Mutual Inductance of Two Long Coaxial Solenoids

When two coils are placed close to each other, a change in current in one coil induces an EMF in the neighbouring coil. This phenomenon is called mutual induction, and it forms the basis of transformers, induction coils, and wireless charging systems.

Mutual inductance quantifies how effectively one coil can "communicate" a changing magnetic flux to another coil.

Setup:

• Primary solenoid P: N₁​ turns, length l, area A, carries current I₁
• Secondary solenoid S: N₂​ turns, wound coaxially over/around P

Step 1: Magnetic field inside primary solenoid:

Step 2: Flux linked with one turn of secondary:

Step 3: Total flux linkage with secondary (N₂ turns):

Step 4: Since N₂Φ₂ = MI₁:

Alternatively, writing turns per unit length n₁ = N₁/l:

• Number of turns in both coils (N_1​, N₂​) — more turns → higher M
• Common cross-sectional area A — larger overlap area → higher M
• Distance/separation between coils — greater separation → lower M
• Relative orientation — maximum M when coils are coaxial, and one is wound directly over the other
• Core material — inserting a ferromagnetic (iron) core greatly increases M due to higher permeability

Current changes from 2 A to 6 A in 2 s in coil P, inducing an EMF of 20 mV in coil S. Find M.

Answer: M = 10 mH

A solenoid has 600 turns, length 1 m, radius 2 cm; a coil of 100 turns is wound over it. Find M.

Using M = μ₀n₁N₂A, with n₁ = 600, N₂ = 100, A = π(0.02)²:

Answer: M ≈ 9.47 × 10^-5 H

• Transformers use mutual inductance between primary and secondary windings to step up/down voltage.
• Wireless charging pads transfer energy between coils via mutual induction, similar to the coaxial solenoid model.
• Analogy: Think of the primary coil as a “broadcasting antenna” and the secondary as a “receiver” — the closer and more aligned they are, the stronger the signal (flux) received.