Mutual inductance

One coil’s current, another coil’s flux

Narrated video. Transcript:

Remember Faraday’s two coils. A current in one coil sends magnetic flux through the other. Change that current, and an emf appears in the other coil.

The flux through the second coil is proportional to the current in the first. Double the current, and the field doubles, so the flux doubles too.

So we write: the flux linkage of coil 1, N₁ Φ₁, equals M₁₂ times I₂, the current in coil 2. The constant M₁₂ is the mutual inductance of coil 1 with respect to coil 2.

Let us find it for the simplest case: two long coaxial solenoids of the same length l. The inner one, S₁, has radius r₁ and n₁ turns per unit length. The outer one, S₂, has n₂ turns per unit length.

Pass a current I₂ through the outer solenoid. Inside it the field is uniform, B equals μ₀ n₂ I₂. This field passes through every turn of the inner solenoid, through the area π r₁ squared.

S₁ has n₁ l turns in all. So the flux linkage is n₁ l times π r₁ squared times μ₀ n₂ I₂. Divide by I₂: M₁₂ equals μ₀ n₁ n₂ π r₁ squared l.

Now do it the other way round: current I₁ in the inner solenoid. Its field, μ₀ n₁ I₁, exists only inside S₁, across the area π r₁ squared. It links the n₂ l turns of S₂. We get the same answer: M₂₁ equals M₁₂.

So we simply call it M, the mutual inductance of the pair. Fill the solenoids with a material of relative permeability μr, and M becomes μr μ₀ n₁ n₂ π r₁ squared l.

When the current in one coil changes, the emf in the other is ε₁ equals minus M dI₂ by dt. The SI unit of inductance is the henry, H. One henry is one weber per ampere.

Numbers and uses

Narrated video. Transcript:

Let us put in numbers. Two solenoids, each 0.5 metre long, with 500 turns each, so n₁ and n₂ are both 1000 turns per metre. The inner radius is 2 centimetres.

M equals μ₀ n₁ n₂ π r₁ squared l: 4π × 10⁻⁷ × 1000 × 1000 × π × 4 × 10⁻⁴ × 0.5. That is about 7.9 × 10⁻⁴ henry, or 0.79 millihenry.

Now let the current in one solenoid rise steadily at 2 amperes per second. The emf in the other is M dI by dt: 0.79 millihenry times 2, about 1.6 millivolts.

Watch the graphs. While the current rises steadily, the emf is steady. When the current stays constant, the emf is zero. When the current falls, the emf reverses.

Mutual inductance depends on the shapes of the coils, their numbers of turns, how far apart they are, how they are turned, and what fills them.

Two loops far apart, or at right angles, share very little flux, so M is tiny. A small loop of radius 1 centimetre at the centre of a loop of radius 20 centimetres has M equal to μ₀ π r₁ squared over 2 r₂, only about 1 nanohenry.

Now Faraday’s third experiment makes sense. Pressing the key makes the current rise quickly, so dI by dt is large and the needle kicks. A steady current gives no emf. An iron rod increases μr, and with it M, so the kick is much bigger.

Mutual induction is used every day. A transformer, which you will meet in the next chapter, is two coils on one iron core. A phone’s wireless charger is a pair of flat coils, one in the pad and one in the phone.

Try it: two coils

Interactive simulation. Things to try:

• Close the key (or tap it in the circuit). The secondary’s needle kicks while the current grows, then returns to zero. Open it: it kicks the other way.
• Tick the soft-iron rod. M is 20 times larger, and the kicks grow, even though the current also changes more slowly.
• With the key closed, drag the secondary coil towards and away from the primary. Changing M with a steady current also gives an emf.
• Switch to “Two solenoids” and try the numbers from the video: N₁ = N₂ = 500, l = 0.5 m, r₁ = 2 cm gives M ≈ 0.79 mH.

Change of flux linkage

A pair of adjacent coils has a mutual inductance of 1.5 H. The current in one coil changes from 0 to 20 A in 0.5 s. What is the change of flux linkage with the other coil?

1. 13.3 Wb
2. 15 Wb
3. 30 Wb
4. 60 Wb

Rewinding the solenoids

For two long coaxial solenoids, the number of turns per unit length of both is doubled, and the radius of the inner solenoid is halved. The length stays the same. What happens to M?

1. It becomes 4 times larger
2. It becomes 2 times larger
3. It stays the same
4. It becomes half

Summary

• Definition: N₁Φ₁ = M I₂
• Induced emf: ε₁ = −M dI₂/dt
• Two long coaxial solenoids: M = μr μ₀ n₁ n₂ π r₁² l
• Unit: 1 henry (H) = 1 Wb/A = 1 V s/A

• The flux linking one coil is proportional to the current in another: the constant is the mutual inductance M, and M₁₂ = M₂₁.
• A changing current in one coil induces an emf in the other; a steady current induces none.
• M depends on geometry (size, turns, separation, orientation) and on the medium (μr); an iron core increases it greatly.
• Mutual induction is the working principle of transformers and wireless chargers.