Side of square is \( L \) and \( R \ll L \). Find mutual inductance of the system shown in the figure.
Show Hint
In problems involving mutual inductance, use the formula involving the geometry of the coils and the distance between them. For small loops in large systems, approximations can simplify the calculation.
Step 1: Understanding the system.
In this question, we are asked to find the mutual inductance of a system involving a square of side length \( L \) and a loop with radius \( R \). The given condition \( R \ll L \) implies that the radius of the loop is much smaller than the side of the square.
Step 2: Formula for mutual inductance.
The mutual inductance between two coils is given by:
\[
M = \frac{\mu_0}{4 \pi} \int \int \frac{d\ell_1 d\ell_2}{r}
\]
Where:
- \( d\ell_1 \) and \( d\ell_2 \) are differential length elements of the two coils,
- \( r \) is the distance between these elements,
- \( \mu_0 \) is the permeability of free space.
Step 3: Applying the formula to the given system.
For the given configuration with a square of side \( L \) and a small loop with radius \( R \), we can use the following approximation for mutual inductance:
\[
M = \frac{2 \sqrt{2 \mu_0} R^2}{L}
\]
Step 4: Conclusion.
Thus, the mutual inductance of the system is \( \frac{2 \sqrt{2 \mu_0} R^2}{L} \).
Final Answer: \( \frac{2 \sqrt{2 \mu_0} R^2}{L} \)
