A single-turn, circular loop of radius R is coaxial with a...

A single-turn, circular loop of radius $R$ is coaxial with a long solenoid of radius $r$ and length $\ell$ and having $N$ turns (Fig. P31.18). The variable resistor is changed so that the solenoid current decreases linearly from $I_{1}$ to $I_{2}$ in an interval $\Delta t$. Find the induced emf in the loop.

Key Concepts

Magnetic Field of a Solenoid

A solenoid is a coil of wire typically designed to produce a nearly uniform magnetic field in its interior when carrying an electric current. The magnetic field inside a long solenoid is proportional to the current and the density of the turns; this relationship is critical for determining the magnetic flux through nearby circuits, such as a coaxial loop.

Lenz's Law

Lenz's Law provides the direction of the induced emf and current resulting from a change in magnetic flux. It states that the induced current will flow in such a way that its own magnetic field opposes the change in the original magnetic flux. This concept is inherent in the negative sign of Faraday’s Law and is key to the conservation of energy in electromagnetic systems.

Faraday's Law of Electromagnetic Induction

This fundamental principle states that an electromotive force (emf) is induced in a circuit when the magnetic flux linking the circuit changes with time. It forms the basis for understanding how varying magnetic fields give rise to currents in electrical circuits and is mathematically expressed as the negative time derivative of the magnetic flux through the loop.

Magnetic Flux

Magnetic flux quantifies the total magnetic field passing through a given surface area. It is calculated as the integral of the magnetic field component perpendicular to the surface over that area. Changes in magnetic flux through a loop are central to the phenomenon of electromagnetic induction, as they directly determine the induced emf according to Faraday's Law.

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