An engineer doubles the current flowing through coil 1, while keeping everything else the same. What happens to the mutual inductance M between the two coils?
AM doubles, because flux through coil 2 doubles when current in coil 1 doubles
BM stays the same, because M is determined by geometry, not by the magnitude of the current
CM halves, because the induced EMF in coil 2 must remain constant
DM increases by √2, following the same relationship as self-inductance
Mutual inductance M is a purely geometric quantity — it depends on the sizes, shapes, and relative positions of the coils, not on what current happens to be flowing. The Explainer states this explicitly: 'The mutual inductance M is a purely geometric quantity.' When current doubles, the flux Φ₂₁ = MI₁ also doubles, and the induced EMF ε₂ = −M(dI₁/dt) changes as well — but M itself is unchanged. Confusing 'M stays fixed while the flux and EMF change' is one of the most common errors in this topic.
Question 2 Multiple Choice
Two coils are placed far apart in free space, with only a tiny fraction of coil 1's flux threading coil 2. What does this imply about the coupling coefficient k?
Ak ≈ 1, because the coils are physically separated and experience no interference
Bk ≈ 0, because almost none of coil 1's flux links coil 2, indicating very weak coupling
Ck depends only on the ratio of self-inductances L₁/L₂, not on the shared flux
Dk = M/√(L₁L₂) = 0.5 for coils in free space, by convention
The coupling coefficient k = M/√(L₁L₂) measures what fraction of coil 1's flux reaches coil 2. When coils are far apart, very little flux is shared, M is tiny relative to √(L₁L₂), and k approaches 0. Conversely, k ≈ 1 means nearly all the flux links both coils — achieved in a transformer wound on a high-permeability iron core that guides the flux. Option A confuses physical separation with strong coupling; physical separation gives weak coupling (k → 0), not strong.
Question 3 True / False
The mutual inductance M₁₂ (flux through coil 2 per unit current in coil 1) equals M₂₁ (flux through coil 1 per unit current in coil 2), even if the coils have very different sizes.
TTrue
FFalse
Answer: True
The Explainer calls this 'a non-obvious but powerful result' and attributes it to 'the reciprocity of the magnetic vector potential.' Geometrically, it seems strange: driving current through a small coil near a large coil and driving current through the large coil near the small one seem like different configurations with different amounts of flux threading the other coil. But the symmetry holds rigorously. In practice, it means you can calculate M from whichever configuration is mathematically easier, and the result applies to both directions.
Question 4 True / False
If you increase the current flowing through coil 1 more rapidly (increase dI₁/dt), the mutual inductance M between the coils increases.
TTrue
FFalse
Answer: False
Increasing dI₁/dt increases the induced EMF in coil 2 (ε₂ = −M·dI₁/dt), but M itself is unchanged. M depends entirely on the geometric configuration of the coils — their sizes, shapes, separation, and orientation. The rate of change of current affects how much EMF is induced, but it does not alter the proportionality constant M. This is analogous to self-inductance: increasing dI/dt in a single coil increases its back-EMF without changing L.
Question 5 Short Answer
Why is the symmetry M₁₂ = M₂₁ described as 'non-obvious,' and what makes it true despite the apparent asymmetry between differently sized coils?
Think about your answer, then reveal below.
Model answer: It is non-obvious because the geometry seems asymmetric: if coil 1 is small and coil 2 is large, driving current through small coil 1 sends a concentrated magnetic field that threads a small area of large coil 2, while driving current through large coil 2 sends a dispersed field over the area of small coil 1. Intuitively, the flux linkages seem different. The symmetry holds because of the reciprocity of the magnetic vector potential — a deep result from electromagnetic theory that shows the mutual flux Φ₂₁/I₁ and Φ₁₂/I₂ are always equal, regardless of size mismatch. This allows engineers to calculate M from whichever direction is easier and trust the result applies both ways.
The practical value of this symmetry is significant in circuit design. It means M is a single number characterizing the coupling between two inductors, not two different numbers depending on which coil is the 'source.' This simplifies transformer analysis, coupled-resonator circuits, and wireless power transfer calculations. The deeper reason — reciprocity of the vector potential — is a consequence of the linearity of Maxwell's equations and is a prototype of broader reciprocity theorems in physics.