A loop of wire in the shape of a rectangle of width w and...

A loop of wire in the shape of a rectangle of width $w$ and length $L$ and a long, straight wire carrying a current $I$ lie on a tabletop as shown in Figure P31.8. (a) Determine the magnetic flux through the loop due to the current $I$ (b) Suppose the current is changing with time according to $I=a+b t,$ where $a$ and $b$ are constants. Determine the emf that is induced in the loop if $b=10.0 \mathrm{A} / \mathrm{s}, h=$ 1.00 $\mathrm{cm}, w=10.0 \mathrm{cm},$ and $L=100 \mathrm{cm} .$ What is the direction of the induced current in the rectangle?

Key Concepts

Faraday's Law of Electromagnetic Induction

Faraday's law relates the time rate of change of magnetic flux through a loop to the induced electromotive force (emf) in that loop. It states that the induced emf is proportional to the negative of the rate of change of the magnetic flux. This law is critical for understanding how changing magnetic fields generate currents in circuits.

Magnetic Field of a Long Straight Current-Carrying Wire

This concept describes the magnetic field produced by a current flowing in a long, straight conductor. The magnetic field at a distance from the wire can be calculated using Ampère's law or the Biot–Savart law, and it decreases with increasing distance from the wire. The field is directed tangentially around the wire according to the right-hand rule.

Magnetic Flux

Magnetic flux represents the total magnetic field passing through a given area. For a loop, it is calculated by integrating the magnetic field component perpendicular to the surface over the area enclosed by the loop. It quantifies how much magnetic field 'threads' the loop, making it a fundamental quantity in electromagnetic induction.

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 a direction that opposes the change in flux that produced it. This principle is consistent with the conservation of energy and is essential for determining the behavior of circuits under varying magnetic conditions.

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