Discussion
A circular loop of radius r is placed in a magnetic field \(\vec{B}(t) = B_{0} e^{-\alpha t} \hat{z}\), \(B_{0}\) is constant. If the loop is in xy plane, then the induced emf in the loop is :
Answer: B
The induced emf is \(\pi r^{2}\alpha B_{0} e^{-\alpha t}\).
The field is along z and the loop lies in the xy plane, so the flux is field times area:
\[ \begin{aligned} \Phi &= \pi r^{2} B_{0} e^{-\alpha t} \\ \varepsilon &= -\frac{d\Phi}{dt} = \pi r^{2}\alpha B_{0} e^{-\alpha t} \end{aligned} \]
The emf is not zero because the flux changes with time. The options with \(2\pi r\) use the circumference where Faraday's law needs the enclosed area; one of them also omits the factor \(\alpha\) that comes from differentiating the exponential.
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