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AB is an infinitely long wire placed in the plane of rectangular coil of dimensions as shown in the figure. Calculate the mutual inductance of wire AB and coil PQRS

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a

$\frac{μ_{0} a b c}{2 π {(b - a)}^{2}}$

b

$\frac{μ_{0} c}{2 π} \ln \frac{b}{a}$

c

$\frac{μ_{0} b}{2 π} \ln \frac{a}{b}$

d

None of these

answer is B.

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Detailed Solution

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$\begin{array}{rcl} B & = & \frac{μ_{0}}{2 π} \frac{i}{x} \Rightarrow d S = c d x \\ d ϕ & = & B d S = \frac{μ_{0}}{2 π} \frac{i c}{x} d x \\ ϕ & = & \int_{a}^{b} d ϕ = \frac{μ_{0} i c}{2 π} \ln (\frac{b}{a}) \\ M & = & \frac{ϕ}{i} = \frac{μ_{0} c}{2 π} \ln (\frac{b}{a}) \end{array}$

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