A rectangular coil in the plane of the page has dimensions  $a$ and  $b$. A long wire that carries a current  $I=2t$ is placed directly on the coil as shown. Choose the CORRECT options.

$$\begin{gathered} \varphi =\underset{b-x}{\overset{x}{\int }}\frac{{\mu }_{0}Iadr}{2\pi }=\frac{{\mu }_{0}Ia}{2\pi }\ln \left(\frac{x}{b-x}\right) \\ x\to b \varphi \to \infty \\ x=\frac{b}{2}\varphi =0 \\ {E}_{in}=-\frac{d\varphi }{dt}=-\frac{d}{dt}\left\{\frac{{\mu }_{0}a}{2\pi }\ln \left(\frac{1}{3}\right).2t\right\} \\ =\left(\frac{{\mu }_{0}a\ln 3}{\pi }\right) \end{gathered}$$