Electromagnetics • Power • Motion • Materials

Magnetic Fields Refresher (Engineering): E, H, B, Materials, and Practical Design

Fred Fisher President & Principal Engineer | Validus Group | U.S. Precision Manufacturing Industrial troubleshooting, controls integration, and custom equipment

Clear meaning of E, H, B Maxwell’s equations in practice Cores, gaps, losses, and scaling

Executive Summary Field Variables: E, D, H, B Constitutive Relations Maxwell’s Equations Boundary Conditions Magnetic Circuits Energy & Inductance Materials & B–H Eddy Currents & Skin Depth Worked Example (Gapped Core)

Executive Summary

This paper is a compact engineering refresher on magnetic fields with emphasis on the symbols used in real design work: E, H, B, and how materials change the story. It connects the “textbook” formulation (Maxwell’s equations and constitutive laws) to practical tasks like sizing coils, understanding air-gaps, predicting saturation, and estimating loss mechanisms.

Key takeaway: H is primarily set by free current (windings / conductors), while B is the total flux density that causes force and induction. In materials, B includes both the applied field and the material response (magnetization).

Field Variables: E, D, H, B

In electromagnetics, it helps to remember the “intensity vs flux density” pairing: E and H are field intensities (how strongly the field is “driven”), while D and B are flux densities (how much “flux” exists in the medium).

Symbol	Name	Units	Practical meaning / where it shows up
E	Electric field	V/m	Voltage gradients, insulation stress, induced fields from changing B
D	Electric flux density	C/m²	Free charge boundary behavior; capacitance and dielectrics
H	Magnetic field intensity	A/m	Set by free current; “magnetizing” field in cores and gaps
B	Magnetic flux density	T (Wb/m²)	Force on currents/charges, flux linkage, induced voltage (transformer action)

The three magnetic “players” (B, H, M)

When materials are involved, introduce magnetization M (dipole moment per unit volume, A/m). A clean conceptual split is:

H: what the winding / free current produces (“applied” magnetizing field)
M: what the material contributes (alignment of domains / dipoles)
B: the combined result that exists in space and couples to circuits

B = μ₀ ( H + M )

In vacuum, M = 0 so B = μ₀H. In ferromagnets, M can dominate.

Constitutive Relations (what the material does)

For linear, isotropic materials (useful approximation for many non-ferromagnets):

M = χₘ H
B = μ₀ (1 + χₘ) H = μ H
μ = μ₀ μᵣ

Engineering caution: For steels and many ferrites, μ is not constant. Use the manufacturer’s B–H curve and consider hysteresis and temperature dependence.

Maxwell’s Equations (magnetics-focused)

These four laws define classical electromagnetics. For most magnetic design and troubleshooting work, the two that come up constantly are Faraday’s and Ampère–Maxwell.

Gauss’s law for magnetism (no magnetic monopoles)

∇ · B = 0

Flux lines are continuous loops; they do not “start” or “end” in space.

Faraday’s law (changing B induces E)

∇ × E = − ∂B/∂t

This is transformer action, back-EMF, inductive kick, and the driver of eddy currents.

Ampère–Maxwell (H is driven by free current and changing D)

∇ × H = J_f + ∂D/∂t

In many industrial power/motion applications (typical dimensions and frequencies in the Hz–kHz range), the displacement term ∂D/∂t is small and a magnetoquasistatic approximation is used:

∇ × H ≈ J_f

Force and induction (why B matters)

If you care about force on a conductor or induced voltage in a loop, the governing quantity is B.

F = q ( E + v × B )
dF = I dℓ × B
∮ E · dℓ = − d/dt ∫ B · dA

Boundary Conditions (interfaces you design around)

Boundary conditions tell you what must be continuous when fields cross materials. They are essential for core/air-gap intuition and for understanding why shielding and grounding matter.

Normal component of B is continuous

n̂ · ( B₂ − B₁ ) = 0

Tangential component of H jumps with surface free current K_f

n̂ × ( H₂ − H₁ ) = K_f

Practical note: If current flows on a conductor surface, the tangential H field changes across that surface. This is one reason return-path geometry and conductor placement directly influence EMI/EMC behavior.

Magnetic Circuits (engineering shortcut that works)

When flux follows a defined path (core + gap), use the magnetic circuit analogy:

MMF:  𝓕 = N I
Reluctance: 𝓡 = ℓ / ( μ A )
Flux: Φ = 𝓕 / 𝓡
B = Φ / A

Why air-gaps dominate

Even a small air-gap often dominates the reluctance because μᵣ of air is ~1:

𝓡_gap = g / ( μ₀ A )

In many gapped inductors and actuators, most magnetic energy is stored in the gap, not the core.

Energy, Inductance, and “Where the Work Is”

For linear media, magnetic energy density is:

u = 1/2 ( B · H )

Total energy in a volume:

W = ∫∫∫ 1/2 ( B · H ) dV

For a lumped inductor, the same energy is written:

W = 1/2 L I²

Design intuition: If the gap dominates reluctance, it also dominates energy storage. That’s why adding a gap makes an inductor more predictable (and less sensitive to core μ variability), at the cost of needing more ampere-turns for the same B.

Materials: μ, B–H Curves, Hysteresis, Saturation

Ferromagnets are nonlinear and history-dependent. A few engineering implications:

Permeability is not constant: incremental μ changes with operating point
Hysteresis: B depends on prior magnetization; the loop area is energy lost per cycle
Saturation: beyond a point, increasing H yields diminishing B; magnetizing current rises sharply

Phenomenon	What you see	Engineering consequence
Hysteresis	Different B for same H depending on history	Loss per cycle; affects transformer efficiency and heating
Saturation	B “flattens” as H rises	Magnetizing current spikes; inductance collapses; overheating risk
Temperature dependence	μ and loss change with temperature	Derating; core selection; stability in hot enclosures

Eddy Currents and Skin Depth (why frequency matters)

Changing magnetic fields induce circulating electric fields, driving eddy currents that oppose changes (Lenz’s law) and dissipate power. A common first-order metric is skin depth:

δ = √( 2 / ( ω μ σ ) )

Higher frequency ω, higher permeability μ, or higher conductivity σ → smaller δ → more current crowding → more loss.

60 Hz power: laminated steel reduces eddy current paths
kHz–MHz power: ferrites reduce eddy currents via high resistivity
Conductors: skin and proximity effect drive litz wire or foil winding decisions

Worked Example: Coil + Gapped Core (fast design estimate)

Assume a core of cross-sectional area A, a single air-gap g, N turns carrying current I. If the gap dominates the reluctance (common in gapped inductors), then:

𝓡_total ≈ 𝓡_gap = g / ( μ₀ A )
Φ ≈ ( N I ) / 𝓡_total = ( N I μ₀ A ) / g
B ≈ Φ / A = μ₀ N I / g

Interpretation: With a dominant gap, B is set primarily by N·I and the gap g. The core permeability matters much less (until you approach saturation).

Practical checklist (what to verify next)

Check fringing at the gap (effective area increases; local loss/heat can rise)
Check saturation margin at peak current (use vendor B–H and temperature)
Estimate copper loss and AC effects (skin/proximity) at operating frequency
For EMI: review loop areas and return paths; field containment is geometry-driven

Note: This document is provided for informational and educational purposes. Always follow site safety procedures, OEM documentation, and applicable electrical/mechanical standards when designing, testing, or troubleshooting equipment.