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Ch 29: Electromagnetic Induction
Young & Freedman Calc - University Physics 15th Edition
Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook
Chapter 29, Problem 10

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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1
First, understand that the problem involves electromagnetic induction, where a change in the magnetic flux through a coil induces an electromotive force (emf). The formula for the average induced emf (ε) is given by Faraday's law: ε = -ΔΦ/Δt, where ΔΦ is the change in magnetic flux and Δt is the time interval.
Calculate the initial magnetic flux (Φ_initial) through the coil using the formula Φ = B * A * cos(θ), where B is the magnetic field strength, A is the area of the coil, and θ is the angle between the magnetic field and the normal to the plane of the coil. Here, A = length * width = 0.25 m * 0.40 m.
Determine the final magnetic flux (Φ_final) when the coil is perpendicular to the magnetic field. In this position, θ = 0°, so cos(θ) = 1. Therefore, Φ_final = B * A.
Calculate the change in magnetic flux (ΔΦ) as ΔΦ = Φ_final - Φ_initial. Substitute the values obtained from the previous steps to find ΔΦ.
Finally, use Faraday's law to find the average induced emf: ε = -ΔΦ/Δt. Substitute ΔΦ and Δt = 0.0600 s into the formula to calculate the average emf induced in the coil.

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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 coil induces an electromotive force (emf) in the coil. The average emf is calculated by the rate of change of magnetic flux, which is the product of the number of turns, the change in magnetic flux, and the time over which the change occurs.
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Magnetic Flux

Magnetic flux is the measure of the quantity of magnetism, considering the strength and extent of a magnetic field. It is calculated as the product of the magnetic field strength, the area through which the field lines pass, and the cosine of the angle between the field lines and the normal to the surface.
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Angular Motion and Geometry

Understanding the geometry of the coil's rotation is crucial. The coil rotates from an angle to perpendicular to the magnetic field, affecting the magnetic flux. The change in angle alters the cosine component in the magnetic flux calculation, which is essential for determining the change in flux and thus the induced emf.
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Related Practice
Textbook Question

A circular loop of wire with radius r = 0.0480 m and resistance R = 0.160 Ω is in a region of spatially uniform magnetic field, as shown in Fig. E29.22. The magnetic field is directed out of the plane of the figure. The magnetic field has an initial value of 8.00 T and is decreasing at a rate of dB/dt = -0.680 T/s. Is the induced current in the loop clockwise or counterclockwise?

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Textbook Question

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of 12.0 cm/s due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 T. Find the emf induced in the loop at the instant when 9.0 s have passed.

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Textbook Question

Shrinking Loop. A circular loop of flexible iron wire has an initial circumference of 165.0 cm, but its circumference is decreasing at a constant rate of 12.0 cm/s due to a tangential pull on the wire. The loop is in a constant, uniform magnetic field oriented perpendicular to the plane of the loop and with magnitude 0.500 T. Find the direction of the induced current in the loop as viewed looking along the direction of the magnetic field.

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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?

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

In a physics laboratory experiment, a coil with 200 turns enclosing an area of 12 cm2 is rotated in 0.040 s from a position where its plane is perpendicular to the earth's magnetic field to a position where its plane is parallel to the field. The earth's magnetic field at the lab location is 6.0 × 10-5 T. What is the average emf induced in the coil?