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SarkariResultOutVidyut Sahayak (Junior Engineer - Electrical) · 2024

Previous Year Question

If two coils have self inductance L₁ and L₂ , then mutual inductance M between them is given by:
  1. M = √(L₁/L₂)
  2. M = k/√(L₁L₂)
  3. M = k√(L₁L₂)
  4. M = √(L₁L₂)

Correct answer: C

Explanation

C — M = k√(L₁L₂). The coupling coefficient k accounts for the fraction of magnetic coupling between the two coils.

Formula

M = k√(L₁L₂)
0 ≤ k ≤ 1; M, L₁ and L₂ in H

Step-by-step solution

Correct answer: C — M = k√(L₁L₂)

Relation between self and mutual inductance

Self-inductance describes the flux linkage a coil produces in itself for a given current. Mutual inductance describes the flux linkage produced in one coil by current in the other. Not all the flux generated by one coil necessarily links the second coil; some may follow leakage paths.

The coupling coefficient k expresses the degree of magnetic coupling. For the magnitude of mutual inductance in the usual passive two-coil model:

k = M/√(L₁L₂)
M = k√(L₁L₂), with 0 ≤ k ≤ 1

L₁, L₂ and M are measured in henrys, while k is dimensionless. When k = 0, the coils have no magnetic coupling. When k = 1, coupling is perfect and M reaches √(L₁L₂). Intermediate values represent partial coupling.

Example of using the relation

If L₁ = 4 H, L₂ = 9 H and k = 0.5, first find the geometric mean of the two self-inductances, then apply the coupling factor:

√(L₁L₂) = √(4 × 9) = 6 H
M = 0.5 × 6 = 3 H

The induced voltage due to mutual coupling has magnitude M|di/dt|. Its polarity is determined by the dot convention and reference directions, rather than by changing the magnitude relation above.

Other options

A is dimensionally not generally correct. B puts k in the numerator of an inverse-inductance form and has wrong units. D is only true for perfect coupling k=1. C is the general relation.

Common mistake

Forgetting the coupling coefficient and choosing √(L₁L₂) for all coils.

Exam tip

Use a units check: mutual inductance must be in henries.

Quick method

M = k × geometric mean of L₁ and L₂.