section of the string above the magnet has its own magnetic field. As the string oscillates, the induced current changes direction at the same frequency as the oscillations of the strings. This process relays to the amplifier and speakers causing the distinctive sound to be heard. Most electric guitars have three groups of pickups. The group closest to the near end better detect high frequency oscillations, while the group farthest from the near end better detects the low-frequency oscillations. By activating a switch (located on the guitar) the musician is able to determine which group sends the signals to the amplifier, allowing for a wide range of sounds.

Induction and Energy Transfers

_3.bmpNo matter how current is induced in a loop, energy is ALWAYS transferred to thermal energy during the process. The figure to the right illustrates a situation concerning induced current. As the hand removes the conducting loop from the magnetic field at constant velocity (marked as V→), a clockwise current is induced in the loop (indicated as i). The loop segments that are still within the magnetic field experience force F1-F3.



To pull the loop at a constant velocity ( V), constant force must be applied as well. This is because the magnetic force acts in the opposite direction, but in equal magnitude to counter your pull. The rate at which you do work is power, shown by P = Fv, where F is the magnitude of the force. To find the current, Faraday’s law must be applied. When x = length of loop still in the magnetic field, the area still in the magnetic field = Lx. The magnitude of the flux through the loop is determined by:
_4.bmp
As the length of the loop still in the magnetic field decreases, the flux decreases. Faraday’s law explains that the flux decrease causes an emf to be induced in the loop. By using a similar equation (changing the equation only by dropping the (-) sign in Faraday’s Law) we are able to write the magnitude of the emf:
_5.bmp
Where dx/dt is replaced with v (velocity).

In the figure above, the segments marked F1, F2, and F3 are deflecting forces. F2 and F3 are both pulling in opposite directions and are equal in magnitude. This causes them to cancel each other out, leaving only the force of F1, which is the force pulling opposite your pull (_6.bmp)

The actual rate at which you do work on the loop as you pull it from the magnetic field can be determined through:
__1.bmp
B, L, and R are constants, so v at which the loop is moved is constant if the magnitude you apply is constant as well. Below is the rate at which thermal energy appears in the loop.
__2.bmp
(thermal energy rate)

Induced Electric Fields

_+1.bmp

In (a): Place a copper ring in a uniform external magnetic field. Then increase the strength of the field at a steady rate.


As previously discussed, if there is a current in the copper ring, then there also must be an electric field must also be present. This is because the conduction electrons are moved by the electric field. This field is known as an induced electric field.


If the magnetic field is increasing with time, the field represented by circular lines in Figure C will be present. If the field is decreasing, the lines will be present, but in the opposite direction. Because a changing magnetic field produces an electric field, the electric field lines will not be present if the magnetic field remains constant.

Inductance of a Solenoid

A long solenoid like the one shown to the right is the basic type of inductor needed in order to produce a desired magnetic field. The inductance of the inductor is:
_01.bmp
The Unit is: 1 henry =1H = 1 T m2/A

Inductance of a solenoid is…
_02.bmp

_03.bmp

_04.bmp

_05.bmp


This equation represents the rate at which the emf device delivers energy to the rest of the circuit:
_05.bmp

While this equation represents the rate at which energy appears at thermal energy in the resistor:
_06.bmp

Energy delivered to the circuit, but is doesn’t appear as thermal energy must be stored in the magnetic field of the inductor.

Mutual Induction

As discussed earlier, if two coils are close together, a steady current i in one coil will set up a magnetic flux through the other coil. In doing so, this links the other coil. When current is changed with time, an emf given by Faraday’s law becomes apparent in the other coil. This process is referred to as induction. We use the term mutual induction to describe the interaction between the two coils, and to differentiate between self-induction (where only one coil is involved).

The mutual inductance M21 of coil 2 with respect to coil 1 as:
_0001.bmp align=”center”

_0002.bmp is a magnetic flux through coil 2 associated with the current in coil 1.*

The emf induced in either coil is proportional to the rate of change of current in the other coil. The proportionality constants M21 and M12 seem to be different.