Question

Find the mutual inductance in the arrangement, when a small circular loop of radius R is placed inside a large square loop of side \(L\) (\(L \gg R\)). The loops are coplanar and their centers coincide:

• A. \(M = \frac{\sqrt{2} \mu_0 R^2}{L}\)
• B. \(M = \frac{2 \sqrt{2} \mu_0 R}{L^2}\)
• C. \(M = \frac{2 \sqrt{2} \mu_0 R^2}{L}\)
• D. \(M = \frac{\sqrt{2} \mu_0 R}{L^2}\)

Hint:

 For mutual inductance in coplanar loops:
• Use the magnetic field from the larger loop and calculate flux through the smaller loop.
• Divide flux by the current to obtain mutual inductance.

Answer 1

The Correct Option is C

\[ \phi = M i \] 
\[ \phi = (BA) \]
\[ \phi = \pi R^2 \left( \frac{4\mu_0}{4\pi} \cdot i \cdot \frac{L}{2} \right) \sqrt{2} \]
\[ \implies M = \frac{2\sqrt{2} \mu_0 R^2}{L} \]