Magnetic field

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

Image: Magnetic field of a dipole, Public domain — field lines of a magnetic dipole — the same topology as the electric dipole.

• 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) uses 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 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.