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

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(B) To find the induced electric field at a point $P$ outside the circular region $(r > a)$,we use Faraday's law of induction in integral form: $\oint

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

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(B) To find the induced electric field at a point $P$ outside the circular region $(r > a)$,we use Faraday's law of induction in integral form: $\oint \vec{E} \cdot d\vec{l} = -\frac{d\phi_B}{dt}$.Consider a concentric circular path of radius $r$ passing through point $P$. Due to symmetry,the magnitude of the induced electric field $E$ is constant along this path,and $\vec{E}$ is tangential to the circle.Thus,$\oint \vec{E} \cdot d\vec{l} = E(2\pi r)$.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 Faraday's law:$E(2\pi r) = \left| \frac{d}{dt} (B(t) \cdot \pi a^2) \right|$$E(2\pi r) = \pi a^2 \left| \frac{dB}{dt} \right|$$E = \frac{a^2}{2r} \left| \frac{dB}{dt} \right|$Since $a$ and $\frac{dB}{dt}$ are constant,we have $E \propto \frac{1}{r}$.Therefore,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 is:

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$A$ magnetic field at a distance $r$ from the $z$-axis is given by $\vec{B} = B_0 r t \hat{k}$,where $B_0$ is a constant and $t$ is time. The magnitude of the induced electric field at a distance $r$ from the $z$-axis is:

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