In the figure two circular regions and have been shown. Their radii are and , respectively. In there is a uniform magnetic field into the page and in there is a uniform magnetic field out of the page (ignoring any fringing of these fields). Both fields are decreasing at the rate . We have to calculate the integral for each of the three indicated paths.

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According to Faraday’s law of induction

,

where is the flux enclosed by the closed curve.

(a)

We will calculate first for the path (a). It is given that the magnetic field is decreasing in both regions and at the uniform rate of . We note that the uniform magnetic field in the region is into the plane of the figure. Therefore, induced current will flow in the clockwise direction to compensate for the decrease in flux in the region . But the contour integral is being calculated in the counter-clockwise sense; therefore, its value will be negative. Therefore,

(b)

We will calculate first for the path (b). It is given that the magnetic field is decreasing in both regions and at the uniform rate of . We note that the uniform magnetic field in the region is coming out of the plane of the figure. Therefore, induced current will flow in the counter-clockwise direction to compensate for the decrease in flux in the region . But the contour integral is being calculated in the clockwise sense; therefore, its value will be negative. Therefore,

(c)

We will calculate first for the path (c). It is given that the magnetic field is decreasing in both regions and at the uniform rate of . As is greater than the net decrease of flux in the region enclosed by the contour (c) will be

Therefore, induced current will flow in the counter-clockwise direction to compensate for the decrease in flux enclosed. The sense of the contour integral is that of the flow of the induced current; therefore, its value will be negative. Therefore,