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 $r$ from the centre of the circular region [2000]

Question

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(2)

$\oint \vec{E} \cdot d \vec{ℓ} = \frac{d ϕ}{d t} = \frac{d}{d t} (\vec{B} \cdot \vec{A}) = \frac{d}{d t} (B A \cos 0 °) = π a^{2} \frac{d B}{d t}$

$\Rightarrow E (2 π r) = π a^{2} \frac{d B}{d t}$ For $r \geq a$ ,

$\Rightarrow E = \frac{a^{2}}{2 r} \frac{d B}{d t}$ $\Rightarrow E \propto \frac{1}{r}$

Hence, the magnitude of the induced electric field at a distance $r$ from the centre of the circular region decreases as $\frac{1}{r} .$

Solution

Q.
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 $r$ from the centre of the circular region [2000]

1 is zero

2 decreases as $1 / r$

3 increases as $r$

4 decreases as $1 / r^{2}$

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Solution

Q. 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 from the centre of the circular region [2000]

1 is zero

2 decreases as 1/r

3 increases as r

4 decreases as 1/r2

Ans. (2) ∮E→·dℓ→=dϕdt=ddt(B→·A→)=ddt(BAcos0°)=πa2dBdt ⇒E(2πr)=πa2dBdt For r≥a, ⇒E=a22rdBdt⇒E∝1r Hence, the magnitude of the induced electric field at a distance r from the centre of the circular region decreases as 1r.

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Q.
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 $r$ from the centre of the circular region [2000]

2 decreases as $1 / r$

3 increases as $r$

4 decreases as $1 / r^{2}$