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Loop A of radius r <<R moves toward loop B with a constant velocity V in such a way that their planes are always parallel. What is the distance between the two loops (x) when the induced emf in loop A is maximum?

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$R (1 - \frac{1}{\sqrt{2}})$

$\frac{R}{2}$

$\frac{R}{\sqrt{2}}$

answer is C.

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Detailed Solution

$ϕ_{A} = \frac{μ_{0} i π R^{2}}{2 π {(R^{2} + x^{2})}^{3 / 2}} π r^{2}$

$\Rightarrow E_{A} = - \frac{d ϕ}{d t} = \frac{- μ_{0} i π}{2} R^{2} r^{2} (- 3 / 2) {(R^{2} + x^{2})}^{- 5 / 2} 2 x$

$E_{A} i s m a x i m u m w h e n \frac{d E_{A}}{d x} = 0$

$\Rightarrow \frac{d}{d x} \frac{x}{{(R^{2} + x^{2})}^{5 / 2}} = 0$

$o r {(R^{2} + x^{2})}^{5 / 2} - \frac{5 x}{2} {(R^{2} + x^{2})}^{3 / 2} 2 x = 0$

$o r, R^{2} + x^{2} - 5 x^{2} = 0$

$o r, x = \frac{R}{2}$

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