A flat metal spiral (with a constant distance $h$ between coils) and a total number of coils N is placed in a uniform magnetic field $B = B_{0} \cos ω t$ perpendicular to the plane of the spiral (see figure). The total electromotive force induced in the spiral (between points A and C) is approximately. (Assume $N > > 1$ )

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$\frac{2 π B_{0} ω h^{2} N^{3}}{3}$

$\frac{π B_{0} ω h^{2} N^{3} \sin ω t}{1}$

$\frac{π B_{0} ω h^{2} N^{2} \sin ω t}{3}$

$\frac{π B_{0} ω h^{2} N^{3} \sin ω t}{3}$

answer is D.

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Detailed Solution

Emf is induced in each turn of the spiral which depends on rate at which magnetic flux is changing within the area bounded by that turn.

$d ε = - \frac{d ϕ}{d t} F o r a t u r n l o c a t e d a t a d i s \tan c e n h f r o m t h e c e n t r e d ϕ = π n^{2} h^{2} B ∴ d ε = \frac{d}{d t} (π n^{2} h^{2} B) = π n^{2} h^{2} \frac{d B}{d t} ∴ ε = \sum_{n = 0}^{N} π n^{2} h^{2} ω B_{0} \sin ω t = \frac{π B_{0} ω h^{2} N^{3} \sin ω t}{3} [\sum_{n = 0}^{N} n^{2} = \frac{N (N + 1) (2 N + 1)}{6} \approx \frac{N^{3}}{3}]$