Magnetisation and magnetic intensity

Magnetisation M and intensity H

Narrated video. Transcript:

Zoom into an atom. Electrons orbit the nucleus, and each electron also spins. Each of these motions is a tiny current loop, so an atom can have a tiny magnetic moment.

In an ordinary piece of material, the atomic moments point in all directions, and they cancel. The piece as a whole has no magnetic moment.

Put the material in a magnetic field, and it develops a net magnetic moment. We describe this with the magnetisation, M: the net magnetic moment per unit volume. M = m_net/V.

M is a vector. Its unit is A m² divided by m³, which is ampere per metre, A m⁻¹.

Now take a long solenoid with n turns per unit length, carrying current I. With nothing inside, the field in it is B₀ = μ₀nI.

Fill the solenoid with a material. The field inside becomes B = B₀ + B_m. B_m is the extra field made by the material, and it turns out that B_m = μ₀M.

It is useful to define one more vector, the magnetic intensity H: H = B/μ₀ − M. Turn it round, and B = μ₀(H + M). H has the same unit as M, ampere per metre.

This splits the total field into two parts. μ₀H is due to external causes, like the current in the solenoid. μ₀M is due to the material itself.

For the solenoid, H = nI. It depends only on the turns and the current, not on what is inside. Push an iron core in, keep the current the same, and H does not change, but B does.

Susceptibility and permeability

Narrated video. Transcript:

How strongly does a material respond? For most materials, the magnetisation is proportional to the magnetic intensity: M = χH.

χ, the Greek letter chi, is the magnetic susceptibility. Since M and H have the same unit, χ is a pure number. It is small and positive for some materials, small and negative for others, and very large for iron.

Put M = χH into B = μ₀(H + M). We get B = μ₀(1 + χ)H. We write this as B = μ₀μ_r H = μH.

μ_r = 1 + χ is the relative magnetic permeability, also a pure number. μ = μ₀μ_r is the magnetic permeability of the material. For vacuum, χ = 0 and μ_r = 1.

Worked example. A solenoid has 1000 turns per metre and carries a current of 2 A. Its core has relative permeability 400. First, H = nI = 1000 × 2 = 2 × 10³ A m⁻¹.

The field: B = μ_r μ₀ H = 400 × 4π × 10⁻⁷ × 2 × 10³. That is about 1.0 tesla. Without the core, B₀ = μ₀nI would be only 2.5 × 10⁻³ tesla. The core multiplies the field 400 times.

The magnetisation: M = χH = (μ_r − 1)H = 399 × 2 × 10³, which is about 8 × 10⁵ A m⁻¹. Almost all of B comes from μ₀M, the part due to the core.

Here is a way to picture it. B = μ₀n(I + I_m), as if an extra current I_m = M/n were flowing in the winding. That is 8 × 10⁵ ÷ 1000, about 800 A, carried not by the wire but by the atoms of the core.

Try it: a solenoid with a core

Interactive simulation. Things to try:

• Start with the worked example: n = 1000 m⁻¹, I = 2 A, iron with μ_r = 400. Check H, M and B against the video.
• Switch the core between vacuum, bismuth, aluminium and iron. H = nI never changes; M and B do.
• Bismuth makes B very slightly smaller than B₀ (χ < 0); aluminium makes it very slightly larger (χ > 0).
• Watch the dipoles in the core: they line up with H in aluminium and iron, and point against it in bismuth.

Field with a core

A solenoid with 500 turns per metre carries a current of 4 A. Its core has relative permeability μ_r = 500. What is the magnetic field B inside the core?

1. 2.5 × 10⁻³ T
2. 1.26 T
3. 0.63 T
4. 1.0 × 10⁶ T

What stays the same?

An iron core is pushed into a solenoid while the current in it is kept the same. Which quantity inside the solenoid does NOT change?

1. The magnetic field B
2. The magnetisation M
3. The magnetic intensity H
4. The field due to the material, μ₀M

Summary

• Magnetisation: M = m_net / V (A m⁻¹)
• Magnetic intensity: H = B/μ₀ − M, so B = μ₀(H + M)
• Susceptibility: M = χH
• Permeability: B = μ₀μ_r H = μH, μ_r = 1 + χ

• Magnetisation M is the net magnetic moment per unit volume of a material. Its unit is A m⁻¹.
• Magnetic intensity H comes from external causes, such as currents. In a solenoid H = nI, whatever the core.
• The material adds B_m = μ₀M to the field. The total is B = μ₀(H + M).
• χ and μ_r are pure numbers. χ is small and negative, small and positive, or very large, depending on the material.