Two co-planar, concentric circular coils of radii a and b are placed in the y-z plane such that the common center of the coils is origin (a >> b). A current i flows in the outer coil. Now the inner coil is moved along the x-axis with a constant speed v keeping its plane unchanged. The emf induced in the inner coil is maximum at

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$t = \frac{3 a}{v}$

$t = \frac{a}{v}$

$t = \frac{4 a}{v}$

$t = \frac{a}{2 v}$

answer is C.

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Detailed Solution

$\begin{array}{l} ϕ = B ({πb}^{2} =) \frac{μ_{0}}{2 π} \frac{({iπa}^{2})}{{(a^{2} + x^{2})}^{3 / 2}} {πb}^{2} \\ = \frac{{πμ}_{0} a^{2} b^{2} i}{2 {(a^{2} + x^{2})}^{3 / 2}} \end{array}$
At time t,
$\begin{array}{l} ϕ = \frac{{πμ}_{0} a^{2} b^{2} i}{2 {(a^{2} + v^{2} t^{2})}^{3 / 2}} = k {(a^{2} + v^{2} t^{2})}^{- \frac{3}{2}} \\ \frac{dϕ}{dt} = k (- \frac{3}{2}) {(a^{2} + v^{2} t^{2} \times v^{2})}^{- \frac{5}{2}} (2 t) \\ ε = - \frac{3 {kv}^{2} t}{{(a^{2} + v^{2} t^{2})}^{\frac{5}{2}}} \end{array} \begin{array}{l} \frac{dε}{dt} = - 3 {kv}^{2} [\frac{{(a^{2} + v^{2} t^{2} 1 - t)}^{\frac{5}{2}} {(a^{2} + v^{2} t^{2} v^{2} (2 t))}^{\frac{3}{2}}}{{(a^{2} + v^{2} t^{2})}^{5}}] \\ = \frac{- 3 {kv}^{2}}{{(a^{2} + v^{2} t^{2})}^{5}} {(a^{2} + v^{2} t^{2})}^{\frac{3}{2}} [a^{2} + v^{2} t^{2} - 5 v^{2} t^{2}] \\ For e_{max} \Rightarrow 4 v^{2} t^{2} = a^{2} \\ t^{2} = \frac{a^{2}}{4 v^{2}} \\ t = \frac{a}{2 v} \end{array}$