The strange phenomenon of magnetism has been recognized since antiquity. The ancient Chinese, the Greeks, Romans, and Europeans during the Age of Exploration all used the properties of lodestone in the construction of compasses and other navigational tools--important in the time when humanity didn't have Google maps or the Global Positioning System.
Elementary physics says that magnetic fields are caused by a moving charge; for a single point charge, the Biot-Savart law says the magnetic field vector B at a point that is distance r away from a point charge q is given by
B = μ0qvsinϑ/[4πr2]
Where μ0 is the permittivity of free space, q is the elementary particle charge, v is the particle's velocity, and ϑ is the angle that the velocity vector makes with its radius vector r. When the charge velocity is 0, there is no B.
This begs the question; what exactly is it about a moving charge that creates a magnetic field? The answer is somewhat complicated. Consider a single point charge next to a line of moving charges--a current. The current moves with a velocity v. Now, because it is literally electromagnetic phenomena, the current moves at great speeds, up to and including the speed of light. Classical electrostatics says that the charge will experience an attractive (or repulsive) force equal to the net force from all the charges in the line of charge, which have a have a distance r away from the point charge. At least, that's looking at it from the current's frame of reference, anyway. From the point of view of the point charge, the fast-moving current experiences Lorentz contraction and so sees a different r than the charges "actually" have and thus experiences a different net force from them. This is completely incongrous with the fact that even in different frames of reference, the forces and accelerations in a system must be the same. The idea of "magnetism" is a mathematical model for electrostatics (because magnetism turns out to be, simply, electrostatics for a moving reference frame) that corrects for special relativistic effects in mechanics.
Thankfully, one does not need special relativity to use the ideas of magnetism. The two main ideas of magnetism are Faraday's Law and Lenz's Law. Faraday's Law of Induction states that a time-variant (and only a time-variant) magnetic field induces a voltage (and thus, a current) in a closed loop proportional to the change in the field. Now, sometimes, electronics are ambiguous and open ended, and their design is somewhat open-ended. However, there's really only one way to make a loop of wire, and once you wind it, the only thing to do is to wind it some more. Hence, we are naturally led to making a coil of wire. The coil creates a magnetic field
B = μ0NI/L
Where N is the number of turns in the coil, I is the current through the coil, and L is is the wire length. The net magnetic field through a solenoid (not pictured to the left; that is a toroid) is fairly uniform, which is largely where it derives its usefulness. The amount of magnetic flux through the coil is equal to the amount of flux through each coil turn, multiplied by the number of turns. The flux through one coil is
Φturn = AB = Aμ0NI/L; Φm = NΦturn = Aμ0N2I/LWhere Φm is the flux through all the coils.
Because we love symmetry in physics, we say that the coil, which we will now refer to as an inductor, stores energy in the form of the magnetic field it generates outside of it. Similar to how the capacitance of a material is the amount of charge it takes to produce some voltage across the plates, the inductance of an inductor is the amount of magnetic flux produced per some amount of current. The inductance of a solenoid is
L = Φm/I = Aμ0N2/LWhich shows that the inductance of an inductor depends only on its geometry.
(Actually, that's not entirely true; just as one can add a dielectric between capacitor plates to increase its capacitance, one can wind a coil around a core of some material to change its inductance; for example, winding around a ferromagnetic core gives a core much more receptive to magnetic flux than air, increasing the flux through the coil and thus its inductance.)
When a changing magnetic field comes through the inductor (or, when time-variant current tries to change the magnetic field of the inductor) Lenz's Law states that the inductor will self-induce a voltage that resists a change in its magnetic field. The voltage produced is equal to
V = L[di/dt]The more the current tries to change, the greater the voltage induced in the inductor to resist that change.
The unit of inductance is the Henry, measured in volt-seconds over amps. One Henry is equal to 1 V*s/A. The unit is named in honor of Joseph Henry, an American scientist who discovered the principles of magnetism and inductance independent of Michael Faraday, who already has a unit named after him.
Transformer
A transformer is a basic ferromagnetic circuit; two coils of wire wrapped around a loop of ferromagnetic core. The energy (magnetic flux) in the inductor flows through the core loop and through a second inductor; this inductor has less coil turns Nthan the other inductor, and thus less voltage is induced. In this way, we can turn a large voltage, which might destroy delicate electronics or send too much current through residential wiring, into smaller, safer voltages.
