A wire is bent into three circular segments, each of radius r= 10 cm, as shown in Fig. 30-48. E

step 1 In this problem on the topic of induction, we are shown a wire in the figure bent into the circu

A wire is bent into three circular segments, each of radius...

A wire is bent into three circular segments, each of radius $r=$ $10 \mathrm{~cm}$, as shown in Fig. $30-48$. Each segment is a quadrant of a circle, $a b$ lying in the $x y$ plane, $b c$ lying in the $y z$ plane, and $c a$ lying in the $z x$ plane. (a) If a uniform magnetic field $\vec{B}$ points in the positive $x$ direction, what is the magnitude of the emf developed in the wire when $B$ increases at the rate of $3.0 \mathrm{mT} / \mathrm{s}$ ? (b) What is the direction of the current in segment $b \bar{c}$ ?

A wire is bent into three circular segments, each of radius $r=$ $10 \mathrm{~cm}$, as shown in Fig. $30-48$. Each segment is a quadrant of a circle, $a b$ lying in the $x y$ plane, $b c$ lying in the $y z$ plane, and $c a$ lying in the $z x$ plane. (a) If a uniform magnetic field $\vec{B}$ points in the positive $x$ direction, what is the magnitude of the emf developed in the wire when $B$ increases at the rate of $3.0 \mathrm{mT} / \mathrm{s}$ ?
(b) What is the direction of the current in segment $b \bar{c}$ ?

Key Concepts

Lenz’s Law

Lenz’s Law is a principle that provides the direction of the induced emf and current resulting from a changing magnetic flux. It asserts that the induced current will flow in such a way that the magnetic field it creates opposes the change in the original magnetic flux. This opposition is represented by the negative sign in Faraday’s Law, and it ensures the conservation of energy in electromagnetic processes by resisting the change that produced it.

Vector Area of a Closed Loop

In the analysis of nonplanar or complex-shaped loops, the concept of a vector area is useful. The vector area not only has a magnitude equal to the effective area enclosed by the loop but also a direction, which is defined by the right?hand rule applied to the loop’s line integral (specifically, (1/2)? r × dr). This vector area allows one to calculate the magnetic flux through an arbitrarily oriented loop by taking the dot product with the magnetic field, ensuring that the directional nature of both the field and the loop is properly accounted for.

Right-Hand Rule

The right?hand rule is a mnemonic used to determine the direction of a vector resulting from a cross product in electromagnetism. In the context of induced currents and magnetic moments, if you curl the fingers of your right hand in the direction of the current’s circulation, your thumb points in the direction of the resulting magnetic dipole moment (or vector area). This rule is essential for visualizing and predicting the orientation of fields and currents in various electromagnetic situations.

Faraday’s Law of Electromagnetic Induction

Faraday’s Law states that an electromotive force (emf) is induced in a closed circuit whenever the magnetic flux through the circuit changes with time. The law is mathematically expressed as emf = -d?/dt, where ? is the magnetic flux. This concept is central to understanding how time?varying magnetic fields can drive currents in conductive loops, and it forms the basis for technologies such as transformers, electric generators, and induction heating devices.

Magnetic Flux

Magnetic flux is a measure of the amount of magnetic field passing through a given surface. It is formally defined as the surface integral of the magnetic field B over an area A, taking into account the angle between B and the area’s normal vector. Magnetic flux is a scalar quantity that quantifies the linkage between magnetic field lines and a surface, and its change in time is what, according to Faraday’s Law, gives rise to an induced emf.

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