A

To calculate the mutual inductance of a rectangular coil placed near a long current-carrying wire, we need to use the formula for mutual inductance, which can be expressed as:
$$ M = \frac{\mu_0 N A}{l} $$
where $\mu_0$ is the permeability of free space, $N$ is the number of turns in the coil, $A$ is the area of the coil, and $l$ is the distance between the wire and the coil.

• Step 1: Identify the values from the image. Let's assume the coil has width $b$, height $h$, and laminated along a distance $l$. Hence, the area $A = b \times h$.
• Step 2: Determine the number of turns $N$. If the coil is a single loop, then $N = 1$.
• Step 3: Substitute into the formula. Assuming $r$ is the distance from the wire to the coil, i.e., $l = r$. Then, we can write:
  $$ M = \frac{\mu_0 (1) A}{r} = \frac{\mu_0 (b \times h)}{r} $$

Therefore, the mutual inductance for the rectangular coil near the long wire is given by the final expression.