A wire is bent into three circular segments of radius , as shown in the figure. Each segment is a quadrant of a circle, ab lying in the xy plane, bc lying in the yz plane, and ca lying in the zx plane. (a) If a uniform magnetic field points in the positive x direction, we have to find the emf developed in the wire when increases at the rate of . (b) We have to find the direction of the emf in the segment bc.

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A wire is bent into three circular segments of radius 10.4 cm. Each segment is a quadrant of a circle, ab lying in the xy plane, bc lying in the yz plane, and ca lying in the zx plane. A uniform magnetic field points in the positive x direction.

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We have to find the emf developed in the wire when B increases at the rate of .

For calculating the flux through the surface bounded by the segments ab, bc and ca, we will use spherical polar coordinates.

The unit vector normal to the spherical surface is

Therefore, the flux through the surface indicated as shown in the figure will be given by the integral

By Faraday’s law of induction,

It is given that

and

We have to find the direction of the emf in the segment bc. We set up a circuit by joining the loop bc to the origin of the coordinate system; join b to the origin and from origin to c. As the magnetic field is in the positive x direction and is increasing with time, by the Lenz’ law induced current will flow from c to b, i.e. in the clockwise direction to resist the increase in flux with time.