A uniform magnetic field is restricted within a region of radius . The magnetic field changes with time at a rate . Loop 1 of radius encloses the region and loop 2 of radius is outside the region of magnetic field as shown in the figure. Then the e.m.f

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Solution

Q.
A uniform magnetic field is restricted within a region of radius $r$ . The magnetic field changes with time at a rate $\frac{d \vec{B}}{d t}$ . Loop 1 of radius $R > r$ encloses the region $r$ and loop 2 of radius $R$ is outside the region of magnetic field as shown in the figure. Then the e.m.f. generated is [2016]

1 zero in loop 1 and zero in loop 2

2 $- \frac{d \vec{B}}{d t} π r^{2}$ in loop 1 and $- \frac{d \vec{B}}{d t} π r^{2}$ in loop 2

3 $- \frac{d \vec{B}}{d t} π R^{2}$ in loop 1 and zero in loop 2

4 $- \frac{d \vec{B}}{d t} π r^{2}$ in loop 1 and zero in loop 2

Ans.
(4)

Emf generated in loop 1,

$ε_{1} = - \frac{d ϕ}{d t} = - \frac{d}{d t} (\vec{B} \cdot \vec{A}) = - \frac{d}{d t} (B A) = - A \times \frac{d B}{d t}$

$ε_{1} = - (π r^{2} \frac{d B}{d t}) (∵ A = π r^{2} because \frac{d B}{d t} is restricted up to radius r .)$

Emf generated in loop 2, $ε_{2} = - \frac{d}{d t} (B A) = - \frac{d}{d t} (0 \times A) = 0$

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