A spatially uniform magnetic field is restricted in a circular region of space with radius of 2.5 cm and has a constant ...

$$\begin{gathered} Given, \\ Radius of within which the magnetic flux exist, r_1 = 2.5 cm \\ Time varying magnetic field, B =\left(25 \raisebox{1ex}{$mT$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.\right)t^2 \\ {r}_2 = 4.00 cm \\ t = 2 s \end{gathered}$$

We know the magnetic flux changes from the loop is,

$$\begin{gathered} \phi = \int \overrightarrow{B} .d\overrightarrow{a} \\ \phi = \left(25 \raisebox{1ex}{$mT$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.\right)t^2\int da \\ \phi = \left(25 \raisebox{1ex}{$mT$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.\right)t^2{\mathrm{\pi r}}_1^2 \end{gathered}$$

Now according to Faraday's law,

$$\begin{gathered} \int \overrightarrow{E}.d\overrightarrow{l} = -\frac{d\phi }{dt} \\ E\left(2{\mathrm{\pi r}}_2\right) = -\frac{d}{dt}\left(\left(25 \raisebox{1ex}{$mT$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.\right)t^2{\mathrm{\pi r}}_1^2\right) \\ E\left(2{\mathrm{\pi r}}_2\right) = -\left(\left(25 \raisebox{1ex}{$mT$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right.\right)t^2{\mathrm{\pi r}}_1^2\right)\frac{d}{dt}{t}^2 \\ E = -\frac{\left(25\times {10}^{-3}\right)r_1^2}{2r_2}\left(2t\right) \\ E = -\frac{\left(25\times {10}^{-3}\right)t{r}_1^2}{r_2} \end{gathered}$$

Now the induced electric field in the loop of radius r = 4 cm at time t = 2 s

$$\begin{gathered} E = -\frac{25t{r}_1^2}{r_2} \\ E = -\frac{25\times {10}^{-3}\times 2\times {\left(2.50 \times {10}^{-2}\right)}^2}{4.00\times {10}^{-2}} \\ \mathit{E}\mathbf{ }\mathbf{=}\mathbf{ }\mathbf{-}\mathbf{78}\mathbf{.}\mathbf{125}\mathbf{\times }{\mathbf{10}}^{\mathbf{-}\mathbf{5}}\mathbf{ }\raisebox{1ex}{$\mathbf{N}$}\!\left/ \!\raisebox{-1ex}{$\mathbf{C}$}\right. \end{gathered}$$

Negative sign indicates that the induced emf oppose the change of the magnetic flux.

magnitude of the electric field,

$E = 78.125\times {10}^{-5} \raisebox{1ex}{$N$}\!\left/ \!\raisebox{-1ex}{$C$}\right.$