A coil of area 0.100 m^2 is rotating at 60.0 rev / s with...

A coil of area $0.100 \mathrm{~m}^{2}$ is rotating at $60.0$ rev $/ \mathrm{s}$ with the axis of rotation perpendicular to a $0.200$ -T magnetic field. (a) If there are 1000 turns on the coil, what is the maximum voltage induced in it? (b) What is the orientation of the coil with respect to the magnetic field when the maximum induced voltage occurs?

Key Concepts

Angular Frequency

Angular frequency quantifies the rate of rotation of the coil in terms of radians per second. It is a critical parameter in determining the frequency of the alternating voltage generated by the rotating coil. A higher angular frequency results in a more rapid change in flux, thereby inducing a higher EMF.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that an electromotive force (EMF) is induced in a circuit when there is a change in magnetic flux through the circuit. This principle is essential for understanding how motion through a magnetic field, such as the rotation of a coil, generates an electrical voltage. The induced voltage is directly proportional to the rate at which the magnetic flux changes over time.

Magnetic Flux

Magnetic flux is the measure of the amount of magnetic field passing through a given area. It is calculated as the product of the magnetic field strength, the area through which the field lines pass, and the cosine of the angle between the magnetic field and the area’s normal vector. This concept is central to analyzing how changes in a coil’s orientation alter the flux and, consequently, induce voltage.

Rotating Coil

A rotating coil in a magnetic field experiences a periodic change in magnetic flux due to its changing orientation relative to the field. This motion causes the induced EMF to vary cyclically, achieving a maximum when the plane of the coil is in the optimal position relative to the field. The rotation provides a dynamic system to apply Faraday’s Law effectively.

Orientation and Maximum Induced EMF

The induced EMF in a rotating coil reaches its maximum when the rate of change of magnetic flux is greatest. This occurs when the coil is oriented such that its plane is perpendicular to the magnetic field lines, meaning that the normal to the coil is parallel to the field. This orientation ensures that the sinusoidal variation of flux reaches its peak rate of change at the optimal moment, maximizing the induced voltage.

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