Inductance - Mutual Inductance

Every time you charge your phone wirelessly, use a transformer, or start a car's ignition, mutual inductance is at work. It is the fundamental principle behind transformers, induction cooktops, RFID cards, and wireless charging pads.

The property of two coils by which a change in current in one coil induces an emf in the other coil — equal to the magnetic flux linked with one circuit per unit current in the other, or the value of induced emf produced in the secondary circuit per unit rate of change in current in the primary circuit — is called mutual inductance.

OR

Mutual Inductance (M) of a pair of coils is defined as the ratio of the total magnetic flux linkage in the secondary coil to the current in the primary coil that produces it.

The coefficient of coupling K between two coils is the fraction of the total magnetic flux produced by one coil that links with the other coil.

N₂​ϕ₂₁ ​∝ I_1​ ⟹ N_2​ϕ_21​ = M ⋅ I₁

Therefore:

M = K\[\sqrt {L_1L_2}\]

Where:

• L₁, L₂​ = Self-inductances of coil 1 and coil 2
• K = Coefficient of coupling (dimensionless, no units)
• Range: 0 ≤ K ≤ 1

Therefore:

M ≤ \[\sqrt {L_1​L_2}\]​​

From Faraday's Second Law, the EMF induced in the secondary coil is:

\[e_2=-N_2\frac{d\phi_{21}}{dt}=-M\frac{dI_1}{dt}\]

If the rate of change of current in the primary is 1 A/s and the induced EMF in the secondary is 1 V, then M = 1 H (one Henry).

1 H = 1 V·s/A = 1 Wb/A

Statement: The mutual inductance of coil 1 with respect to coil 2 equals the mutual inductance of coil 2 with respect to coil 1.

This is called the Reciprocity Theorem of Mutual Inductance.

Implication: It does not matter which coil drives the current — the mutual inductance M between the pair is always the same property of the system, not just one coil.youtube

The value of M between two coils depends on the following:

• Number of turns N₁​ and N_2​ — M increases as turns increase (M ∝ N₁N₂)
• Size and geometry of the coils — larger cross-sectional area → more flux linkage → larger M
• Distance between coils — as distance increases, M decreases (less flux links the secondary)
• Relative orientation — M is maximum when coils are coaxial (same axis); M = 0 when coil axes are perpendicular to each other
• Permeability of the core medium (μr\mu_rμr​) — winding coils on a soft iron core greatly increases M
• Coupling factor K — quantifies how efficiently flux is shared between the coils

Setup:

• Solenoid S₁ (primary): length l, N_1​ turns, n₁ = N₁/l turns/m, radius r₁ (larger)
• Solenoid S₂ (secondary): N₂​ turns, n₂ = N₂/l​ turns/m, radius r₂ (smaller, wound inside S₁)
• Current I₁​ passed through S₁

Step 1: Magnetic field inside S₁:

Step 2: Flux through each turn of S₂ (area = \[\pi r_2^2\], the smaller radius):

Step 3: Total flux linkage of S₂:

Step 4: Mutual Inductance:

Or equivalently (using N₁ = n₁l and N₂ = n₂l):

where A = \[\pi r_2^2\]​ (cross-sectional area of the inner solenoid).

Similarly: M12 = μ₀n₁n₂\[\pi r_2^2\]l, confirming Reciprocity: M₁₂ = M₂₁ = M.

For a small coil of radius r₁​ placed at the centre of a large coil of radius r₂​ (where r₁ ≪ r_2​):

M = \[\frac{\mu_0\pi r_1^2}{2r_2}\]

Two concentric circular coils are placed with the same centre and axis. The small coil has radius r₁=​ and the large coil has radius r₂​, with r₁ ≪ r₂​.

A current I₂​ is passed through the outer large coil. This creates a magnetic field at its centre:

Since the inner coil is very small compared to the outer coil, this field B₂ can be assumed uniform across the entire area of the inner coil. This is the key approximation in the problem.

The magnetic flux passing through the inner small coil is then:

By definition, Φ₁ = M₁₂ ⋅ I₂. Comparing both sides gives the mutual inductance:

Finally, by the Reciprocity Theorem, M₁₂ = M₂₁ = M, so this single value describes the mutual inductance of the entire arrangement regardless of which coil carries the current.