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To monitor the breathing of a hospital patient, a thin belt is girded around the patient’s chest as in Figure P20.21. The belt is a 200-turn coil. When the patient inhales, the area encircled by the coil increases by 39.0 $\mathrm{cm}^{2} .$ The magnitude of Earth's magnetic field is 50.0$\mu \mathrm{T}$ and makes an angle of $28.0^{\circ}$ with the plane of the coil. Assuming a patient takes 1.80 s to inhale, find the magnitude of the average induced emf in the coil during that time.

1.017 \times 10^{-5} \text { Volts }

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Rutgers, The State University of New Jersey

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McMaster University

All right. So induced. The IMF is given by number of loops times, uh, rate of change off flux. Magnetic flux. So this is end times the rate of change off, um, be a co sign theater on be coast. Ada can go out here since ah, really, it's the area that's changing its magnetic field strength, forming the same. Let's pull it out. Over here is this is equal to end be co sign data times D A duty. Ah, and so bean and so fado. Well, actually not be 28 data will be 90 minus 28 degrees because, remember, theta is the angle between the magnetic field and the and the plane in the direction perpendicular to the plane of the loop. So it's 90 minus the given angle. Uh, so you have 200 loops. Magic field strength of the Earth is 50 times 10 to the negative. Six Tesler, 15 micro Tesla, uh, Times CO signed 62 degrees right times to change in area A Delta Air, which is, uh, 39 centimeters squared. So that's 39 times 10 to the negative four meter squared over the time taken for this change which is 1.8 seconds, and this gives us about 10.2 times 10 to the negative six volts. In other words, uh, 10.2 micro volts induced.