A certain elastic conducting material is stretched into a...

A certain elastic conducting material is stretched into a circular loop of $12.0 \mathrm{~cm}$ radius. It is placed with its plane perpendicular to a uniform $0.800$ T magnetic field. When released, the radius of the loop starts to shrink at an instantaneous rate of $75.0 \mathrm{~cm} / \mathrm{s}$. What emf is induced in the loop at that instant?

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Key Concepts

Faraday’s Law of Electromagnetic Induction

Faraday’s Law states that the induced electromotive force (emf) in a closed circuit is equal to the negative of the rate of change of the magnetic flux through the loop. This fundamental principle explains how changes in a magnetic environment can create electric currents in conductive materials.

Magnetic Flux

Magnetic flux is the measure of the quantity of magnetic field passing through a given area, defined as the product of the magnetic field strength, the area perpendicular to the field, and the cosine of the angle between the field and the perpendicular. It is a central concept in understanding electromagnetic induction.

Area of a Circle

The area of a circle is given by the formula A = ?r², where r is the radius of the circle. In problems involving circular loops in magnetic fields, changes in the radius directly affect the area and therefore the magnetic flux through the loop.

Rate of Change and Differentiation

Understanding the rate at which a physical quantity changes, typically via differentiation with respect to time, is crucial in applying Faraday’s Law. The induced emf depends on the time derivative of the magnetic flux, so analyzing how the area (and thus flux) changes over time is essential.

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