A conducting wire formed in the shape of a right triangle with base b = 31 cm an

Question

A conducting wire formed in the shape of a right triangle with base b = 31 cm and height h = 60 cm and having resistance R = 2.6 %u03A9, rotates uniformly around the y-axis in the direction indicated by the arrow (clockwise as viewed from above (loooking down in the negative y-direction)). The triangle makes one complete rotaion in time t = T = 1.9 seconds. A constant magnetic field B = 1.4 T pointing in the positive z-direction (out of the screen) exists in the region where the wire is rotating.

2) At time t = 0, the wire is positioned as shown. What is the magnitude of the magnetic flux %u03A61 at time t = t1 = 0.7125 s (It is not .13 since the flux is not max at that time)?

3) What is I1, the induced current in the loop at time t = 0.7125s? I1 is defined to be positive if it flows in the negative y-direction in the segment of length h.

Explanation / Answer

(1)

Time period = 1.9 s

w = 2*pi/T = 3.307 (angular velocity)

Area of triangle = 1/2*0.31*0.6 = 0.093

Flux at any time is given by = B.A = B*A*cos(e)

e is the angle between normal vector of triangle and magnetic field lines

de/dt is the rate of change of angle which is equal to angular velocity w

I = -d(flux)/dt

= -B*A*-sin(e)*de/dt

= BA sin(e)*w

Imax is when sin(e) = 1

Imax = BAw = 1.4*0.093*3.307 = 0.431 A

(2)

AT t = 0.7125 s

e = wt = 3.307*0.7125 = 2.356

Flux = B*A*cos(2.356) = -0.092

(3)

I1 = -d(flux)/dt

= -B*A*-sin(e)*de/dt

= BA sin(e)*w

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