Step 1: Understanding the Concept:
Mutual inductance is the property of two coils such that a change in current in one induces an EMF in the other.
To find mutual inductance, we pass a current through one loop, calculate the magnetic field it produces, and then find the magnetic flux linked with the second loop.
Step 2: Key Formulas or Approach:
Magnetic field at the center of a circular loop: \( B = \frac{\mu_0 I}{2R} \).
Magnetic flux: \( \Phi = B \times A \), where \( A \) is the area.
Mutual inductance definition: \( M = \frac{\Phi_2}{I_1} \).
Step 3: Detailed Explanation:
Let's pass a current \( I_1 \) through the larger outer loop of radius \( R_1 \).
The magnetic field produced by this loop at its center is:
\[ B_1 = \frac{\mu_0 I_1}{2R_1} \]
Since \( R_1>R_2 \) (typically implying \( R_1 \gg R_2 \) in such standard textbook problems for the formula to be highly accurate, but the proportionality holds under the approximation that the field is roughly uniform over the small central area), we assume this field \( B_1 \) is uniform over the area of the smaller inner loop.
The area of the inner loop is \( A_2 = \pi R_2^2 \).
The magnetic flux \( \Phi_2 \) linked with the smaller inner loop due to the field of the outer loop is:
\[ \Phi_2 = B_1 \times A_2 \]
\[ \Phi_2 = \left( \frac{\mu_0 I_1}{2R_1} \right) \times (\pi R_2^2) \]
\[ \Phi_2 = \left( \frac{\mu_0 \pi R_2^2}{2R_1} \right) I_1 \]
From the definition of mutual inductance \( \Phi_2 = M \cdot I_1 \), we get:
\[ M = \frac{\mu_0 \pi R_2^2}{2R_1} \]
Looking at the terms, \( \mu_0 \) and \( \pi \) and \( 2 \) are constants.
Therefore, the mutual inductance \( M \) is directly proportional to \( \frac{R_2^2}{R_1} \).
\[ M \propto \frac{R_2^2}{R_1} \]
Step 4: Final Answer:
The mutual inductance \( M \) is directly proportional to \( \frac{R_2^2}{R_1} \).