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Ch 29: Electromagnetic Induction
All textbooksYoung & Freedman Calc 14th EditionCh 29: Electromagnetic InductionProblem 15
Chapter 29, Problem 15

A circular loop of wire is in a region of spatially uniform magnetic field, as shown in Fig. E29.15. The magnetic field is directed into the plane of the figure. Determine the direction (clockwise or counterclockwise) of the induced current in the loop when (a) B is increasing; (b) B is decreasing; (c) B is constant with value B0. Explain your reasoning.

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To determine the direction of the induced current, we use Lenz's Law, which states that the direction of the induced current is such that it opposes the change in magnetic flux through the loop.
Consider case (a) where the magnetic field B is increasing. The increasing magnetic field into the plane means the magnetic flux through the loop is increasing. According to Lenz's Law, the induced current will create a magnetic field that opposes this increase. Therefore, the induced magnetic field must be out of the plane, which requires a counterclockwise current.
For case (b), where the magnetic field B is decreasing, the magnetic flux through the loop is decreasing. The induced current will create a magnetic field to oppose this decrease, meaning it will try to maintain the magnetic field into the plane. Thus, the induced current must be clockwise to create a magnetic field into the plane.
In case (c), where the magnetic field B is constant, there is no change in magnetic flux through the loop. According to Faraday's Law of Induction, no change in flux means no induced electromotive force (emf) and therefore no induced current in the loop.
Summarizing, the direction of the induced current is counterclockwise when B is increasing, clockwise when B is decreasing, and there is no induced current when B is constant.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Faraday's Law of Electromagnetic Induction

Faraday's Law states that a change in magnetic flux through a loop induces an electromotive force (EMF) in the loop. The induced EMF is proportional to the rate of change of the magnetic flux. This principle is crucial for understanding how varying magnetic fields can generate currents in conductive loops, as seen in the given problem.
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Faraday's Law

Lenz's Law

Lenz's Law provides the direction of the induced current, stating that the induced current will flow in a direction that opposes the change in magnetic flux. This means if the magnetic field through the loop is increasing, the induced current will create a magnetic field opposing this increase, and vice versa for a decreasing field.
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Magnetic Flux

Magnetic flux is a measure of the magnetic field passing through a given area, such as a loop of wire. It is calculated as the product of the magnetic field strength, the area of the loop, and the cosine of the angle between the field and the normal to the loop. Understanding magnetic flux is essential for applying Faraday's Law to determine the induced EMF and current.
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Related Practice
Textbook Question

A closely wound rectangular coil of 80 turns has dimen-sions of 25.0 cm by 40.0 cm. The plane of the coil is rotated from a position where it makes an angle of 37.0° with a magnetic field of 1.70 T to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

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Textbook Question

The armature of a small generator consists of a flat, square coil with 120 turns and sides with a length of 1.60 cm. The coil rotates in a magnetic field of 0.0750 T. What is the angular speed of the coil if the maximum emf produced is 24.0 mV?

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Textbook Question

A flat, rectangular coil of dimensions l and w is pulled with uniform speed v through a uniform magnetic field B with the plane of its area perpendicular to the field (Fig. E29.14). (a) Find the emf induced in this coil. (b) If the speed and magnetic field are both tripled, what is the induced emf?

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Textbook Question

The current in Fig. E29.18E29.18 obeys the equation I(t)=I0e−btI(t)=I_0e^{-bt}, where b>0b > 0. Find the direction (clockwise or counterclockwise) of the current induced in the round coil for t>0t > 0.

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Textbook Question

Using Lenz's law, determine the direction of the current in resistor ab of Fig. E29.19 when (a) switch S is opened after having been closed for several minutes; (b) coil B is brought closer to coil A with the switch closed; (c) the resistance of R is decreased while the switch remains closed.

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Textbook Question

A cardboard tube is wrapped with two windings of insulated wire wound in opposite directions, as shown in Fig. E29.20. Terminals a and b of winding A may be connected to a battery through a reversing switch. State whether the induced current in the resistor R is from left to right or from right to left in the following circumstances: (a) the current in winding Ais from a to b and is increasing; (b) the current in winding A is from b to a and is decreasing; (c) the current in winding A is from b to a and is increasing.

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