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(A, B, C, D) For $r < R$, using Faraday's law of induction: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\phi_B}{dt}$.$E(2\pi r) = \frac{d}{dt}[(4t^2 - 2t

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Electric field varies linearly with $r$ if $r < R$, where $r$ is the radial distance from the centerline of the cylinder

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(A, B, C, D) For $r $E(2\pi r) = \frac{d}{dt}[(4t^2 - 2t + 6) \pi r^2] = (8t - 2) \pi r^2$.Thus, $E = \frac{(8t - 2)r}{2} = (4t - 1)r$. Since $E \propto r$, the electric field varies linearly with $r$ for $r For $r > R$, the magnetic flux is confined to the region $r \le R$: $\oint \vec{E} \cdot d\vec{l} = -\frac{d}{dt}[B \cdot \pi R^2]$.$E(2\pi r) = (8t - 2) \pi R^2 \implies E = \frac{(8t - 2)R^2}{2r}$.At $r = 2R$ and $t = 2 	ext{ s}$:$E = \frac{(8(2) - 2)R^2}{2(2R)} = \frac{14R^2}{4R} = \frac{7R}{2}$.The acceleration $a = \frac{Eq}{m} = \frac{7qR}{2m}$.Induced electric fields are non-conservative and form closed loops. By Lenz's law, since $\vec{B}$ is increasing for $t > 0.25 	ext{ s}$, the induced field opposes the change, resulting in a clockwise direction for a positive charge.

$A$ charged particle of charge $q$ and mass $m$ is placed at a distance $2R$ from the centre of a vertical cylindrical region of radius $R$ where the magnetic field varies as $\vec{B} = (4t^2 - 2t + 6) \hat{k}$, where $t$ is time. Then which of the following statement$(s)$ is/are true?

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The figure shows a uniform magnetic field $B$ confined to a cylindrical volume,which is increasing at a constant rate. The instantaneous acceleration experienced by an electron placed at point $P$ is:

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