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Question 7.13: A square loop of wire, with sides of length a, lies in the f...

A square loop of wire, with sides of length a, lies in the first quadrant of the xy plane, with one corner at the origin. In this region, there is a nonuniform time-dependent magnetic field B(y, t) = k y^{3} t^{2} \hat{ z } (where k is a constant). Find the emf induced in the loop .

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\Phi=\int B d x d y=k t^{2} \int_{0}^{a} d x \int_{0}^{a} y^{3} d y=\frac{1}{4} k t^{2} a^{5} . \quad ε =-\frac{d \Phi}{d t}=-\frac{1}{2} k t a^{5} .

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