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A square loop PQRS of side ‘ $a$ ’ and resistance ‘ $r$ ’ is placed near an infinitely along wire carrying a constant current $I$ . The sides PQ and RS are parallel to the wire. The wire and the loop are in the same plane. The loop is rotated by $180 °$ about an axis parallel to the long wire and passing through the mid points of the side QR and PS. The total amount of charge which passes through any point of the loop during rotation is __________coulombs.

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$\frac{μ_{0} I a}{2 π r} l n 2$

$\frac{μ_{0} I a}{π r} l n 2$

$\frac{μ_{0} I a^{2}}{2 π r}$

$\frac{μ_{0} i a}{2 π r}$

answer is B.

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Detailed Solution

Flux $d ϕ = \frac{μ_{0} i}{2 π x} a d x$

$\int d ϕ = \frac{μ_{0} i a}{2 π} \int_{a}^{2 a} \frac{1}{x} d x ϕ = \frac{μ_{0} i a}{2 π} l n 2$

$Δ ϕ = \frac{μ_{0} i a}{π} l n 2$

Charge = $\frac{Δ ϕ}{r} = \frac{μ_{0} i a}{π r} l n 2$

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