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Problem3 Your textbook calls the electric generator "probably the most important technological application of induction." Let's consider an example of such a generator, consisting of a single round loop of wire of R 100 Ω. The loop is rotated at a frequency f of 60 turns per second (and thus at 60x 2π radians per second). As shown in the figure below, the rotation axis of the loop is through its diameter and perpendicular to the page, and the loop rotates clockwise. We place the loop in a uniform magnetic field of strength B 0.50 T that points to the right. Note that the loop is arranged so that its rotation axis is perpendicular to the field. The angle θ measures the angle between the field and the normal onto the plane of the loop. An oscillating current with an amplitude of lo 1.5 A is induced Pivot Circular loop (a) Assuming we let t 0 correspond to θ 0° (i.e., the loop is vertical), show that the flux through the loop as a function of time is given by where r is the (unknown) radius of the loop. Hint: Since the coil is rotating steadily, 0 increases linearly with time. Thus, θ θ(t) wt, where w 2nf is the angular velocity. (b) Using your result from part (a), show that the magnitude of the induced emf as a function of time is given by do -BTr (2T f) sin(2r ft). Using your result from part (b), Ohm's law, and the numerical values given in the problem, find the radius of the loop. (d) If the loop is rotated more slowly, then if we wanted to get the same induced current we would need a larger loop. Explain why this is true, and check that your solution is consistent with this prediction

Problem3 Your textbook calls the electric generator "probably the most important technological application of induction." Let's consider an example of such a generator, consisting of a single round loop of wire of R 100 Ω. The loop is rotated at a frequency f of 60 turns per second (and thus at 60x 2π radians per second). As shown in the figure below, the rotation axis of the loop is through its diameter and perpendicular to the page, and the loop rotates clockwise. We place the loop in a uniform magnetic field of strength B 0.50 T that points to the right. Note that the loop is arranged so that its rotation axis is perpendicular to the field. The angle θ measures the angle between the field and the normal onto the plane of the loop. An oscillating current with an amplitude of lo 1.5 A is induced Pivot Circular loop (a) Assuming we let t 0 correspond to θ 0° (i.e., the loop is vertical), show that the flux through the loop as a function of time is given by where r is the (unknown) radius of the loop. Hint: Since the coil is rotating steadily, 0 increases linearly with time. Thus, θ θ(t) wt, where w 2nf is the angular velocity. (b) Using your result from part (a), show that the magnitude of the induced emf as a function of time is given by do -BTr (2T f) sin(2r ft). Using your result from part (b), Ohm's law, and the numerical values given in the problem, find the radius of the loop. (d) If the loop is rotated more slowly, then if we wanted to get the same induced current we would need a larger loop. Explain why this is true, and check that your solution is consistent with this prediction

College Physics
College Physics
11th Edition
ISBN:9781305952300
Author:Raymond A. Serway, Chris Vuille
Publisher:Raymond A. Serway, Chris Vuille
Chapter1: Units, Trigonometry. And Vectors
Section: Chapter Questions
Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...
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Problem3 Your textbook calls the electric generator "probably the most important technological application of induction." Let's consider an example of such a generator, consisting of a single round loop of wire of R 100 Ω. The loop is rotated at a frequency f of 60 turns per second (and thus at 60x 2π radians per second). As shown in the figure below, the rotation axis of the loop is through its diameter and perpendicular to the page, and the loop rotates clockwise. We place the loop in a uniform magnetic field of strength B 0.50 T that points to the right. Note that the loop is arranged so that its rotation axis is perpendicular to the field. The angle θ measures the angle between the field and the normal onto the plane of the loop. An oscillating current with an amplitude of lo 1.5 A is induced Pivot Circular loop (a) Assuming we let t 0 correspond to θ 0° (i.e., the loop is vertical), show that the flux through the loop as a function of time is given by where r is the (unknown) radius of the loop. Hint: Since the coil is rotating steadily, 0 increases linearly with time. Thus, θ θ(t) wt, where w 2nf is the angular velocity. (b) Using your result from part (a), show that the magnitude of the induced emf as a function of time is given by do -BTr (2T f) sin(2r ft). Using your result from part (b), Ohm's law, and the numerical values given in the problem, find the radius of the loop. (d) If the loop is rotated more slowly, then if we wanted to get the same induced current we would need a larger loop. Explain why this is true, and check that your solution is consistent with this prediction
Transcribed Image Text:Problem3 Your textbook calls the electric generator "probably the most important technological application of induction." Let's consider an example of such a generator, consisting of a single round loop of wire of R 100 Ω. The loop is rotated at a frequency f of 60 turns per second (and thus at 60x 2π radians per second). As shown in the figure below, the rotation axis of the loop is through its diameter and perpendicular to the page, and the loop rotates clockwise. We place the loop in a uniform magnetic field of strength B 0.50 T that points to the right. Note that the loop is arranged so that its rotation axis is perpendicular to the field. The angle θ measures the angle between the field and the normal onto the plane of the loop. An oscillating current with an amplitude of lo 1.5 A is induced Pivot Circular loop (a) Assuming we let t 0 correspond to θ 0° (i.e., the loop is vertical), show that the flux through the loop as a function of time is given by where r is the (unknown) radius of the loop. Hint: Since the coil is rotating steadily, 0 increases linearly with time. Thus, θ θ(t) wt, where w 2nf is the angular velocity. (b) Using your result from part (a), show that the magnitude of the induced emf as a function of time is given by do -BTr (2T f) sin(2r ft). Using your result from part (b), Ohm's law, and the numerical values given in the problem, find the radius of the loop. (d) If the loop is rotated more slowly, then if we wanted to get the same induced current we would need a larger loop. Explain why this is true, and check that your solution is consistent with this prediction
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