30.7.1 Generators

A generator is a device that transforms mechanical energy into electric energy. FIGURE 30.35 shows a generator in which a coil of wire, perhaps spun by a windmill, rotates in a magnetic field. Both the field and the area of the loop are constant, but the magnetic flux through the loop changes continuously as the loop rotates. The induced current is removed from the rotating loop by brushes that press up against rotating slip rings.

The figure illustrates an alternating current generator and the Epsilon versus time graph associated with it.

FIGURE 30.35 Full Alternative Text

A photograph of the generators of a hydroelectric dam. — A generator inside a hydroelectric dam uses electromagnetic induction to convert the mechanical energy of a spinning turbine into electric energy.

The flux through the coil is

Φ_{m} = \vec{A} \cdot \vec{B} = A B cos θ = A B cos ω t

(30.28)

where $ω$ is the angular frequency $(ω = 2 π f)$ with which the coil rotates. The induced emf is given by Faraday’s law,

ε_{coil} = - N \frac{d Φ_{m}}{d t} = - A B N \frac{d}{d t} (cos ω t) = ω A B N sin ω t

(30.29)

where N is the number of turns on the coil. Here it’s best to use the signed version of Faraday’s law to see how $ε_{coil}$ alternates between positive and negative.

Because the emf alternates in sign, the current through resistor R alternates back and forth in direction. Hence the generator of Figure 30.35 is an alternating-current generator, producing what we call an AC voltage.

EXAMPLE 30.11 An AC generator

A coil with area $2.0 m^{2}$ rotates in a 0.010 T magnetic field at a frequency of 60 Hz. How many turns are needed to generate a peak voltage of 160 V?

SOLVE The coil’s maximum voltage is found from Equation 30.29:

ε_{max} = ω A B N = 2 π f A B N

The number of turns needed to generate $ε_{max} = 160 V$ is

N = \frac{ε_{max}}{2 π f A B} = \frac{160 V}{2 π (60 Hz) (2.0 m^{2}) (0.010 T)} = 21 turns

REVIEW A 0.010 T field is modest, so you can see that generating large voltages is not difficult with large $(2 m^{2})$ coils. Commercial generators use water flowing through a dam, rotating windmill blades, or turbines spun by expanding steam to rotate the generator coils. Work is required to rotate the coil, just as work was required to pull the slide wire in Section 30.2, because the magnetic field exerts retarding forces on the currents in the coil. Thus a generator is a device that turns motion (mechanical energy) into a current (electric energy). A generator is the opposite of a motor, which turns a current into motion.