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_(1)`
D. `R_(2)^(2)//R_(1)`

Answer

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

### Step-by-Step Solution:

1. **Understanding the Setup**:
- We have two circular loops: an outer loop with radius \( R_1 \) and an inner loop with radius \( R_2 \). The centers of both loops coincide.

2. **Magnetic Field Due to the Outer Loop**:
- The magnetic field \( B \) at the center of a circular loop carrying current \( I_1 \) is given by the formula:
\[ B = \frac{\mu_0 I_1}{2 R_1} \]
- Here, \( \mu_0 \) is the permeability of free space.

3. **Calculating the Magnetic Flux Through the Inner Loop**:
- The magnetic flux \( \Phi \) through the inner loop due to the magnetic field created by the outer loop is given by:
\[ \Phi = B \cdot A \]
- The area \( A \) of the inner loop is:
\[ A = \pi R_2^2 \]
- Substituting the expression for \( B \):
\[ \Phi = \left(\frac{\mu_0 I_1}{2 R_1}\right) \cdot \pi R_2^2 \]

4. **Relating Flux to Mutual Inductance**:
- The mutual inductance \( M \) is defined as:
\[ M = \frac{\Phi}{I_1} \]
- Substituting the expression for \( \Phi \):
\[ M = \frac{\left(\frac{\mu_0 I_1}{2 R_1}\right) \cdot \pi R_2^2}{I_1} \]
- The \( I_1 \) terms cancel out:
\[ M = \frac{\mu_0 \pi R_2^2}{2 R_1} \]

5. **Identifying Proportionality**:
- From the final expression for \( M \), we can see that:
\[ M \propto \frac{R_2^2}{R_1} \]
- This indicates that the mutual inductance \( M \) is directly proportional to \( R_2^2 \) and inversely proportional to \( R_1 \).

### Final Answer:
The mutual inductance \( M \) between the two loops is directly proportional to \( \frac{R_2^2}{R_1} \).