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Ch. 29 - Electromagnetic Induction and Faraday's Law
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 29 - Electromagnetic Induction and Faraday's LawProblem 9
Chapter 28, Problem 9

A circular loop in the plane of the paper lies in a 0.65-T uniform magnetic field pointing into the paper. The loop’s diameter changes from 20.0 cm to 8.0 cm in 0.50 s. What is (a) the direction of the induced current, (b) the magnitude of the average induced emf, and (c) the average induced current if the coil resistance is 2.5Ω?

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Step 1: Understand the problem. The problem involves electromagnetic induction, specifically Faraday's Law of Induction. The magnetic flux through a circular loop changes as its diameter decreases, inducing an emf and current. We need to determine the direction of the induced current, the magnitude of the induced emf, and the induced current given the resistance of the loop.
Step 2: Use Faraday's Law of Induction to calculate the induced emf. Faraday's Law states that the induced emf (ε) is given by: ε=−dΦdt, where Φ is the magnetic flux. Magnetic flux is defined as Φ=BA, where B is the magnetic field and A is the area of the loop. Since the magnetic field is uniform and perpendicular to the loop, the change in flux is due to the change in the loop's area.
Step 3: Calculate the initial and final areas of the loop. The area of a circle is given by A=πr2. The initial diameter is 20.0 cm, so the initial radius is 10.0 cm (or 0.10 m). The final diameter is 8.0 cm, so the final radius is 4.0 cm (or 0.04 m). Calculate the initial and final areas using these radii.
Step 4: Determine the change in flux and the induced emf. The change in flux is given by ΔΦ=B(Afinal−Ainitial). Substitute the values of the magnetic field, initial area, and final area. Then, divide the change in flux by the time interval (0.50 s) to find the magnitude of the induced emf.
Step 5: Determine the direction of the induced current and calculate the average induced current. Use Lenz's Law to determine the direction of the induced current. Lenz's Law states that the induced current will flow in a direction that opposes the change in magnetic flux. Since the magnetic field is pointing into the paper and the area is decreasing, the induced current will create a magnetic field pointing into the paper to oppose the decrease. Finally, use Ohm's Law (I=εR) to calculate the average induced current, where R is the resistance of the loop (2.5 Ω).

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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, which depends on the area of the loop and the strength of the magnetic field. This principle is fundamental for understanding how currents are generated in response to changing magnetic environments.
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Lenz's Law

Lenz's Law provides the direction of the induced current resulting from an induced emf. It states that the induced current will flow in a direction that opposes the change in magnetic flux that produced it. This law ensures the conservation of energy and helps predict the behavior of circuits in changing magnetic fields.
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Ohm's Law

Ohm's Law relates the voltage (V), current (I), and resistance (R) in an electrical circuit, expressed as V = IR. This relationship is crucial for calculating the average induced current when the resistance of the coil is known. Understanding Ohm's Law allows for the determination of how much current flows in response to the induced emf.
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Related Practice
Textbook Question

(III) Suppose a conducting rod (mass m, resistance R) rests on two frictionless and resistanceless parallel rails a distance ℓ apart in a uniform magnetic field B→\(\overrightarrow{B}\) (⊥ to the rails and to the rod) as in Fig. 29–53. At t = 0, the rod is at rest and a source of emf is connected to the points a and b. Determine the speed of the rod as a function of time if (a) the source puts out a constant current I, (b) the source puts out a constant emf ε₀. (c) Does the rod reach a terminal speed in either case? If so, what is it?

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Textbook Question

(II) A conducting rod rests on two long frictionless parallel rails in a magnetic field B→\(\overrightarrow{B}\) (⊥ to the rails and rod) as in Fig. 29–53. (a) If the rails are horizontal and the rod is given an initial push, will the rod travel at constant speed even though a magnetic field is present? (b) Suppose at t = 0, when the rod has speed v = v0, the two rails are connected electrically by a wire from point a to point b. Assuming the rod has resistance R and the rails have negligible resistance, determine the speed of the rod as a function of time. Discuss your answer.

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Textbook Question

The magnetic flux through a coil of wire containing two loops changes at a constant rate from -68 Wb to +48 Wb in 0.42 s. What is the emf induced in the coil?

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