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A closely wound rectangular coil of 80 turns has dimensions of 25.0 $\mathrm{cm}$ by 40.0 $\mathrm{cm}$ . The plane of the coil is rotated from a position where it makes an angle of $37.0^{\circ}$ with a magnetic field of 1.10 $\mathrm{T}$ to a position perpendicular to the field. The rotation takes 0.0600 s. What is the average emf induced in the coil?

the induced emf in the coil is $[58.4 \mathrm{V}]$

Physics 102 Electricity and Magnetism

Chapter 21

Electromagnetic Induction

Current, Resistance, and Electromotive Force

Direct-Current Circuits

Magnetic Field and Magnetic Forces

Sources of Magnetic field

Inductance

Alternating Current

University of Michigan - Ann Arbor

Hope College

McMaster University

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for this problem. We have a 25 by 40 centimeters. Rectangular Coyle does 80 turns, and it's in a 1.1 test, a magnetic field. Our goal is to find the average E m f when this plane is rotated from 37 degrees, two perpendicular with the field, and that is done in a matter of 0.6 seconds. So on this depiction, I've drawn these two different times on the left hand side. It's when it's 37 degrees from the magnetic field and on the right hand side after points, your six seconds have elapsed. It's perpendicular over the field. So if we're looking at the average IMF, we're looking at our equation for the induced IMF, which for a coil situation we have that is number of turns and multiplying by the change in flux through one of the coils over the change in time and this change in time. We're considering these two end points, so let's let's see where this equation takes us. We have the flux being equal. Thio are perpendicular component of our field, multiplied by our area over a change in time. The area is not changing, so we can pull that out. So we have n a and I'm just gonna put our delta t down there. And how could we really are perpendicular component of our Byfield? Well, we know that it's gonna be the co sign fee where that he is the angle between the perpendicular of our area and the Byfield. So over on this diagram that's perpendicular is gonna be these red arrows. And so the angle between the 1st 1 it's gonna be given. And I'll call that initial angle initial that's gonna be with the 53 degrees. And here we see on this one, the final angle. That's just your degrees. So if we hold the be out, the magnitude of the Byfield is not changing. So this is equal to N A B over Delta T and then the change and that coastline fee it's just our change. Cosan Fi final minus co sign fi initial. And now we have all the information we can we can plug in and get what our average your math is. So we have 80 turns here. The area is 25 by 40 and you want to multiply that I attended the negative four to put it into meters, and we have a magnetic field of 1.1 pestle. This dealt the tea is 0.6 seconds, and then our final angle is co sign of zero. That's one. And then the initial angle is 53. And if we plug all of this in, we get 58.4 bolts, that is our average induced the IMF.

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