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

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Question

(B) According to Faraday's law of electromagnetic induction,the induced electric field $\vec{E}$ satisfies the relation $\oint \vec{E} \cdot d\vec{l}

Vedclass · Accepted Answer

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

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(B) According to Faraday's law of electromagnetic induction,the induced electric field $\vec{E}$ satisfies the relation $\oint \vec{E} \cdot d\vec{l} = -\frac{d\phi_B}{dt}$.For a point $P$ at a distance $r > a$ from the center,we consider a circular path of radius $r$ centered at the origin.The magnetic flux $\phi_B$ through this circular path is limited to the region of radius $a$ where the magnetic field exists: $\phi_B = B(t) \cdot \pi a^2$.Applying the integral form of Faraday's law:$E(2\pi r) = \left| \frac{d}{dt} (B(t) \cdot \pi a^2) \right| = \pi a^2 \frac{dB}{dt}$.Solving for $E$,we get:$E = \frac{a^2}{2r} \frac{dB}{dt}$.Since $a$ and $\frac{dB}{dt}$ are constants,the magnitude of the induced electric field $E$ is proportional to $\frac{1}{r}$.Therefore,the induced electric field decreases as $\frac{1}{r}$.

Column-$I$ (Common name)	Column-$II$ (Taxonomic category-Order)
$A$. Wheat	$I$. Primata
$B$. Mango	$II$. Diptera
$C$. Housefly	$III$. Sapindales
$D$. Man	$IV$. Poales

$A$ 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 $r > a$ from the centre of the circular region is:

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