The conducting circular loops of radii `R_(1) and R_(2)` are placed in the same plane with their centres coinciding. If `R_(1) gt gt R_(2)`, the mutual inductance M between them will be directly proportional to

Options

A. `R_(1)//R_(2)`
B. `R_(2)//R_(1)`
C. `R_(1)^(2)//R_(2)`
D. `R_(2)^(2)//R_(1)`

Explanation

To solve the problem of determining the mutual inductance \( M \) between two conducting circular loops of radii \( R_1 \) and \( R_2 \) (where \( R_1 \gg R_2 \)), we can follow these steps:

### Step 1: Understanding Mutual Inductance
Mutual inductance \( M \) is defined as the ratio of the magnetic flux \( \Phi \) through one coil due to the current in another coil. In this case, we will consider the larger loop (radius \( R_1 \)) carrying a current \( I_1 \) and the smaller loop (radius \( R_2 \)).

### Step 2: Magnetic Field at the Center of the Larger Loop
The magnetic field \( B \) at the center of a circular loop of radius \( R_1 \) carrying a current \( I_1 \) is given by the formula:
\[ B = \frac{\mu_0 I_1}{2 R_1} \]
where \( \mu_0 \) is the permeability of free space.

### Step 3: Area of the Smaller Loop
The area \( A_2 \) of the smaller loop (radius \( R_2 \)) is:
\[ A_2 = \pi R_2^2 \]

### Step 4: Magnetic Flux through the Smaller Loop
The magnetic flux \( \Phi_2 \) through the smaller loop due to the magnetic field produced by the larger loop is given by:
\[ \Phi_2 = B \cdot A_2 = \left(\frac{\mu_0 I_1}{2 R_1}\right) \cdot (\pi R_2^2) \]

### Step 5: Expressing Mutual Inductance
The mutual inductance \( M \) can be expressed in terms of the magnetic flux and the current in the larger loop:
\[ M = \frac{\Phi_2}{I_1} = \frac{\left(\frac{\mu_0 I_1}{2 R_1}\right) \cdot (\pi R_2^2)}{I_1} \]
This simplifies to:
\[ M = \frac{\mu_0 \pi R_2^2}{2 R_1} \]

### Step 6: Identifying the Proportionality
From the expression derived for mutual inductance \( M \), we can see that:
\[ M \propto \frac{R_2^2}{R_1} \]
Thus, the mutual inductance \( M \) is directly proportional to \( R_2^2 \) and inversely proportional to \( R_1 \).

### Conclusion
The mutual inductance \( M \) between the two loops is directly proportional to \( R_2^2 \) and inversely proportional to \( R_1 \).