A

To calculate the mutual inductance of a long wire and a rectangular coil, we can follow these steps:

**Step 1: Understand the Configuration**
We have a long straight wire carrying a current I₁, and a rectangular coil with dimensions h (height) and b (width) placed parallel to the wire. The distance between the wire and the nearest side of the coil is r.

**Step 2: Magnetic Field due to the Long Wire**
The magnetic field (B) at a distance r from a long straight wire carrying a current I₁ is given by Ampère’s Law:
$$B = \frac{{\mu_0 I_1}}{{2\pi r}}$$
where \( \mu_0 \) is the permeability of free space.

**Step 3: Calculate the Magnetic Flux through the Coil**
The total magnetic flux (Φ) through the rectangular coil is the integral of the magnetic field (B) over the area (A) of the coil. The flux is given by:
$$\Phi = \int B \cdot dA$$
In this case, the area (dA) is the width (b) times an infinitesimal height (dr):
$$\Phi = \int_0^h B(r) b \, dr$$
Substituting B:
$$\Phi = b \int_0^h \frac{{\mu_0 I_1}}{{2\pi (r + y)}} \, dy$$
where y is a displacement along the height of the coil.

**Step 4: Mutual Inductance Definition**
The mutual inductance (M) can be calculated using the relation:
$$M = \frac{\Phi}{I_1}$$
After evaluating the integral and simplifications, the mutual inductance would depend on the dimensions of the coil (b, h) and the distance (r).

**Conclusion**
If exact numerical values are provided, we could compute a precise value for M, but in general, the mutual inductance will depend on the geometry and configuration provided. Thus, without specific numerical options, we conclude this process here.