The figure shows a conducting loop consisting of a half circle of radius r = 1 m

Question

The figure shows a conducting loop consisting of a half circle of radius r = 1 m and three straight sections. The half circle lies in a spatially uniform, time-dependent magnetic field that is directed perpendicular to the page. The magnitude of the field is given by B = 0. 5 sin( 10 phi t) i with B in Teslas, t in seconds and T is the unit vector pointing out of the page. The resistance of the whole loop is R = 2 omh. Calculate the magnetic flux through the loop as a function of time t. What is the magnitude of the emf induced around the loop at t = 0. 1 s? Calculate the induced current in the loop at t = 0. 1 s . What is the direction (clockwise or counterclockwise) of the induced current at t = 0. 1 s?

Explanation / Answer

a. Magnetic flux phi = Area * dB

flux phi = pi r^2/2 * 0.5 * sin 10 pi t

flux = 3.14 * 1 *1 * sin 10 pi t

flux = pi sin 10 pi t

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induced emf e = A dB/dt

emf e = 3.14 d(0.5 sin 10 pi t)/dt

emf e = 3.14* 0.5* 10 * cos 10 pi t

emf e at t = 0.1 s

emf e = 3.14* 0.5* 10 * cos (3.14*10* 0.1)

emf e = -15.69 Volts

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induced current i = emf/R

i = 15.69/2 = -7.89 AMps
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Direction will be Counterclock wise direction

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