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 $B = B_{0} \cos ωt$ 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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$\frac{{πB}_{0} {ωh}^{2} N^{2} sinωt}{3}$

$\frac{{πB}_{0} {ωh}^{2} N^{3} sinωt}{1}$

$\frac{{πB}_{0} {ωh}^{2} N^{3} sinωt}{3}$

$\frac{2 {πB}_{0} {ωh}^{2} N^{3} sinωt}{3}$

answer is D.

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Detailed Solution

$\begin{array}{l} dε = - \frac{dϕ}{dt} \\ dϕ = {πn}^{2} h^{2} B \\ ∴ dε = \frac{d}{dt} {({πn}^{2} h^{2} B = πn)}^{2} h^{2} \frac{dB}{dt} \\ ∴ ε = \sum_{n = 0}^{N} {πn}^{2} h^{2} {ωB}_{0} \sin ωt \end{array} \begin{array}{l} = \frac{{πB}_{0} {ωh}^{2} N^{3} \sin ωt}{3} \\ [\sum_{n = 0}^{N} n^{2} = \frac{N (N + 1) (2 N + 1)}{6} \approx \frac{N^{3}}{3}] \end{array}$