Electromagnetics: Magnetostatics

Magnetostatics is the branch of electromagnetics that deals with magnetic fields produced by steady currents and time-invariant magnetization. It underpins much of practical electrical engineering, from how a solenoid generates force to why transformers and motors rely on carefully shaped magnetic circuits. In magnetostatics, the geometry of conductors, the distribution of current, and the magnetic properties of materials determine the magnetic field pattern and the energy stored in that field.

A key simplifying assumption runs through the subject: currents are steady in time, so electric charge is not accumulating anywhere. Mathematically, that means the current density satisfies $\nabla \cdot \mathbf{J}=0$. Under this condition, the magnetic field is not driving electromagnetic waves; it is establishing a static field configuration.

Core quantities and what they mean

Magnetostatics uses a small set of field quantities with distinct roles:

• Magnetic flux density $\mathbf{B}$ (tesla, T): the field that directly enters the magnetic force law and is often what you visualize as “magnetic field lines.”
• Magnetic field intensity $\mathbf{H}$ (A/m): a convenient field for separating the effect of free currents from material response.
• Magnetization $\mathbf{M}$ (A/m): a measure of how strongly a material’s microscopic magnetic dipoles are aligned.

In materials, these are linked by the constitutive relation $$\mathbf{B}={\mu }_0(\mathbf{H}+\mathbf{M}).$$ For many engineering problems, especially in linear, isotropic media, magnetization is captured using a relative permeability ${\mu }_r$, giving $$\mathbf{B}=\mu \mathbf{H},\mu ={\mu }_0{\mu }_r.$$ This “linear material” approximation is powerful, but it breaks down in ferromagnets near saturation and in the presence of hysteresis.

Biot-Savart law: building fields from currents

The Biot-Savart law gives the magnetic flux density produced by a steady current distribution. For a filamentary current $I$ along a path $C$, the field at position $\mathbf{r}$ is $$\mathbf{B}(\mathbf{r})=\frac{{\mu }_{0}I}{4\pi }{\int }_C\frac{d\mathbf{\ell }\times \mathbf{R}}{R^3},$$ where $\mathbf{R}=\mathbf{r}-{\mathbf{r}}^{\prime }$ points from the source element to the observation point.

Biot-Savart is most useful when geometry is finite or irregular, where symmetry methods are limited. It is the starting point for calculating the field of a circular loop, a finite straight segment, or a complex wire shape. In practice, engineers often use it to estimate magnetic fields around busbars and current-carrying conductors in power electronics, especially when field exposure or interference matters.

The trade-off is computational: the integral can be tedious by hand and is often evaluated numerically for real layouts.

Ampere’s law: symmetry as a shortcut

Ampere’s law relates the circulation of $\mathbf{H}$ around a closed path to the free current enclosed: $$\oint \mathbf{H}\cdot d\mathbf{\ell }=I_{\text{free, enclosed}}.$$ Combined with symmetry, it yields clean closed-form results.

Classic examples

Infinite straight wire

For a long wire carrying current $I$, the field is azimuthal and depends only on radius $r$: $$H_{\varphi }(r)=\frac{I}{2\pi r},B_{\varphi }(r)=\mu \frac{I}{2\pi r}.$$

Long solenoid

Inside an ideal long solenoid with $n$ turns per unit length carrying current $I$, the internal field is approximately uniform: $$H\approx nI,B\approx \mu nI.$$ This approximation is the basis for many actuator and sensor designs. Real solenoids have fringing fields at the ends, and short coils require more careful analysis.

Toroid

A toroidal core confines flux well. For a toroid with $N$ turns, current $I$, and mean magnetic path length $2\pi r$, the field in the core is approximately $$H(r)\approx \frac{NI}{2\pi r},B(r)\approx \mu \frac{NI}{2\pi r}.$$ This is why toroids are favored in inductors when low external magnetic field leakage is important.

Magnetic materials: permeability, saturation, and hysteresis

Materials shape magnetic fields by providing a path of higher permeability than air. In many devices, the goal is to guide flux through a ferromagnetic core, concentrating $\mathbf{B}$ where it is useful (for force, voltage induction, or energy storage) and minimizing leakage elsewhere.

Linear materials and permeability

In linear regions, permeability $\mu$ describes how much $\mathbf{B}$ results from a given $\mathbf{H}$. Paramagnetic and diamagnetic materials have ${\mu }_r$ near 1, so their influence is modest. Ferromagnetic materials can have very large effective ${\mu }_r$ at low field strengths, dramatically altering flux distribution.

Saturation

Ferromagnets cannot increase magnetization indefinitely. As $\mathbf{H}$ increases, $\mathbf{B}$ eventually grows more slowly and approaches saturation. In design terms, saturation limits:

• maximum force in solenoids and relays
• maximum flux in transformers and inductors
• linearity in sensors and magnetic circuits

Ignoring saturation can lead to overheating (due to higher currents needed for a desired flux) or loss of control authority in actuators.

Hysteresis and loss

Many ferromagnets exhibit hysteresis: the relationship between $\mathbf{B}$ and $\mathbf{H}$ depends on magnetic history. While magnetostatics focuses on steady fields, real devices often operate with slowly varying currents, and hysteresis then translates into energy loss per cycle. Even in “DC” applications, ripple currents can create meaningful hysteresis and eddy-current losses.

Inductance: linking currents to flux

Inductance quantifies how a current produces magnetic flux linkage. For a coil with $N$ turns, magnetic flux through one turn $\Phi$, and current $I$, the flux linkage is $\lambda =N\Phi$. The inductance is defined as $$L=\frac{\lambda }{I},$$ for linear magnetic systems where $\lambda$ is proportional to $I$. In nonlinear cores, $L$ becomes current-dependent, and engineers often use incremental inductance around an operating point.

Inductance is not just a property of the coil’s geometry; it is heavily influenced by the magnetic circuit, especially the presence of an air gap. Adding a small gap in a high-permeability core can dominate the magnetic reluctance, making the inductance more stable and preventing saturation in power inductors.

Energy storage in magnetic fields

A magnetic field stores energy. In linear media, the energy density is $$u=\frac{1}{2}\mathbf{B}\cdot \mathbf{H}.$$ In free space, where $\mathbf{B}={\mu }_0\mathbf{H}$, this becomes $u=\frac{B^2}{2{\mu }_0}$.

For an inductor with inductance $L$ carrying current $I$, the total stored energy is $$W=\frac{1}{2}L{I}^2,$$ again assuming linearity.

A practical insight follows from where the energy actually resides. In many gapped-core inductors, most of the energy is stored in the air gap, not in the ferromagnetic material. This is why gaps are used to create energy-storing inductors for power conversion: the gap supports high field energy density without the same saturation constraints as the core.

Practical design perspective: magnetic circuits

Engineers often model cores and gaps using a magnetic-circuit analogy:

• magnetomotive force (MMF) $NI$ drives flux $\Phi$
• reluctance $\mathcal{R}$ opposes flux
• $\Phi \approx \frac{NI}{\mathcal{R}}$

While not a replacement for field analysis, it is an effective first-order tool. It captures why longer path length reduces flux, why a narrow cross-section increases flux density (and risk of saturation), and why air gaps dominate reluctance.

Where magnetostatics applies and where it stops

Magnetostatics is appropriate when dimensions are small compared to electromagnetic wavelengths and currents are steady or change slowly enough that displacement currents and wave propagation can be neglected. It provides accurate guidance for:

• DC electromagnets, relays, and solenoids
• current distribution effects in conductors (when skin effect is negligible)
• inductance estimation for low-frequency circuits
• field containment and leakage in magnetic cores

When time variation becomes significant, Faraday’s law, induced electric fields, eddy currents, and full electromagnetic wave effects must be considered. Still, magnetostatics remains the foundation: most real systems start with a magnetostatic model, then add corrections for frequency-dependent behavior and losses.

Understanding Biot-Savart law, Ampere’s law, magnetic material behavior, inductance, and magnetic energy is what turns “magnetic field lines” into a working engineering