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

• A

  $\frac{\mu_0}{2 \pi} a b$
• B

  $\frac{\mu}{4 \pi} \frac{b}{a}$
• C

  $\frac{\mu_0 a}{2 \pi} \ln 2$
• Correct answer
  D

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

Solution

Magnetic field through the wire, $B=\frac{\mu_0 I b}{2 \pi a}$.
$\begin{array}{ll}\therefore \text { Magnetic flux induced, } \phi & =\oint_s \mathbf{B} \cdot d \mathbf{a} \\ \Rightarrow & \phi=\frac{\mu_0 I b}{2 \pi} \int_a^{a+a} \frac{d a}{a} \\ \Rightarrow & \phi=\frac{\mu_0 I b}{2 \pi} \ln 2\end{array}$ $\therefore$ Mutual inductance of the wire and the frame is $$ M=\frac{\phi}{I}=\frac{\mu_0 b}{2 \pi} \ln 2 $$

Hence, the correct answer is option: 'D'.