In Fig. 30-37, a rectangular loop of wire with length a=2.2 cm, width b=0.80 cm, and resistanc

step 1 In this particular problem we have given a rectangular loop of wire and the length is this A is

In Fig. 30-37, a rectangular loop of wire with length a=2.2...

In Fig. 30-37, a rectangular loop of wire with length $a=2.2 \mathrm{~cm}$, width $b=0.80 \mathrm{~cm}$, and resistance $R=0.40 \mathrm{~m} \Omega$ is placed near an infinitely long wire carrying current $i=6.9 \mathrm{~A}$. The loop is then moved away from the wire at constant speed $v=3.2 \mathrm{~mm} / \mathrm{s}$. When the center of the loop is at distance $r=1.5 b$, what are (a) the magnitude of the magnetic flux through the loop and (b) the current induced in the loop?

Key Concepts

Magnetic Field of a Current-Carrying Wire

This concept involves calculating the magnetic field produced by an infinitely long wire carrying a steady current. Using Ampere's law, the magnetic field at a distance from the wire is given by B = (??I)/(2?r), where ?? is the permeability of free space. This relation is essential for determining how the magnetic field varies with distance, which in turn affects the magnetic flux through nearby loops.

Magnetic Flux

Magnetic flux quantifies the total magnetic field passing through a given area. When the magnetic field is not uniform over the area, the flux must be calculated by integrating the magnetic field over the area of the loop. This concept is key when analyzing systems where the loop or the source of the magnetic field is moving, thus changing the flux over time.

Faraday’s Law of Electromagnetic Induction

Faraday's law states that a change in magnetic flux through a closed loop induces an electromotive force (EMF) in the loop. The magnitude of the induced EMF is equal to the rate of change of the magnetic flux, and the direction of the induced current opposes the change in flux, as described by Lenz's law. This principle underpins the operation of electrical generators and many sensors.

Ohm’s Law in Induced Circuits

When an EMF is induced in a circuit, the current that flows is determined by Ohm's law, which relates the current to the voltage (EMF) and the total resistance of the circuit (I = EMF/R). This concept is vital for calculating the magnitude of the induced current in scenarios where the resistance of the circuit is known.

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