Compute mutual inductance: Given coupling coefficient k = 0.65, L1 = 2 µH, and L2 = 5 µH, find the mutual inductance M.

Introduction / Context:
Mutual inductance quantifies how strongly two inductors couple magnetically. It is central to transformer action and coupled inductor behavior in filters and converters.

Given Data / Assumptions:

• k = 0.65 (dimensionless coupling coefficient).
• L1 = 2 µH, L2 = 5 µH.
• Ideal linear conditions; standard mutual inductance formula applies.

Concept / Approach:
The mutual inductance is M = k * sqrt(L1 * L2). Use consistent units (here, microhenries) and take the square root carefully.

Step-by-Step Solution:
Compute product: L1 * L2 = 2 * 5 = 10 (µH^2)sqrt(10) ≈ 3.1623 µHM = 0.65 * 3.1623 ≈ 2.055 µHRounded to listed options: 2 µH

Verification / Alternative check:
Because k < 1, M must be less than sqrt(L1 * L2) ≈ 3.162 µH; 2 µH fits this bound and the calculation.

Why Other Options Are Wrong:

• 4 µH or 8 µH: Exceed sqrt(L1 L2) without even applying k, impossible with k ≤ 1.
• 2 mH: Wrong unit and magnitude by 10^3.

Common Pitfalls:
Forgetting to multiply by k or mixing units (mH vs µH). Always verify that M ≤ sqrt(L1 L2).

Final Answer:
2 µH