Mutual inductance from coupling coefficient: Two coils have L1 = 75 mH and L2 = 105 mH with coefficient of coupling k = 0.45. What is the mutual inductance M between the coils?

Introduction / Context:
Mutual inductance quantifies how effectively a changing current in one coil induces voltage in another via shared magnetic flux. In coupled-coil problems, the coupling coefficient k (0 ≤ k ≤ 1) summarizes how much of the flux links both coils. Computing M from L1, L2, and k is a common task in transformer design and filter analysis.

Given Data / Assumptions:

• L1 = 75 mH.
• L2 = 105 mH.
• Coefficient of coupling k = 0.45.
• Linear magnetic operation (no saturation) and sinusoidal excitation.

Concept / Approach:

The relationship between mutual inductance and the self-inductances is M = k * sqrt(L1 * L2). This follows from defining k as the fraction of flux linkage shared between the coils. Units must be consistent (use H, not mH) during calculations, then convert the result to mH for the final answer.

Step-by-Step Solution:

Verification / Alternative check:

Since k < 1, M must be less than sqrt(L1 * L2) ≈ 88.9 mH; 39.9 mH satisfies this bound. Also, M must be greater than 0; the computed value is reasonable given moderate coupling.

Why Other Options Are Wrong:

• 3.54 mH / 7.88 mH: Too small; would imply an unrealistically tiny k for the given L1 and L2.
• 189.3 mH / 450 mH: Exceed sqrt(L1 * L2) or even L values, impossible for k ≤ 1.

Common Pitfalls:

• Mixing mH and H units, leading to errors by factors of 1000.
• Using M = k * (L1 + L2) incorrectly; the correct relation uses the geometric mean.

Final Answer:

39.9 mH