Problem 4Your textbook calls the electric generator "probably the most important technological application ofinduction." Let's consider an example of such a generator, consisting of a single round loop of wire ofR = 100. The loop is rotated at a frequency f of 60 turns per second (and thus at 60 × 2 radiansper second). As shown in the figure below, the rotation axis of the loop is through its diameter andperpendicular to the page, and the loop rotates clockwise. We place the loop in a uniform magneticfield of strength B = 0.50 T that points to the right. Note that the loop is arranged so that itsrotation axis is perpendicular to the field. The angle measures the angle between the field and thenormal onto the plane of the loop. An oscillating current with an amplitude of Io = 1.5 A is induced.BPivotCircular loopAssuming we let t = 0 correspond to 0 = 0° (i.e., the loop is vertical), the flux through the loop asa function of time is given byPB (t) = Bлr² сos (2π ft),where r is the (unknown) radius of the loop. (a) Using the expression provided for the flux, show that the magnitude of the induced emf as afunction of time is given by|E| = Bπr² (2π f) sin(2π ft).(b) Using your result from part (a), Ohm's law, and the numerical values given in the problem, findthe radius of the loop.(c) If the loop is rotated more slowly, then if we wanted to get the same induced current we wouldneed a larger loop. Explain why this is true, and check that your solution is consistent with thisprediction.

Answered: Image /qna-images/answer/4e429de8-0279-432b-b24d-e1f2b0733aec.jpg

(a) Using the expression provided for the flux, show that the magnitude of the induced emf as a function of time is given by |E| = Bπr² (2π f) sin(2π ft). (b) Using your result from part (a), Ohm's law, and the numerical values given in the problem, find the radius of the loop. (c) 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.

(a) Using the expression provided for the flux, show that the magnitude of the induced emf as a function of time is given by |E| = Bπr² (2π f) sin(2π ft). (b) Using your result from part (a), Ohm's law, and the numerical values given in the problem, find the radius of the loop. (c) 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.