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(C) The magnetic field $B$ at a distance $r$ from a long straight wire carrying current $I$ is given by $B = \frac{\mu_0 I}{2 \pi r}$.Consider a small

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$\frac{\mu_0 b}{2 \pi} \ln 2$

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(C) The magnetic field $B$ at a distance $r$ from a long straight wire carrying current $I$ is given by $B = \frac{\mu_0 I}{2 \pi r}$.Consider a small rectangular strip of width $dr$ at a distance $r$ from the wire within the frame. The area of this strip is $dA = b \cdot dr$.The magnetic flux $d\phi$ through this strip is $d\phi = B \cdot dA = \left( \frac{\mu_0 I}{2 \pi r} \right) (b \cdot dr) = \frac{\mu_0 I b}{2 \pi} \frac{dr}{r}$.To find the total magnetic flux $\phi$ through the frame,we integrate from $r = a$ to $r = 2a$:$\phi = \int_a^{2a} \frac{\mu_0 I b}{2 \pi} \frac{dr}{r} = \frac{\mu_0 I b}{2 \pi} [\ln r]_a^{2a} = \frac{\mu_0 I b}{2 \pi} \ln \left( \frac{2a}{a} \right) = \frac{\mu_0 I b}{2 \pi} \ln 2$.The mutual inductance $M$ is defined as $M = \frac{\phi}{I}$.Therefore,$M = \frac{\mu_0 b}{2 \pi} \ln 2$.

$A$ long straight wire carrying current $I$ and a rectangular frame with side lengths $a$ and $b$ lie in the same plane as shown in the figure. The mutual inductance of the wire and frame is

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