Magnetic Field

A magnetic field is a physical field that describes magnetic interactions at each position in space and time. It acts on moving electric charges and magnetic dipoles, and is produced by electric currents, changing electric fields, and the magnetic properties of matter. Usually represented by the vector B, it has both magnitude and direction. Together with the electric field, it forms the electromagnetic field, central to electromagnetism. (openstax.org)

Definition and units

The operational definition of B follows from the Lorentz force on a particle of charge $q$ moving with velocity $\mathbf v$:

The magnetic contribution has magnitude $|q|vB\sin\theta$, where $\theta$ is the angle between velocity and field. It vanishes when the charge is stationary or moves parallel to the field. Its direction is perpendicular to both velocity and field, with opposite directions for positive and negative charges. (openstax.org)

In the International System of Units, magnetic flux density is measured in teslas (T). One tesla equals one newton per ampere-metre, or one weber per square metre. The older unit gauss equals $10^{-4}$ T. Magnetic flux, defined by

measures the field passing through an oriented surface and is expressed in webers. Flux depends on field direction as well as magnitude and surface area. (openstax.org)

Sources and field geometry

A steady electric current produces a magnetic field around its path. For an ideal infinitely long straight wire in vacuum,

where $I$ is current, $r$ is distance from the wire, and $\mu_0$ is vacuum permeability. The field circles the wire according to the right-hand rule. Fields from current distributions can be calculated using the Biot–Savart law or, where sufficient symmetry exists, Ampère’s law. A long, tightly wound solenoid produces an approximately uniform interior field, $B\approx\mu_0 nI$, away from its ends. (openstax.org)

Field lines provide a visualization: their tangent gives the local field direction, while their relative density indicates field magnitude. Outside a bar magnet they run from its north pole toward its south pole; inside, they return toward the north pole. They are not physical threads or necessarily particle trajectories. The magnetic flux through any closed surface is zero, so magnetic field lines have no isolated sources or sinks in conventional electromagnetic theory. (openstax.org)

Forces and particle motion

In a uniform magnetic field, a freely moving charged particle follows a circle if its velocity is perpendicular to the field, or a helix if it also has a parallel velocity component. In the nonrelativistic limit, the circular radius is

The magnetic force alone does no work on a point charge because it is perpendicular to its instantaneous velocity. It changes the direction of motion, not the particle’s kinetic energy. This distinction separates magnetic deflection from acceleration by an electric field. (openstax.org)

A straight conductor carrying current in a uniform field experiences $\mathbf F=I\mathbf L\times\mathbf B$. A current loop behaves as a magnetic dipole and experiences a torque tending to align its magnetic moment with the field. These effects underpin the operation of electric motors. (openstax.org)

Magnetic fields in matter

Microscopic magnetism arises principally from electron orbital motion and spin angular momentum. Spin is an intrinsic quantum property, not literal rotation of a miniature charged sphere. The combined magnetic moments of electrons determine much of the magnetic behavior of atoms and solids; a complete explanation requires quantum mechanics. (openstax.org)

Macroscopic descriptions distinguish B from H, the magnetic field strength, measured in amperes per metre. Their relation is

where $\mathbf M$ is magnetization, or magnetic moment per unit volume. For a linear, isotropic material, this reduces to $\mathbf B=\mu\mathbf H$, with $\mu$ the magnetic permeability. H is therefore not simply another name or unit for B. (nvlpubs.nist.gov)

Diamagnetism produces a response opposing the applied field, whereas paramagnetism produces a weak response along it. In ferromagnetism, interactions can maintain aligned moments within magnetic domains. Changes in domain arrangement can leave persistent magnetization and produce hysteresis, meaning that the response depends on the material’s magnetic history. (openstax.org)

Time dependence and electromagnetic theory

Maxwell’s equations describe the coupling between electric and magnetic fields. A changing magnetic field produces a circulating electric field through electromagnetic induction. Conversely, electric currents and changing electric fields contribute to magnetic fields through the Ampère–Maxwell law. These relationships permit electromagnetic waves to propagate through vacuum, including visible light. (openstax.org)

Under the theory of relativity, electric and magnetic fields are interrelated components of one electromagnetic field. Observers moving relative to one another can assign different electric and magnetic components to the same physical situation. A distinction between purely electric and purely magnetic behavior therefore depends partly on the observer’s reference frame. (openstax.org)

Natural occurrence and applications

The geomagnetic field is generated mainly by dynamo action in Earth’s electrically conducting liquid outer core. Fluid motion induces currents that sustain the field through feedback; Earth is not simply a permanent bar magnet. (usgs.gov)

Technological applications include magnetic information storage and electromagnetic machinery. Strong magnetic fields are also essential to magnetic resonance imaging, where radiofrequency excitation and detected signals from hydrogen nuclei provide information about tissues. MRI combines a strong main field with controlled field variations to construct spatially resolved images. (openstax.org)