A small square loop of wire of side l is placed inside a large square loop of wire of side `L(Lgtgtl)`. The loops are co-planer and their centres coincide. The mutual inductance of the system is proportional to

Options

A. `l/L`
B. `(l^(2))/L`
C. `L/l`
D. `(L^(2))/l`

Explanation

To solve the problem of finding the mutual inductance of a small square loop placed inside a larger square loop, we will follow these steps:

### Step 1: Understand the Configuration
We have two square loops:
- A small loop with side length \( l \)
- A large loop with side length \( L \) where \( L \gg l \)

Both loops are coplanar and their centers coincide.

### Step 2: Determine the Magnetic Flux
The magnetic flux (\( \Phi \)) linked with the larger loop due to the smaller loop can be expressed as:
\[ \Phi = B \cdot A \]
where \( B \) is the magnetic field due to the smaller loop and \( A \) is the area of the larger loop.

The area \( A \) of the larger loop is:
\[ A = L^2 \]

### Step 3: Calculate the Magnetic Field
The magnetic field \( B \) at the center of the smaller loop carrying a current \( I \) can be derived from Ampere's Law or Biot-Savart Law. For a square loop, the magnetic field at its center is given by:
\[ B = \frac{2 \sqrt{2} \mu_0 I}{\pi l} \]
where \( \mu_0 \) is the permeability of free space.

### Step 4: Substitute into the Flux Equation
Now substituting \( B \) into the flux equation:
\[ \Phi = \left(\frac{2 \sqrt{2} \mu_0 I}{\pi l}\right) \cdot L^2 \]

### Step 5: Calculate Mutual Inductance
The mutual inductance \( M_{12} \) is defined as the ratio of the magnetic flux linked with one loop to the current in the other loop:
\[ M_{12} = \frac{\Phi}{I} \]
Substituting the expression for \( \Phi \):
\[ M_{12} = \frac{2 \sqrt{2} \mu_0 L^2}{\pi l} \]

### Step 6: Analyze Proportionality
From the equation \( M_{12} = \frac{2 \sqrt{2} \mu_0 L^2}{\pi l} \), we can see that the mutual inductance is proportional to:
\[ M_{12} \propto \frac{l^2}{L} \]
Thus, the mutual inductance of the system is proportional to \( \frac{l^2}{L} \).

### Final Answer
The mutual inductance of the system is proportional to \( \frac{l^2}{L} \).

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