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(B) The magnetic field $B$ at a distance $x$ from an infinitely long wire carrying current $I$ is given by $B = \frac{{\mu _0}I}{2\pi x}$.Consider a s

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$\frac{{\mu _0}c}{{2\pi }}\ln \frac{b}{a}$

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(B) The magnetic field $B$ at a distance $x$ from an infinitely long wire carrying current $I$ is given by $B = \frac{{\mu _0}I}{2\pi x}$.Consider a small rectangular strip of width $dx$ and length $c$ at a distance $x$ from the wire $AB$.The magnetic flux $d\phi$ through this strip is $d\phi = B \cdot dA = \left( \frac{{\mu _0}I}{2\pi x} \right) (c \cdot dx)$.The total magnetic flux $\phi$ through the coil $PQRS$ is obtained by integrating from $x = a$ to $x = b$:$\phi = \int_{a}^{b} \frac{{\mu _0}Ic}{2\pi x} dx = \frac{{\mu _0}Ic}{2\pi} \int_{a}^{b} \frac{1}{x} dx$.$\phi = \frac{{\mu _0}Ic}{2\pi} [\ln x]_{a}^{b} = \frac{{\mu _0}Ic}{2\pi} \ln \left( \frac{b}{a} \right)$.The mutual inductance $M$ is defined as $M = \frac{\phi}{I}$.Therefore,$M = \frac{{\mu _0}c}{2\pi} \ln \left( \frac{b}{a} \right)$.

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

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