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Magnetic field at a distance of r from z axis is $\mathrm{B}_0 r t \hat{k}$ present in the region. '…
Magnetic field at a distance of r from z axis is $\mathrm{B}_0 r t \hat{k}$ present in the region. ' $\mathrm{B}_0$ ', is constant and ' t ' is time. The magnitude of induced electric field at a distance of $r$ from z -axis is
$\frac{\mathrm{B}_0 r^3}{3}$ $\frac{2 \pi \mathrm{~B}_0 r}{3}$ $\frac{\mathrm{B}_0 r^2}{2 \pi}$ $\frac{\mathrm{B}_0 r^2}{3}$ Solution
$\mathrm{B}=\mathrm{B}_{\mathrm{0}} \mathrm{rt} \hat{\mathrm{k}}$
By Maxwell's equation,
$\begin{aligned}
& \oint \overrightarrow{\mathrm{E}} \cdot \mathrm{~d} \overrightarrow{\mathrm{l}}=\left|-\frac{\mathrm{d} \phi}{\mathrm{dt}}\right| \\
& \Rightarrow \mathrm{E}(2 \pi \mathrm{r})=\frac{\mathrm{d}}{\mathrm{dt}}\left(\int \mathrm{~B}(2 \pi \mathrm{r}) \mathrm{dr}\right)=\frac{\mathrm{d}}{\mathrm{dt}}\left(\int 2 \pi \mathrm{~B}_{\mathrm{o}} \mathrm{tr}^2 \mathrm{dr}\right) \\
& \Rightarrow \mathrm{E}(2 \pi \mathrm{r})=\frac{\mathrm{d}}{\mathrm{dt}}\left(2 \pi \mathrm{~B}_{\mathrm{o}} \mathrm{t} \cdot \frac{\mathrm{r}^3}{3}\right)=\frac{2 \pi \mathrm{~B}_0 \mathrm{r}^3}{3} \\
& \therefore \mathrm{E}=\frac{\mathrm{B}_0 \mathrm{r}^2}{3}
\end{aligned}$
Asked in: AP EAMCET 2024 (19 May Shift 2)
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