A square conducting loop of side `L` is situated in gravity free space. A small conducting circular loop of redius `r (rlt lt L)` is placed at the center of the square loop, with its plane perpendicular to the plane of the square loop. The mutual inductance of the two coils is

Options

A. `(2sqrt(2)mu_(0)I)/(L)r^(2)`
B. `(sqrt(2)mu_(0)I_(0))/(L)r^(2)`
C. `0`
D. None of these

Explanation

To find the mutual inductance \( M \) between a square conducting loop and a small circular loop placed at its center, we can follow these steps:

### Step 1: Understand the Configuration
We have a square loop with side length \( L \) and a small circular loop with radius \( r \) (where \( r \ll L \)) placed at the center of the square loop. The plane of the circular loop is perpendicular to the plane of the square loop.

### Step 2: Analyze the Magnetic Field
The square loop generates a magnetic field due to the current flowing through it. The magnetic field produced by a square loop at its center is directed perpendicular to the plane of the loop.

### Step 3: Determine the Orientation
Since the circular loop is placed perpendicular to the plane of the square loop, the magnetic field produced by the square loop is parallel to the plane of the circular loop.

### Step 4: Apply the Mutual Inductance Formula
The mutual inductance \( M \) between two coils can be defined as:
\[ M = \frac{\Phi}{I} \]
where \( \Phi \) is the magnetic flux through one loop due to the current \( I \) in the other loop.

### Step 5: Evaluate the Magnetic Flux
Since the magnetic field produced by the square loop is parallel to the plane of the circular loop, the angle \( \theta \) between the magnetic field and the normal to the circular loop is \( 90^\circ \). The magnetic flux \( \Phi \) through the circular loop can be expressed as:
\[ \Phi = B \cdot A \cdot \cos(\theta) \]
where \( A \) is the area of the circular loop and \( B \) is the magnetic field strength. Since \( \cos(90^\circ) = 0 \), we find that:
\[ \Phi = B \cdot A \cdot 0 = 0 \]

### Step 6: Conclude the Mutual Inductance
Since the magnetic flux \( \Phi \) is zero, the mutual inductance \( M \) is also zero:
\[ M = 0 \]

### Final Answer
Thus, the mutual inductance of the two coils is:
\[ \boxed{0} \]