Magnetic field

The magnetic field is the vector field that mediates magnetic forces between moving charges and currents. EM uses two related fields:

• Magnetic flux density $\mathbf{B}$ (tesla, T = Wb/m²) — the field that produces forces on moving charges via the Lorentz force $\mathbf{F}=q\mathbf{u}\times \mathbf{B}$.
• Magnetic field intensity $\mathbf{H}$ (A/m) — the field whose curl is sourced by free currents.

The constitutive relation in a linear, isotropic, homogeneous medium:

$\mathbf{B}=\mu \mathbf{H},$

with Permeability $\mu ={\mu }_r{\mu }_0$. For free space, ${\mu }_0=4\pi \times 10^{-7}$ H/m. For most non-magnetic materials (dielectrics, metals other than iron/cobalt/nickel), ${\mu }_r\approx 1$ and so $\mathbf{B}\approx {\mu }_0\mathbf{H}$.

Why two fields

Same logic as for D and E in electrostatics. $\mathbf{H}$ is the “free-source” field (Ampère’s law $\nabla \times \mathbf{H}={\mathbf{J}}_{\text{free}}$). $\mathbf{B}$ is the “what charges actually feel” field — its sources include bound magnetization currents in magnetic materials.

For purely non-magnetic problems they’re proportional and interchangeable up to ${\mu }_0$. Electromagnetics mostly considers non-magnetic media and uses both interchangeably depending on the formula.

How magnetic fields are produced

Magnetic fields are produced by moving charges — equivalently, by electric currents. Two equivalent ways to compute them:

Biot-Savart law (analog of Coulomb’s law) — direct integration over the current distribution:

$\mathbf{H}=\frac{1}{4\pi }{\int }_C\frac{Id{\mathbf{l}}^{\prime }\times \hat{\mathbf{R}}}{R^2}.$

Ampère’s law (analog of Gauss’s law) — exploits symmetry to bypass integration:

${\oint }_C\mathbf{H}\cdot d\mathbf{l}=I_{\text{enc}}.$

For problems with cylindrical, axial, or planar symmetry, Ampère is dramatically easier.

Magnetic field of common sources

Infinite straight wire carrying current $I$ on the $z$-axis. By symmetry $\mathbf{H}=H_{\varphi }(r)\hat{\mathbf{\varphi }}$. Ampère around a circle of radius $r$: $H_{\varphi }\cdot 2\pi r=I$, so

$\mathbf{H}=\hat{\mathbf{\varphi }}\frac{I}{2\pi r},\mathbf{B}=\hat{\mathbf{\varphi }}\frac{{\mu }_{0}I}{2\pi r}.$

The field circles around the wire. The right-hand rule: thumb along $\mathbf{I}$, fingers curl in $\hat{\mathbf{\varphi }}$ direction.

Circular current loop of radius $a$, current $I$, in the $xy$-plane. On the $z$-axis:

$\mathbf{H}=\hat{\mathbf{z}}\frac{I{a}^2}{2(a^2+z^2{)}^{3/2}}.$

At the center ($z=0$): $\mathbf{H}=\hat{\mathbf{z}}I/(2a)$.

Long solenoid of length $l$ with $N$ turns carrying current $I$:

$\mathbf{B}=\hat{\mathbf{z}}\frac{\mu NI}{l}\text{(interior)}.$

Uniform inside, ~zero outside. This is why solenoids are the textbook example of “creating a uniform magnetic field in a region of space.”

Gauss’s law for magnetism

Gauss’s law for magnetism states $\nabla \cdot \mathbf{B}=0$ — equivalently, ${\oint }_S\mathbf{B}\cdot d\mathbf{s}=0$ for any closed surface. The physical content: no magnetic monopoles. Every field line is a closed loop. Compare with Gauss’s law for $\mathbf{E}$, where field lines start on positive charges and end on negative ones.

Because $\mathbf{B}$ is divergence-free, it can be written as the curl of a Vector magnetic potential: $\mathbf{B}=\nabla \times \mathbf{A}$.

In Maxwell’s equations

The two equations involving magnetic fields:

• $\nabla \cdot \mathbf{B}=0$ — no monopoles.
• $\nabla \times \mathbf{H}=\mathbf{J}+{\partial }_t\mathbf{D}$ — Ampère’s law with Displacement current.

In magnetostatics (${\partial }_t=0$), the second reduces to $\nabla \times \mathbf{H}=\mathbf{J}$.

Versus electric field

Two key distinctions:

1. Magnetic field acts only on moving charges; electric field acts on all charges. The magnetic force $q\mathbf{u}\times \mathbf{B}$ vanishes for $\mathbf{u}=0$.
2. Magnetic force does no work: ${\mathbf{F}}_m\cdot \mathbf{u}=q(\mathbf{u}\times \mathbf{B})\cdot \mathbf{u}=0$ since $\mathbf{u}\times \mathbf{B}\perp \mathbf{u}$. A magnetic field can change a charge’s direction but never its kinetic energy. Energy changes always come from the electric field component of the Lorentz force.

The deep unification: $\mathbf{B}$ and $\mathbf{E}$ are different components of the same electromagnetic field tensor; they transform into each other under Lorentz boosts. What one observer calls a pure $\mathbf{E}$ field, another (moving) observer sees as a mixture of $\mathbf{E}$ and $\mathbf{B}$. The non-relativistic split into “electric” and “magnetic” is frame-dependent.