From the previous arguments we have deduced that an electric current can produce a magnetic field, or, in field language, that an electric field can produce a magnetic field. What about the reverse? Can a Magnetic field produce a current? As explained below, these questions have affirmative answers.
If you move a magnet in and out of a solenoid attached to an amperometer (figure above), you can see the needle deflection in one direction when the magnet moves in, the needle deflection in the opposite direction when the magnet moves out. This means that a current flows in the coil of wire when the magnetic field produced by the magnet varies. Where does the electromotive force (voltage) responsible for the current in the coil come from? These experiments and many others led physicists to understand that this voltage depends not strictly on variation of the magnetic field but more precisely on the variation of the magnetic field flux through the solenoid.
Magnetic Flux
The magnetic flux through a loop of wire is a physical quantity giving
account for the number of magnetic field lines passing through the surface of
the loop. The magnetic flux is defined as follows:
where "B" is the magnetic field strength and "A" is the area of the loop surface oriented at a right-angle to the field.
The flux depends on the direction between the field and the surface too. For example if the loop surface is parallel to the direction of the field the net flux is zero because no field lines cross the loop surface. Anyway for our purposes the previous definition is enough.
Faraday-Neumann-Lenz law
Returning to our example of the solenoid, if it consists of a number
"N" of turns, and if it is subjected to a constant magnetic field
"B" perpendicular to its cross-sectional area, the total flux through
it will be:
Faraday and Neumann demonstrated that if this flux varies with time, a
potential difference 
is induced
in the solenoid and a current flows. The induced voltage is directly
proportional to the flux variation over time. Later, Lenz demonstrated that the
voltage difference induced opposes the change in the original magnetic flux.
Using maths language, these statements can be shortened by writing:
Faraday-Neumann-Lenz law
In other words, the more rapidly the magnetic flux through the solenoid changes, the higher the voltage in the wire and vice-versa.
This law underlies the most common devices we use everyday, for example generators and transformers.
Generators and Transformer
Generators are used to produce currents. These devices are based on
the movement of a coil of wire in a magnetic field; the mechanical movement
makes the magnetic flux through the coil vary, and in accordance with the
previous law this movement produces a voltage, hence a current.The smallest
generator is a bicycle dynamo, the biggest are the power plants used to supply
energy to our houses, but in spite of their different dimensions all generators
work on the same principle: the Faraday-Neumann-Lenz law.
Transformers are devices transforming the voltage difference from a value to another. For instance, I can’t connect my walkman directly to a wall outlet. My walkman works with a 3 Volt direct-voltage source, while the household power line supplies a 220 Volt alternating voltage. To make the connection we need an opportune adaptor. An adaptor is a device that:
- transforms the voltage from 220 Volts to 3 Volts.
- reverses from an alternating voltage to a steady voltage.
In other word the adaptor of my walkman is something more than a transformer, because it transforms the voltage but changes from A.C. to D.C. too. In any case, it works as a step-down transformer because the voltage changes from a higher value of 220V to a lower value of 3V.
Step-up transformers also exist; for example the ones which pass the voltage produced by power plants from a few volts to 230.000 volts. Such a step-up is desirable for long-distance transmission of electric power because in this way energy losses become negligible (The power in Physics is equal to the voltage times the current, hence, for a given power transmitted, the higher the voltage is, the lower the current and its loss due to resistive effects of the wires).
How does a transformer work?
A transformer can be sketched as in the figure below
A coil (primary) is wrapped around an iron core with a cross sectional area
"A"and an alternate voltage"
"
is fed into it. This sets up a varying magnetic field "B"inside the
iron. Such a varying field produces a varying flux through a second coil
(secondary) and in accordance with the Faraday-Neumann-Lenz law a potential
difference 
is induced in the
second coil.
If we write the law for each coil, we have
for the primary.
for the secondary.
The ratio between the two relations leads to
In other words the two potential differences are in the same ratio of the
number of turns of wire in the two coils. If we make
explicit in the last relation we have:
that means we can transform the input voltage
to a desired output voltage
only by adjusting the numbers of turns 
and
respectively in the primary and in the secondary coil. If is
bigger than the output voltage
will be bigger than the voltage in input: the transformer will be a step-up
transformer. On the other hand, if
is smaller than the output
voltage will be smaller, the transformer will be a step-down transformer. index