Faraday's Law of Induction

Multiply the number of turns N by the rate of change of the magnetic flux and place a minus sign in front: U_ind = −N·dΦ/dt. The flux is Φ = B·A·cos(θ), that is the product of flux density, penetrated area and the cosine of the angle between the field and the surface normal. If the flux of a coil with N = 200 turns changes from 0 to 50 mWb in 0.1 s, you get U = −200·(0.05/0.1) = −100 V. For the magnitude you often drop the sign. Insert the flux in weber and the time in seconds, then the voltage comes out in volts.

The minus sign is Lenz law and describes the direction of the induced voltage and the resulting current. The induced current always flows so that its own magnetic field opposes the change of flux that caused it. If the flux increases, the induced current works against it; if it decreases, the current tries to maintain it. This is a consequence of energy conservation: without this sign you could gain energy from nothing. In calculation problems one often considers only the magnitude of the voltage and uses the sign separately to determine the current direction.

The induced voltage depends on three factors: the number of turns N, the rate of flux change dΦ/dt and, through the flux, also on area and angle. Without a stronger field B you can raise the voltage in several ways. First, through more turns, since U is proportional to N. Second, through a faster flux change, that is by making the process happen in a shorter time; in a generator this corresponds to a higher rotation speed. Third, through a larger penetrated area or a faster tilting of the coil, so that cos(θ) changes more quickly. All of this increases dΦ/dt and thus the magnitude of the voltage.

Because it is not the magnetic field itself that induces a voltage, but only its change of flux over time. If a coil sits in an arbitrarily strong but constant field, then dΦ/dt = 0 and no voltage arises. Only when the flux changes is a voltage induced. This can happen by changing the flux density B, the penetrated area A or the angle θ. A common mistake is to look only at B and forget area and angle. A generator uses exactly this: a coil rotates in a constant field, so that cos(θ) changes continuously and an alternating voltage arises.

The magnetic flux density B is a local field measure and states how strong the magnetic field is at a point, measured in tesla. The magnetic flux Φ, by contrast, is a quantity for an entire area and describes how many field lines penetrate that area: Φ = B·A·cos(θ), measured in weber. The flux therefore depends not only on B but also on the size of the area and the angle at which the field passes through it. For induction the change of flux matters, not the flux density alone. If you confuse the two, you often underestimate the influence of the coil area and orientation.