H, Susceptibility & Permeability
Separating the field you apply from the field matter adds.
Lesson at a glanceShow the key points
- H⃗ = B⃗/μ₀ − I⃗ (37.2); in vacuum H = B/μ₀.
- With no end effects, H depends only on external currents (solenoid: H = ni).
- I⃗ = χH⃗ (37.6): χ > 0 para, χ < 0 dia.
- B⃗ = μH⃗, μ = μ₀(1 + χ), μ_r = 1 + χ (37.7–37.8).
Inside matter the field B has two sources: the currents we control, and the material’s own magnetization. Separating them gives three useful quantities: H, χ and μ.
Magnetic intensity H
In vacuum I = 0 and H = B/μ₀. So the familiar formulas carry over with μ₀ removed:
Where the ends of a magnetized body don’t matter (a ring, or the middle of a long rod), magnetization makes no H: H is set by the external currents alone, magnetized material or not.
A long solenoid has n turns per metre carrying i. Find H at its centre (a) empty, (b) with a long copper rod inside.
Susceptibility
For paramagnetic and diamagnetic materials the magnetization is proportional to H:
| Material | χ (at about 20 °C) |
|---|---|
| Oxygen gas (STP) | +0.18 × 10⁻⁵ |
| Aluminium | +2.2 × 10⁻⁵ |
| Titanium | +7.06 × 10⁻⁵ |
| Hydrogen (1 atm) | −9.9 × 10⁻⁹ |
| Sodium | −0.24 × 10⁻⁵ |
| Copper | −0.96 × 10⁻⁵ |
| Lead | −1.6 × 10⁻⁵ |
| Silver | −2.6 × 10⁻⁵ |
| Gold | −3.6 × 10⁻⁵ |
| Bismuth | −16.6 × 10⁻⁵ |
Permeability
From H = B/μ₀ − I and I = χH: B = μ₀(H + χH) = μ₀(1 + χ)H.
By what percentage does B change when a toroid is filled with copper (χ = −0.96 × 10⁻⁵)?
A long solenoid with 30 turns per cm carries 3.0 A. Find H and B at its centre. An iron core is inserted and becomes magnetized to I = 3.0 × 10⁶ A/m. Find the new H and B.
An iron core of cross-section 4.0 cm² sits in a long solenoid of 1500 turns/m carrying 2.0 A. The field in the core is 1.26 T. Find its magnetization and the pole strength at its ends.
- 1H⃗ = B⃗/μ₀ − I⃗ (37.2); H depends only on external currents when end effects are absent.
- 2I⃗ = χH⃗ (37.6): χ > 0 paramagnetic, χ < 0 diamagnetic.
- 3B⃗ = μH⃗, μ = μ₀(1 + χ), μ_r = 1 + χ (37.7–37.8).
Check yourself
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