A toy electric generator has a 20-turn circular coil with...

A toy electric generator has a 20-turn circular coil with each turn of radius $1.8 \mathrm{cm} .$ The coil resides in a $1.0-\mathrm{T}$ magnitude $\vec{B}$ field. It also has a lightbulb that lights if the potential difference across it is about 1 V. You start rotating the coil, which is initially perpendicular to the $\vec{B}$ field. (a) Determine the time interval needed for a $90^{\circ}$ rotation that will produce an average induced emf of $1.0 \mathrm{V}$. (b) Use a proportion technique to show that the same emf can be produced if the time interval for one rotation is reduced to one-fourth while the radius of the coil is reduced by one-half.

Key Concepts

Scaling and Proportional Reasoning

This involves understanding how modifying parameters such as coil radius and rotation time proportionally affects the induced EMF. It is important for analyzing and predicting how adjustments in the system can maintain or change the performance of electromagnetic devices.

Angular Velocity

Angular velocity is the rate at which an object rotates and is crucial in determining the time rate of change of the magnetic flux through a rotating coil. It directly influences the magnitude of the induced EMF as per Faraday's law.

Coil Geometry

The shape and size of a coil, specifically its area, directly affect the amount of magnetic flux it intercepts. For a circular coil, the radius determines its area, and thus alterations in the coil's dimensions will change the magnetic flux passing through it.

Magnetic Flux

Magnetic flux measures the quantity of magnetic field passing through a given area and is dependent on both the strength of the magnetic field and the orientation of the surface with respect to the field. Changes in flux as the coil rotates are what cause the induced EMF.

Faraday's Law

Faraday's law quantitatively relates the induced EMF in a circuit to the rate of change of magnetic flux through the circuit. It is a fundamental principle that explains how varying magnetic fields, such as those created by a rotating coil, produce an electromotive force.

Electromagnetic Induction

This is the process through which a changing magnetic flux induces an electromotive force (EMF) in a circuit. It forms the basis for the operation of generators and transformers by converting mechanical energy into electrical energy, and vice versa.

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