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How generators work

Current is generated (induced) when a loop of conductive material spins in a magnetic field, like between the poles of a magnet.

Why’s that? In electricity, there’s voltage and current.

the base unit of charge is a coulomb, and electrons have charge: $-1.6 \times 10^{-19}$ coulombs.
Voltage is the potential energy (or difference in potential) that a unit of charge has (not an electron), determined by the ‘pressure’ on it from the power source. Electrons experience that pressure since they have a charge.
Current is the rate at which electricity flows past a point, in coulombs per second (or amperes).

When a battery is ‘uncharged’, it is in equilibrium, with equal charge on both sides. Charging it moves electrons from one side to the other, thereby moving charge. The side that loses electrons becomes positively charged (since electrons have negative charge), and the side that gains them becomes negatively charged. You essentially put energy into the battery to upset the equilibrium, and the electrons naturally want to go back to the positive side to restore equilibrium, which releases a similar amount of energy (slightly less, since you lose some energy as heat and other things).

When you start the flow of electrons between the positive and negative terminals of the battery, they flow from the negative terminal to the positive terminal, because they’re attracted to the positive side to restore equilibrium. That gives you an electromagnetic field. If the wire is coiled around ferrous metal, these moving electrons in the coil repel the electrons in the ferrous metal, shoving them into one side of the metal, making it magnetic (the area towards which the electrons in the coil are moving becomes positively charged, the area away from it negatively charged because that’s where the electrons get shoved). Now for the induction: if there’s another coil around the metal, the fact that one side of the metal becomes positively charged means it attracts electrons in that coil, so they begin to move towards it – that’s the induced current, which only lasts as long as the field changes. Because if the field doesn’t change, there’s nothing to move the electrons, so you get a potential difference (the field is still there) but no current. And the more loops there are on the coil, the more electrons in it get moved, so the stronger the induced current.

The magnetic flux in Webers is $\Phi = \vec{B} \cdot \vec{A} = BAcos\theta$, where $\vec{B}$ is magnetic field, $\vec{A}$ is area vector (pointing out from the area, perpendicular), $\theta$ is angle between field and area of loop. This means that the current is alternating, because sometimes the cosine will be negative, and sometimes positive.

Why $\cos{\theta}$?

Rotating the loop is in effect the same as changing its area - if you look at the loop from the top in 2D, when you spin the loop, it'll look like its area shrinks and grows. The $\cos{\theta}$ will be between 0 and ±1, so you'll multiply the magnetic field's vector by some fraction of the loop's area.

By Faraday’s law of induction, voltage induced in the coil when magnetic flux changes is $-N \frac{d\Phi}{dt} = -N \frac{d(BA\cos{(\omega t)})}{dt}$, with $N$ the number of loops on the coil and $t$ the time. Also called “emf”. If we hold B (the magnetic field) and A (the area of the loop) constant, then the induced voltage is $-NBA(-\sin{(\omega t)}) \omega = NBA\omega \sin{(\omega t)}$.

So how much power is there? Well, power through a resistor is $P = \frac{V^2}{R}$, with V the voltage and R the resistance. So the result is $\frac{N^2 B^2 A^2 \omega^2 \sin^2{(\omega t)}}{R}$. Everything except $\omega$ is constant, so the power generated depends heavily on how quickly we spin the loop.