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
Glencoe Physics: Principles and Problems, Student Edition
1st Edition
ISBN:9780078807213
Author:Paul W. Zitzewitz
Publisher:Paul W. Zitzewitz
Chapter25: Electromagnetic Induction
Section: Chapter Questions
Problem 69A
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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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