A rectangular coil of N turns and of length a and width b is...

A rectangular coil of $N$ turns and of length $a$ and width $b$ is rotated at frequency $f$ in a uniform magnetic field $B,$ as indicated in Fig. $30-40 .$ The coil is connected to co-rotating cylinders, against which metal brushes slide to make contact. (a) Show that the emf induced in the coil is given (as a function of time $t )$ by $$\mathscr{E}_{0}=2 \pi f N a b B \sin (2 \pi f t)=\mathscr{E}_{0} \sin (2 \pi f t)$$ This is the principle of the commercial alternating-current generator. (b) What value of $N a b$ gives an emf with $\mathscr{E}_{0}=150 \mathrm{V}$ when the loop is rotated at 60.0 rev/s in a uniform magnetic field of 0.500 $\mathrm{T} ?$

A rectangular coil of $N$ turns and of length $a$ and width $b$ is
rotated at frequency $f$ in a uniform magnetic field $B,$ as indicated in
Fig. $30-40 .$ The coil is connected to co-rotating cylinders, against
which metal brushes slide to make contact. (a) Show that the emf
induced in the coil is given (as a function of time $t )$ by
$$\mathscr{E}_{0}=2 \pi f N a b B \sin (2 \pi f t)=\mathscr{E}_{0} \sin (2 \pi f t)$$
This is the principle of the commercial alternating-current generator. (b) What value of $N a b$ gives an emf with $\mathscr{E}_{0}=150 \mathrm{V}$
when the loop is rotated at 60.0 rev/s in a uniform magnetic field
of 0.500 $\mathrm{T} ?$

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Key Concepts

Faraday's Law of Electromagnetic Induction

This law states that an emf is induced in a circuit when there is a change in the magnetic flux through the circuit. It quantifies the induced emf as being proportional to the rate of change of flux, and is foundational to understanding how currents are generated by moving magnets or rotating coils in magnetic fields.

Magnetic Flux

Magnetic flux refers to the measure of the amount of magnetic field passing through a given area. In the context of a rotating coil, the flux changes as the orientation of the coil with respect to the magnetic field changes, leading to the generation of an alternating emf.

Alternating Current Generation

The process of generating alternating current (AC) involves producing an emf that periodically reverses direction. In a rotating coil within a magnetic field, the sinusoidal variation in magnetic flux results in a sinusoidal induced emf, which is the operating principle behind AC generators.

Rotational Motion and Angular Frequency

The rotation of the coil causes the magnetic flux through it to vary periodically. The concept of angular frequency, which is 2? times the rotation frequency, is used to describe this periodic variation. It provides a direct link between the mechanical motion of the coil and the frequency of the induced emf.

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