A uniform but time-varying magnetic field B(t | vedclass

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(B) According to Faraday's law of electromagnetic induction,the induced electric field $\vec{E}$ is related to the changing magnetic field by the equa

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Decreases as $\frac{1}{r}$

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(B) According to Faraday's law of electromagnetic induction,the induced electric field $\vec{E}$ is related to the changing magnetic field by the equation:$\oint \vec{E} \cdot d\vec{l} = -\frac{d\Phi_B}{dt}$For a point $P$ at a distance $r > a$ from the center,we consider a circular path of radius $r$ centered at the origin.By symmetry,the magnitude of the electric field $E$ is constant along this path,and $\vec{E}$ is tangential to the path.Thus,$\oint \vec{E} \cdot d\vec{l} = E(2\pi r)$.The magnetic flux $\Phi_B$ through the area enclosed by this path is limited to the region of radius $a$ where the magnetic field exists:$\Phi_B = B(t) \cdot (\pi a^2)$Therefore,$E(2\pi r) = \left| \frac{d}{dt} (B(t) \cdot \pi a^2) \right| = \pi a^2 \left| \frac{dB}{dt} \right|$.Solving for $E$,we get $E = \frac{a^2}{2r} \left| \frac{dB}{dt} \right|$.Since $a$ and $\frac{dB}{dt}$ are constants for a given time,we find that $E \propto \frac{1}{r}$.Thus,the magnitude of the induced electric field decreases as $\frac{1}{r}$.

$A$ uniform but time-varying magnetic field $B(t)$ exists in a circular region of radius $a$ and is directed into the plane of the paper,as shown. The magnitude of the induced electric field at point $P$ at a distance $r$ $(r > a)$ from the centre of the circular region:

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