A magnetic field is a vector (more precisely a pseudovector) field that describes the magnetic influence experienced by moving electric charges, electric currents and magnetized matter. A charge moving through the field is subject to a force perpendicular to both its velocity and the local magnetic-field vector; permanent magnets exert forces on ferromagnetic materials and on other magnets. Magnetic fields persist in the vicinity of sources such as magnetized bodies, steady currents and time-varying electric fields, and their magnitude and direction vary from point to point in space.
Electromagnetism distinguishes two related fields: B, the magnetic flux density, and H, the magnetic field strength. In SI units B is measured in tesla (T) — equivalent to kg·s−2·A−1 or N·m−1·A−1 — while H is measured in ampere per metre (A·m−1). In free space the fields are proportional through the vacuum permeability μ0 (H = B/μ0), but inside magnetized materials the relation is altered by the material’s internal magnetization, so B and H differ according to how the medium responds.
Magnetic fields arise from two fundamental origins: moving electric charge (currents) and the intrinsic magnetic moments of particles, the latter stemming from quantum mechanical spin. Spatially nonuniform fields can also exert small forces on materials usually described as nonmagnetic; these effects reflect paramagnetism, diamagnetism and antiferromagnetism and are often measurable only with sensitive instruments. Electric and magnetic fields are not independent but are interwoven aspects of the single electromagnetic interaction: temporal changes in one field can induce the other under Maxwell’s laws.
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Practically, magnetic fields underpin many technologies and diagnostic methods. Rotating magnetic fields drive electric motors and generators, magnetic circuits provide a framework for analyzing devices such as transformers, and phenomena like the Hall effect probe charge-carrier properties in materials. On a planetary scale, Earth’s magnetic field both deflects charged particles from the solar wind—helping to protect the atmosphere—and furnishes a stable directional reference exploited in compass navigation.
Description
The electromagnetic force on a charged particle depends on where the particle is and on its velocity; accordingly, it is specified by two vector fields that together determine the instantaneous force. One of these, the electric field E, gives the component of force that a charge experiences irrespective of motion and therefore characterizes the force on a stationary charge.
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The other field describes the contribution that depends on the particle’s motion. Quantitatively the total force is expressed by the Lorentz relation F = q(E + v × B), so the velocity‑dependent magnetic term appears as q v × B. Because this term is a cross product, the magnetic force is orthogonal to both the particle’s velocity and the local B vector and its sense is set by the right‑hand rule; it vanishes when v = 0 and varies with both the speed and direction of motion.
Two closely related magnetic fields, conventionally denoted B and H, are used in electromagnetic theory. Their empirical effects are unambiguous, but the most useful terminology and interpretation have been debated historically: older sources sometimes labeled H as the “magnetic field,” whereas many contemporary texts refer to B by that name (and provide distinct roles for H when materials and magnetization are involved). Despite this variety of names and perspectives, the underlying physical relations they represent are well established.
The vector commonly denoted B is the field used to quantify magnetic effects on moving charges and magnetic dipoles. It is variously termed magnetic flux density, magnetic induction, or (ambiguously) the magnetic field; in this section B is treated as the vector quantity that appears in the dynamical laws governing magnetic interactions.
Operationally B is introduced by the Lorentz force law. A particle of charge q moving with velocity v in electromagnetic fields E and B experiences force F = qE + q(v × B). The magnetic contribution is orthogonal to both v and B and has magnitude |F_magnetic| = q |v| |B| sinθ, where θ is the angle between v and B; consequently no magnetic force acts when the velocity is parallel or antiparallel to the local B. The sense of q(v × B) is given by the right-hand rule, and the force reverses sign if the particle charge is reversed. Reversing both q and v leaves q(v × B) unchanged, so a measurement that relies solely on magnetic deflection cannot distinguish a positive charge moving one way from a negative charge moving the opposite way; combining electric and magnetic effects (for example in the Hall effect) permits determination of charge sign and direction.
Practically, B at a point can be determined by using known test charges: first measure the force on a charged probe at rest to infer E, then record forces for known velocities in different directions and solve the Lorentz relation for the vector B that reproduces the observed forces. An alternative operational definition uses magnetic dipoles: a magnetic moment m placed in a B field experiences a torque τ = m × B (SI vector form), so measuring torques on calibrated dipoles yields the local B direction and magnitude.
In SI units B is measured in tesla (T). In Gaussian–cgs units the corresponding unit is the gauss (G), with 1 T = 10,000 G. A nanotesla is also called a gamma (γ).
The magnetic field H (often termed magnetic field intensity, magnetic field strength, auxiliary or magnetizing field) is a vector field that characterizes the portion of magnetic effects attributable to currents and applied magnetization, distinct from the flux carried through space. It is related to the magnetic flux density B and the material magnetization M by
H = (1/μ0) B − M,
where μ0 is the vacuum permeability. This decomposition removes the contribution of a material’s internal magnetic moments (M) from the scaled flux density B/μ0, yielding the auxiliary field that drives magnetization and responds directly to free currents.
In free space M vanishes, so B and H are simply proportional through μ0; within magnetic media, however, internal magnetization alters the balance and B and H differ. The distinction between these two fields is therefore central to describing magnetic behavior inside versus outside materials. In SI units H is measured in amperes per metre (A m−1); in CGS units the corresponding quantity is expressed in oersted (Oe), reflecting H’s interpretation as a field tied to currents and magnetizing effects rather than to flux density alone.
Measurement
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Magnetic-field measurement is performed with magnetometers, instruments that convert the local magnetic environment into a measurable signal. Major classes include induction (search‑coil) devices, rotating‑coil instruments, Hall‑effect sensors, nuclear magnetic resonance (NMR) magnetometers, superconducting quantum interference devices (SQUIDs), and fluxgate magnetometers. Search‑coil (induction) magnetometers respond to changes in magnetic flux and thus detect only time‑varying (AC) components, whereas the other techniques are complementary and, between them, provide capabilities for sensing field magnitude, vector direction and temporal behavior across a wide range of amplitudes and frequencies. In astronomy, magnetic fields of remote objects are typically not sampled directly but inferred from their effects on charged particles; for example, relativistic electrons gyrating around field lines produce synchrotron radio emission, which can be used to diagnose astrophysical magnetic fields. The most sensitive laboratory measurement to date was reported by Gravity Probe B, which achieved an attotesla‑level sensitivity of about 5 × 10−18 T.
Visualization of the magnetic field employs field lines as a graphical device that traces the local orientation of the magnetic vector at each point. Constructed by sampling the field’s direction and magnitude, these lines are drawn so that the tangent at any location aligns with the local field direction and the local spacing of lines reflects field strength: closer lines indicate larger magnitude. Because they are generated by discrete sampling, field-line diagrams are resolution dependent (finer sampling shows more lines) and should be understood as a continuous representation—analogous to streamlines in fluid flow—rather than as physical filaments.
The field-line picture is mathematically useful because counting lines crossing a surface gives the magnetic flux through that surface; this count corresponds directly to the surface integral of the magnetic field and can be expressed precisely in integral form. Field lines can also be revealed experimentally: for example, magnetized iron filings align with the local direction to make visible patterns, and large-scale auroral emissions trace charged-particle motion along Earth’s magnetic geometry, exposing the global topology.
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For qualitative reasoning about forces, field lines admit a simple mechanical analogy: each line can be pictured as having a tension that resists bending and a lateral pressure that acts between neighboring lines. This intuition helps explain magnetostatic behavior—opposite poles appear to attract because they are connected by shared field lines that draw together, while like poles repel because their adjacent lines run parallel and exert mutual lateral pressure.
Despite their pedagogical and geometric value in conveying direction, topology, and relative magnitude, field lines are conceptual tools. The rigorous, quantitative description of magnetism remains the vector field B and its associated integrals (e.g., surface integrals for flux); field-line depictions should not be mistaken for literal physical entities.
Magnetic field of permanent magnets
Permanent magnets are solid bodies, typically made from ferromagnetic materials such as iron or nickel, that sustain an intrinsic magnetic field and present two opposing poles conventionally labeled north and south. At the simplest level a small straight magnet behaves like a magnetic dipole: its external field strength and spatial pattern scale with the dipole moment m, which is directed from the magnet’s south pole toward its north pole. Reversing a bar magnet is therefore equivalent, in dipole terms, to rotating its dipole moment m through 180°; more generally the detailed field at any point depends on both distance from the magnet and the magnet’s orientation, so the analytic field expressions are non‑trivial.
A macroscopic permanent magnet can be modeled as an assembly of many microscopic dipoles, each with its own dipole moment; the observable field is the vector sum (superposition) of the fields produced by those constituents, and any net mechanical force or torque on the body results from summing the forces and torques on the individual dipoles. Two idealized, instructive representations of these microscopic sources are commonly used: the magnetic pole model, which treats magnetization as distributions of fictitious north and south poles, and the Amperian loop model, which represents magnetic moments as small circulating currents. These two pictures give rise to two field vectors used in magnetostatics, conventionally denoted H and B.
In regions outside magnetic material the H and B fields are proportional and hence practically indistinguishable up to a constant factor, so for many problems—especially those where fields are generated by macroscopic currents—the distinction is often immaterial. However, both simplified models have limits. The magnetic pole description lacks fundamental experimental basis at the microscopic level, and the Amperian loop model, while it captures an orbital contribution by associating magnetism with circulating electronic motion, treats electron motion classically and therefore cannot account for all magnetic behavior. A substantial portion of a material’s magnetic moment arises from electron spin, a quantum mechanical property that neither the pole model nor a purely classical current loop fully explains; spin is therefore a central ingredient in the true microscopic origin of permanent magnetism.
Magnetic pole model
The magnetic pole model represents a magnet macroscopically as two opposite poles—North (+) and South (−)—separated by a distance d and treated as sources of an H-field whose lines run from the north pole to the south pole both outside and inside the magnet. In this picture the H-field is generated by a distribution of fictitious magnetic “charges” on pole surfaces, a mathematical construction closely tied to the material’s magnetization M; for a small pair of opposite poles of strength q_m separated by vector d the model assigns a dipole moment m = q_m d, and this arrangement reproduces the gross H-field pattern inside and outside magnetic bodies in the pole approximation. Mechanically, an isolated north pole would be forced in the direction of H and a south pole oppositely, mirroring the way electric charges experience forces in an electric field and making the H-field directly analogous to the electric field E in its source–line topology.
Historically, the pole picture allowed early texts to treat magnet–magnet forces and torques by Coulomb-like laws for poles because of its mathematical simplicity. However, the model rests on fictitious magnetic charge and therefore conflicts with microscopic evidence: isolated magnetic poles are not observed—cutting a magnet produces new complementary north–south pairs on the fragments’ surfaces—so real magnetism does not arise from free magnetic charge. Consequently the pole model cannot account for magnetism generated by electrical currents (Ampèrian currents) nor for the intrinsic link between angular momentum (electron spin and orbital motion) and magnetic moment; those phenomena require current-based or quantum-mechanical descriptions.
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In modern theory the notion of magnetic charge is treated as a useful abstraction for macroscopic problems rather than a literal particle property. Hypothetical magnetic monopoles—single magnetic charges that would allow field lines to begin or end on a particle—are predicted in some high-energy and grand-unified frameworks, but none have been detected experimentally. Thus the pole model endures as a convenient, phenomenological tool for describing H-fields in magnetized materials at the macroscopic level while its fundamental limitations are recognized.
The elementary magnetic dipole is modeled as a sufficiently small Amperian current loop carrying current I and enclosing area A; its magnetic dipole moment is defined by m = I A, with the vector m normal to the loop plane and oriented according to the right‑hand rule. This current‑loop representation serves as the microscopic building block of macroscopic magnetization and is physically preferred to formulations invoking separated magnetic charges.
Each Amperian loop produces a magnetic B‑field with the characteristic dipolar spatial dependence; in the idealized (point‑dipole) limit the external B‑field assumes the same dipolar form as that of an ideal electric dipole of comparable strength. The ideal magnetic dipole is obtained by shrinking the loop area a → 0 while increasing the current I → ∞ so that the product m = I a remains finite, thereby preserving the dipole moment while removing finite spatial extent.
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The loop model also makes explicit the dynamical origin of magnetism: mechanical rotation of charged matter that increases the effective circulation produces a corresponding increase in I and thus m, directly linking angular momentum and magnetic moment. This intimate relation accounts for reciprocal phenomena such as the Einstein–de Haas effect (mechanical rotation induced by magnetization) and the Barnett effect (magnetization induced by rotation), which demonstrate conversion between macroscopic rotation and magnetic moment.
In summary, the Amperian loop formalism relates current and loop area to dipole moment via m = I A (or m = I a), attains the ideal point dipole by the limit a → 0, I → ∞ with m fixed, and yields the dipolar B‑field that characterizes both microscopic loops and macroscopic magnets.
The force between two small magnets depends simultaneously on each magnet’s magnetic moment, their relative orientations, and the vector separation between them; because magnetic torque can reorient a dipole, rotations strongly affect the forces experienced. In the magnetic-pole picture the H-field produced by one magnet acts on the opposite poles of a second magnet: a spatially uniform H at the two poles produces equal and opposite forces that cancel, whereas a nonuniform H (for example the stronger field near a pole of the source magnet) exerts different forces on the poles that sum to a net translational force toward regions of stronger field and can also produce a torque. The general physical consequence, independent of model details, is that a magnetic dipole tends to move into regions of larger magnetic-field magnitude (or, more precisely, into regions of larger m · B), with attraction or repulsion determined by the dipole’s orientation relative to the field.
The current-loop (Amperian) representation of a dipole is microscopically distinct from the pole model but yields the same macroscopic prediction: dipoles experience a net force toward regions that increase the scalar product of their moment with the ambient field. This result is compactly expressed for an ideal point dipole by the vector relation
\mathbf{F}=\nabla(\mathbf{m}\cdot\mathbf{B}),
where ∇ denotes the spatial gradient of the scalar m · B. Since m · B = m B cos θ (with θ the angle between m and B), alignment (θ = 0) makes m · B positive and the gradient points toward increasing m · B, producing a pull into stronger-field regions.
The formula F = ∇(m · B) is exact only for point dipoles; for real magnets of finite extent one typically subdivides the body into small volume elements, assigns each a local moment, applies the local expression for the force, and integrates the contributions to obtain the net force and torque. Because the gradient ∇(m · B) both sets magnitude and direction, the combined effects of spatial nonuniformity in the external field and the dipole’s orientation determine whether a magnet is attracted or repelled and how large the resulting torque and translational acceleration will be.
Magnetic torque on permanent magnets
When a magnet that can rotate freely is placed in the field produced by another magnet, the fixed magnet’s field applies forces to the separated poles of the free magnet. Those spatially separated, oppositely directed forces produce a turning moment (torque), which is the mechanical cause of a compass needle rotating to align with Earth’s field. In the pole model a magnet is idealized as two equal and opposite magnetic charges separated by a distance; although the forces on the charges are equal in magnitude and opposite in direction, their different lines of action generate a torque proportional to the pole separation. Defining the magnetic moment m as the product of pole strength and pole separation yields the scalar magnitude τ = μ0 m H sin θ, where θ is the angle between m and the applied field intensity H and sin θ picks out the perpendicular component of H. More generally, the torque is a vector given by τ = m × B = μ0 m × H, with B the magnetic flux density and × the vector cross product; this form shows that no torque acts when m is collinear with B or H, while any other orientation produces a rotating tendency that aligns the moment with the field. The vacuum permeability μ0 (μ0 = 4π×10−7 V·s/(A·m)) provides the SI scaling between H and B and therefore appears explicitly in the torque expressions.
Electric currents are sources of magnetic induction: a steady current I in a conductor produces a vector magnetic field B whose lines form closed loops around the conductor. For an ideal infinitely long straight wire the field magnitude at radial distance r is
B = μ0 I / (2π r),
with μ0 = 4π × 10^−7 N·A^−2. The magnetic field from arbitrary current distributions is obtained by superposition using the Biot–Savart law or Ampère’s law. The field direction is given geometrically by the right‑hand rule (thumb along conventional current, fingers show the sense of the circular B‑field); for current loops the resultant field is axial through the loop center.
Moving charges and macroscopic currents immersed in an external B field experience forces perpendicular to both the field and the motion. A point charge q moving with velocity v feels the Lorentz force F = q (v × B), with magnitude |F| = |q||v||B| sin θ, where θ is the angle between v and B. Similarly, a conductor segment represented by a length vector L carrying current I in a uniform B field is subject to F = I (L × B), |F| = I L B sin θ. These relations produce torques on current loops (for a planar loop with area vector A, τ = I (A × B)) and form the mechanical basis of electromechanical devices such as electric motors. Because the magnetic force is always orthogonal to velocity or current, it does no work directly on charged particles (it alters direction but not kinetic energy); nevertheless, when combined with electric fields or with induced electric fields from time‑varying magnetic flux, magnetic interactions enable energy transfer and mechanical work.
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Spatial and vector properties are essential: B is a spatial (and, for time‑varying currents, temporal) vector field; fields from multiple currents add by vector superposition; the sin θ dependence makes force and torque highly geometry‑dependent, determining equilibrium orientations and magnitudes. Standard SI quantities are magnetic induction B in tesla (T), current I in ampere (A), charge q in coulomb (C), and velocity v in metres per second (m·s⁻¹); μ0 appears explicitly in analytic expressions relating currents to B.
Moving electric charges produce magnetic fields: any charged particle in motion generates a magnetic field whose local form depends on the charge, its velocity and acceleration. Thus magnetism in classical electromagnetism originates from charge motion rather than from static charge distributions alone.
For a long, straight, current-carrying conductor the magnetic field lines lie on concentric circles centered on the wire. The sense of these circular B-field lines is given by the right-hand rule (grip): align the thumb with the current I and the curled fingers indicate the direction of B around the conductor. The field magnitude falls off with radial distance r from the wire; for an ideal infinitely long straight wire the dependence is inverse in r, with the standard result |B| = μ0 I/(2π r).
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More generally, the magnetic field of a steady current distribution is given in integral form by the Biot–Savart law,
B = (μ0 I / 4π) ∫(dℓ × r̂) / r^2,
where the integral runs over the current path, dℓ is an element oriented along the current, r is the vector from dℓ to the field point (r = |r|), and r̂ = r/r. The cross product dℓ × r̂ determines the direction of each contribution. For a long straight wire this integral reduces to the closed-form expression above, producing tangential field vectors in the plane perpendicular to the wire consistent with the right-hand rule.
Ampère’s law provides a complementary global relation between currents and magnetic fields:
∮ B · dℓ = μ0 I_enc,
where the line integral is taken around any closed loop and I_enc is the net current penetrating the loop. For steady currents Ampère’s law is exact and is especially effective when applied to highly symmetric geometries (e.g., infinite straight wires or ideal solenoids) to obtain B directly without evaluating Biot–Savart integrals.
Reshaping a current path alters the field topology: a single circular loop concentrates B inside the loop and weakens it outside; stacking many closely spaced loops yields a coil or solenoid that amplifies and nearly homogenizes the interior field. An ideal infinitely long solenoid produces a uniform interior magnetic field and vanishing exterior field; a real solenoid around an iron core functions as an electromagnet whose field strength and polarity are set by the coil current and whose flux is enhanced by the magnetic core.
A finite-length solenoid exhibits an external field with near-dipolar geometry resembling a permanent bar magnet: the interior field is strong and directed by the winding sense, while the outside field closes in lobes of opposite polarity. When electric and magnetic fields vary in time the quasi-static relations must be replaced by the full Maxwell–Ampère law (which adds the displacement current term), thereby coupling time-varying electric fields to magnetic fields and completing the set of Maxwell’s equations that govern electrodynamics.
Force on a charged particle
The motion of a charged particle in electromagnetic fields is governed by the Lorentz force, F = qE + q v × B, in which q is the particle charge, E the electric field, v the instantaneous velocity, and B the magnetic flux density (tesla). The magnetic part q v × B is a vector cross product that vanishes for velocity components parallel to B and is maximal for perpendicular components; its direction is orthogonal to both v and B at each instant. In a time‑invariant (static) magnetic field a particle therefore preserves its velocity component along B while the perpendicular component is forced into circular motion, producing a helical trajectory about the field lines. Because the magnetic force is everywhere perpendicular to the instantaneous velocity of an isolated charge, it does no work on that charge and cannot change its speed, only its direction. Energy transfer associated with time‑varying magnetic fields occurs indirectly: a changing B induces an electric field (Faraday’s law), and that induced electric field can perform work on charges. Apparent counterexamples—such as work on non‑elementary magnetic dipoles or on charges whose motion is constrained—are resolved by recognizing that any net work is supplied by electric forces acting on constituent charges or by constraint forces, not by the magnetic force acting alone.
Force on a current‑carrying wire
A conductor carrying an electric current experiences a magnetic force that is the macroscopic counterpart of the Lorentz force on an individual moving charge; this bulk effect is commonly termed the Laplace force. Consider a straight segment of conductor of length ℓ and cross‑sectional area A, containing charge carriers of charge q and number density n that move with drift speed v; the conductor carries current i and is immersed in a magnetic field of magnitude B that subtends an angle θ with the carrier velocity v. A single carrier is subject to a magnetic force of magnitude F = q v B sin θ, where the sine factor selects the component of velocity perpendicular to the field. The segment contains N = n ℓ A mobile charges, and the current relates to the microscopic quantities by i = n q v A. Multiplying the single‑charge force by N and substituting the expression for i yields the macroscopic force magnitude on the segment f = B i ℓ sin θ, the usual Laplace formula. Thus f scales linearly with B, i and the length ℓ of conductor within the field and varies with the angle as sin θ (maximal when v ⟂ B, vanishing when v ∥ B); the derivation presumes uniform magnetic field, uniform carrier density and uniform drift velocity across the considered segment.
Magnetic field formulas derived for vacuum or current distributions must be applied to the total current in a system; when magnetic material occupies the field region the material’s internal currents alter the macroscopic field and cannot be neglected. At the microscopic level each atom or molecule contributes tiny current loops (orbital motion and spin) whose collective alignment produces a net magnetic dipole moment per unit volume, the magnetization M. The macroscopic currents produced by this magnetization are compactly expressed as a volume bound current density J_b = ∇ × M and a surface bound current density K_b = M × n, where n is the outward normal to the material surface.
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To separate material-produced currents from externally imposed (free) currents one introduces the auxiliary field H, related to B and M by B = μ0(H + M) (equivalently H = (1/μ0)B − M). In macroscopic magnetostatics Ampère’s law then becomes ∇ × H = J_free, so the curl of H depends only on free current density; bound currents are incorporated through M. For linear, homogeneous media this separation yields simple constitutive relations M = χ_m H and B = μμ0 H (with χ_m the magnetic susceptibility and μ the relative permeability), which greatly simplifies boundary-value problems. Omitting bound currents or an appropriate magnetization thus produces incomplete predictions for B; using M and H provides a consistent macroscopic framework that accounts for atomic-scale sources while isolating the influence of free currents.
Magnetization
Magnetization M(r) is the vector field that quantifies the local density of magnetic dipole moment: it equals the net magnetic dipole moment per unit volume in a small region and thus specifies both the magnitude and direction of the local magnetic moment density. For a uniformly magnetized body M is spatially constant and equals the total magnetic moment m divided by the body’s volume, M = m/volume. Because magnetic moment has SI units A·m^2, M has units A·m^−1, the same units as the magnetic field intensity H.
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The direction of M at a point corresponds to the average orientation of the constituent dipoles there. Field lines of the magnetization, considered inside the material, run from the magnetic south toward the magnetic north; by construction the magnetization vector field is defined only within the magnetic material and does not extend into free space.
Two complementary pictures relate M to sources of the magnetic B field. In the microscopic (Amperian-loop) picture, magnetization arises from many small circulating currents; their aggregation produces macroscopic bound currents wherever M varies, and these bound currents generate the magnet’s B field. This connection can be expressed by an Ampère-like integral relation for bound current: ∮ M · dℓ = I_b, where the line integral is around any closed loop and I_b is the bound current enclosed by that loop.
In the magnetic-pole (pole or charge) model, magnetization is represented by effective magnetic charges (poles) at which magnetization lines begin or end. A region with net positive magnetic pole strength (a net north pole) therefore has an excess of magnetization field lines terminating there. This imbalance is captured by a closed-surface flux law for magnetization: ∮_S μ0 M · dA = − q_M, where the surface integral is taken over a closed surface S that completely encloses the region, μ0 is the permeability of free space, and q_M is the enclosed magnetic charge (measured in units of magnetic flux). The minus sign reflects the adopted convention that magnetization lines run from magnetic south to magnetic north.
H is defined so that the macroscopic contribution of the material magnetization is removed from the magnetic flux density: H ≡ B/μ0 − M. By construction H isolates the part of the field produced by free currents and re-expresses the bound-current contribution of magnetized matter in a different mathematical form.
When Ampère’s law is written in terms of H, its circulation around a closed path depends only on free currents: ∮ H · dℓ = I_f. This follows because the bound currents generated by M cancel from the line integral, and the integral statement has an equivalent differential Maxwell equation that yields the tangential boundary condition across an interface, (H1∥ − H2∥) = K_f × n̂, where K_f is the surface free-current density and n̂ points from medium 2 into medium 1. Thus discontinuities in the tangential H component are produced solely by surface free currents.
A surface integral relation shows that μ0 times the net H flux through any closed surface equals the net “magnetic charge” q_M enclosed: μ0 ∮ H · dA = q_M. This equality, independent of free currents, encodes the bound-current effects of magnetization as an effective distribution of magnetic charge, providing a convenient scalar representation of otherwise circulating bound currents.
H can be decomposed uniquely into an applied (source) part and a demagnetizing part, H = H0 + Hd, where H0 is produced only by free currents and Hd arises only from bound currents associated with M. Hd typically opposes M within a magnet and thereby modifies the internal field relative to the externally applied H0; this demagnetizing field depends on body shape and magnetization distribution.
Topologically, H and B differ: because H encodes bound currents as equivalent magnetic charges, H-field lines may begin and end on these effective poles, whereas B is divergence-free (∇·B = 0) and its lines form continuous closed loops that never start or terminate. For a cylindrical bar magnet this distinction is concrete: inside the cylinder bound currents produce a demagnetizing Hd that alters the net H, while outside the H-field appears as if emanating from magnetic poles at the ends. Consequently, line integrals of H around closed loops detect only free currents, whereas closed-surface integrals of μ0H measure the effective magnetic charge distribution associated with the magnetization.
Magnetism
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When a material is exposed to an external magnetic induction (B-field) it typically develops an internal magnetization M that produces additional, induced magnetic fields. This microscopic response — magnetism — defines the material’s magnetic phase and in many materials is a relatively small effect that appears primarily while the external field is applied.
Materials are grouped by their magnetic response. Diamagnetic substances generate a magnetization that opposes the applied B-field, whereas paramagnetic substances produce magnetization in the same sense as the applied field. Ferromagnetic, ferrimagnetic and antiferromagnetic materials can sustain a spontaneous magnetization even in the absence of an applied field and exhibit complex interrelations between B and the magnetic intensity H. Superconductors (including superconductors that also display ferromagnetism) combine zero electrical resistance below a critical temperature with distinctive magnetic behavior, notably perfect diamagnetism under sufficiently low applied fields.
In many diamagnetic and paramagnetic materials the magnetization is proportional to the applied magnetic intensity, so that B and H are related by B = μH, where μ is the material-dependent permeability; in this linear regime both M and B scale linearly with H. In anisotropic media μ is not necessarily a scalar but can be a second-rank tensor, so that B and H need not be collinear and the induced field direction can differ from the applied H.
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Equations of the form B = μH and their tensor generalizations are constitutive relations that encode material-specific responses linking the macroscopic fields B and H; such laws are fundamental for predicting magnetic behavior across material classes.
Ferromagnetic, ferrimagnetic and many antiferromagnetic materials depart from linearity and exhibit history-dependent B–H behavior (magnetic hysteresis). Their magnetization can remain after removal of the applied field (remanence) and follow different trajectories upon field reversal, producing characteristic hysteresis loops in B–H or M–B diagrams.
Superconductors present multiple magnetic regimes: below a lower critical field they expel magnetic flux (the Meissner effect) and behave as ideal diamagnets; above an upper critical field or above the critical temperature superconductivity is lost; between these limits many superconductors enter a mixed state in which vortices and pinning lead to a complex, often hysteretic dependence of M on B.
Stored energy in a magnetic field arises because creating a magnetic field requires work both against the electric field induced by any time-varying magnetic flux and to change the magnetization of material occupying the region. In non‑dispersive, reversible media the work invested is recovered when the field is removed, so it is naturally described as energy stored in the field and quantified by an energy density u (energy per unit volume).
For linear, non‑dispersive materials with the constitutive relation B = μH where μ is frequency‑independent, the magnetic energy density has the closed forms
u = (B · H)/2 = (B · B)/(2μ) = (μ H · H)/2.
Which form is most convenient depends on whether B, H or μ is known. In free space or in the absence of magnetic materials μ is replaced by the vacuum permeability μ0, yielding the same algebraic forms with μ0.
When the material response is nonlinear the simple quadratic formulas no longer hold. The correct incremental work per unit volume required to effect a small change δB is δW = H · δB, and the total work to reach a given magnetic state must be obtained by integrating H · δB along the actual B–H path taken. Materials with hysteresis (e.g., ferromagnets, superconductors) are path dependent; in those cases the stored and dissipated energy depend on the magnetization history and on how the field was produced, because energy is not solely a function of the instantaneous B and H. In the special case of linear, non‑dispersive media the integral of δW = H · δB reproduces the quadratic expressions above, showing consistency between the differential and closed‑form descriptions when μ is constant.
Maxwell’s equations, together with the Lorentz force law, provide the complete classical framework for electrodynamics, unifying electric and magnetic phenomena. Within this framework the magnetic induction B is governed by specific divergence and curl relations that determine its local behavior and sources: electric currents and time‑varying electric fields act as the generators of B, while its global structure is constrained by an absence of isolated magnetic charges.
The divergence operator ∇·A measures the net flux of a vector field A out of an infinitesimal volume and thus identifies local sources or sinks. For the magnetic field this relation reads ∇·B = 0 (Gauss’s law for magnetism). Mathematically this means B is solenoidal: field lines neither begin nor end at points but form continuous closed loops. Physically, ∇·B = 0 encodes the empirical and theoretical statement that magnetic monopoles do not occur in classical electrodynamics.
The curl operator ∇×A characterizes the tendency of A to circulate about a point and thereby serves as a local measure of induced circulation. The curl relations in Maxwell’s system are Faraday’s law, ∇×E = −∂B/∂t, which links time‑varying magnetic fields to induced electric circulation, and the Ampère–Maxwell law, ∇×B = μ0J + μ0ε0∂E/∂t (in SI units), which relates the curl of B to conduction currents J and to changing electric fields. Thus electric currents and evolving electric fields function as the explicit sources driving magnetic circulation.
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Together the two divergence and two curl equations for E and B specify how the fields are generated and how they circulate: B is produced by currents and time‑dependent electric fields while E acquires circulating character from time‑dependent magnetic fields. These relations, complemented by the Lorentz force law, constitute the core of classical electrodynamics.
The magnetic field produced by steady currents or permanent magnets is solenoidal: its lines of force have no origins or terminations. Individual B-field lines therefore either extend without bound, form closed loops, or follow nonterminating trajectories, but they cannot begin or end at isolated points.
This property is manifest around a bar magnet, where field lines appear to emerge from the region near the north pole and re-enter near the south pole; within the magnet the lines continue from the south back to the north. Consequently any line that enters a material region must leave it elsewhere, since true terminal points do not exist for B.
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Globally, this absence of beginnings or endings implies conservation of magnetic flux through any closed region: the total flux entering a closed surface equals the total flux leaving it, so the net flux is zero. Gauss’s law for magnetism expresses this precisely as
∮_S B · dA = 0,
for any closed surface S (with dA taken outward). The dot product B · dA is positive where the field points outward and negative where it points inward, so the surface integral directly measures the balance of exiting and entering field lines. In differential form this condition is written ∇·B = 0, stating that the magnetic field has zero divergence everywhere.
Faraday’s law describes how a time-varying magnetic field interacting with a conductive loop produces an electric field that drives a circulating current, the basic mechanism of electromagnetic induction exploited in generators and motors. In integral form the law quantifies the induced electromotive force (EMF) around a closed contour by the relation ℰ = −dΦ/dt, where Φ is the magnetic flux through any surface bounded by the contour. Magnetic flux is the surface integral of the normal component of the magnetic field, Φ = ∫_S B · dA, which motivates referring to B as the magnetic flux density. The minus sign encodes Lenz’s law: the induced current produces its own magnetic field that opposes the change in flux that created it, ensuring consistency with energy conservation and fixing the direction of the induced EMF. Applying Stokes’ theorem and assuming smoothly varying fields converts the integral statement to the local differential form ∇×E = −∂B/∂t, which states that the curl of the electric field at a point equals the negative time derivative of the magnetic field there. Practically, any time-varying B-field near conductors will produce measurable voltages and currents; controlled variation of magnetic flux underlies transformers, inductors, electric generators and motors.
Maxwell’s correction augments Ampère’s law by adding a term proportional to the time derivative of the electric flux, yielding the full Maxwell–Ampère relation in differential form,
∇ × B = μ0 J + μ0 ε0 ∂E/∂t,
where J is the microscopic current density, μ0 the magnetic constant and ε0 the vacuum permittivity. Physically, the added term μ0ε0∂E/∂t represents the magnetic effect of a time-varying electric field (i.e., the time rate of change of electric flux through a surface) and is formally analogous to Faraday’s law, although it appears with a different sign/constant structure. This “displacement-current” contribution is essential for self-sustaining electromagnetic waves: a varying electric field generates a varying magnetic field, which in turn regenerates the electric field, allowing propagation (for example, light). In many practical integral-law calculations the displacement term is numerically negligible and so receives less emphasis, but it is indispensable in the local differential description and in wave phenomena. In material media microscopic polarization and magnetization produce bound charges and currents that alter E and B; to avoid computing these explicitly one commonly introduces the auxiliary fields D and H and the free current density Jf, giving the macroscopic form
∇ × H = Jf + ∂D/∂t.
This H–D formulation is equivalent in content to the microscopic equations only when supplemented by constitutive relations (B–H and E–D laws) that specify the material response; when such relations are known and simple, they obviate detailed bound-charge/current calculations.
As different aspects of the same phenomenon
Special relativity shows that what is labeled “electric’’ or “magnetic’’ depends on the observer: a purely electric force in one inertial frame can appear as a magnetic force, or as a mixture of electric and magnetic forces, in another. This frame dependence follows quantitatively by applying a Lorentz transformation to the four‑force obtained from Coulomb’s law in the charge rest frame and then identifying fields via the Lorentz‑force law; for a point charge moving at constant velocity this procedure yields a magnetic field that satisfies Maxwell’s equations in the observer frame. For a uniformly translating point charge q the field may be written in closed form as
B = (q / (4π ε0 r^3)) · ((1 − β^2) / (1 − β^2 sin^2 θ)^{3/2}) · (v × r) / c^2 = (v × E) / c^2,
where r locates the field point relative to the source, v is the source velocity, θ is the angle between v and r, β = |v|/c, and the cross product v × r gives the field direction orthogonal to both velocity and displacement. This expression assumes the source moves without acceleration and, in the nonrelativistic limit β ≪ 1, reduces to the familiar Biot–Savart form appropriate for steady currents or slowly moving charges. More generally, electric and magnetic fields are components of a single antisymmetric rank‑2 electromagnetic tensor; Lorentz transformations mix these components just as they mix space and time, so E and B are different projections of one covariant field. The electromagnetic stress–energy tensor similarly unifies field energy and momentum: the energy density ascribed to a magnetic field in one frame corresponds in part to electric field energy in another, reflecting that field energy and stress are frame‑dependent components of a single tensorial energy–momentum content.
Magnetic vector potential
In electrodynamics the magnetic vector potential A and the electric scalar potential φ serve as alternative field variables from which the observable fields are obtained by differential relations. Specifically,
[
\mathbf{B}=\nabla\times\mathbf{A},\qquad
\mathbf{E}=-\nabla\varphi-\frac{\partial\mathbf{A}}{\partial t},
]
so that A and φ generate the magnetic and electric fields respectively. These relations provide the canonical mapping from potentials to the measurable fields B and E.
Physically, the potentials admit a mechanical interpretation: A acts like a generalized momentum per unit charge, while φ plays the role of a generalized potential energy per unit charge. This analogy places the electromagnetic potentials on the same conceptual footing as the potentials that determine forces and energies in classical mechanics for charged particles.
The potentials are not uniquely determined by E and B because of gauge freedom: different pairs (A, φ) can produce the same fields. Imposing a gauge condition (gauge fixing) selects a unique representative within an equivalence class of potentials. A particularly useful choice is the Lorenz gauge, which casts the potential equations into manifestly Lorentz-covariant wave equations and thereby makes the compatibility of classical electrodynamics with special relativity explicit.
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In the relativistic formulation the spatial vector A and scalar φ combine into the electromagnetic four-potential Aμ, a four-vector analogous to a particle’s four-momentum. Using the four-potential simplifies many theoretical expressions, yields compact Lorentz-covariant field equations, and aligns the geometry of electromagnetic quantities with other relativistic four-vectors.
The potential formulation also has practical and conceptual advantages: it integrates naturally with quantum mechanics (e.g., minimal coupling in wave equations), streamlines variational and algebraic treatments, and clarifies the relationship between symmetries and conserved quantities. These benefits must be balanced against the fact that physical observables remain the fields E and B and that consistent handling of gauge freedom is required.
Special relativity requires that causal interactions be confined to time-like or light-like separations so that no influence travels faster than the invariant speed c; this light-cone structure fixes an absolute causal ordering of events. Classical Maxwell theory is consistent with this requirement: electromagnetic disturbances propagate at speed c, so the fields observed at a spacetime point necessarily originate from source behavior in the past, delayed by the finite travel time of signals. Locality in electrodynamics is implemented by expressing the field at (r, t) in terms of the source evaluated at the retarded time t_r, the unique emission instant whose light signal can reach the observation event.
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For a point source with trajectory r_s(t) the retarded time is defined implicitly by the light-travel relation t_r = t − |r − r_s(t_r)|/c, where the term |r − r_s(t_r)|/c is the travel time from the source at emission to the observer at (r, t). Because the source position enters evaluated at t_r, this relation is transcendental in general and must be solved for t_r; geometrically it locates the intersection of the source worldline with the past light cone of the observation event. The subtraction of the travel-time term enforces causality: only source events on or inside the past light cone (signals moving at ≤ c) can influence the field at (r, t). When the source speed satisfies |v| < c the retarded-time equation yields a unique solution, guaranteeing a single causal emission point on the worldline that produced the observed electromagnetic field.
Magnetic field of an arbitrarily moving point charge
Electromagnetic effects propagate at finite speed c, so the fields observed at space-time point (r, t) depend on the source’s state at the retarded time t_r, defined implicitly by t = t_r + |r − r_s(t_r)|/c. All source-dependent quantities in the field expressions — position r_s, velocity β_s = v/c, Lorentz factor γ = 1/√(1 − |β_s|^2), and acceleration ˙β_s = (1/c)dv/dt — are evaluated at this retarded time. The geometry is set by the unit vector n_s = (r − r_s(t_r))/|r − r_s(t_r)|, which points from the source (at emission) toward the field point.
In the Lorenz gauge the magnetic vector potential is given by
A(r,t) = (μ0 c / 4π) [ q β_s / ((1 − n_s·β_s) |r − r_s|) ]_{t_r},
and is related to the scalar potential φ by A(r,t) = [β_s(t_r)/c] φ(r,t). The complete electromagnetic fields follow from the usual relations E = −∇φ − ∂A/∂t and B = ∇×A, with all source quantities taken at t_r.
The exact expression for B naturally separates into a near-field (velocity) contribution that decays as 1/|r − r_s|^2 and a radiation (acceleration) contribution that decays as 1/|r − r_s|. The near-field term is proportional to q c (β_s × n_s) divided by γ^2(1 − n_s·β_s)^3 and represents a generalized Coulomb-like magnetic part, which is suppressed at relativistic speeds by the γ^2 factor. The radiation term is proportional to q times n_s × [ n_s × ( (n_s − β_s) × ˙β_s ) ]/(1 − n_s·β_s)^3 and depends explicitly on the source acceleration ˙β_s; this term carries energy away from the charge as electromagnetic radiation.
For the radiation components the fields form a mutually orthogonal triad: E, B and n_s (evaluated at t_r) are perpendicular to one another, with B = (n_s/c) × E. Finally, because Maxwell’s equations are linear, the fields produced by multiple point charges superpose linearly; each charge contributes independently according to its past-light-cone (retarded-time) data, ensuring causal locality of electromagnetic interactions.
Quantum electrodynamics
The inability of semi-classical radiation theory—where matter is quantized but the electromagnetic field remains classical—to account for phenomena such as spontaneous emission and the Lamb shift motivates treating the electromagnetic field itself quantum mechanically. In the quantum-field description the field at each spacetime point is represented not by ordinary numbers but by operator-valued vector fields; the quanta of these field operators are photons.
Quantum electrodynamics (QED), the quantum field theory of the electromagnetic interaction and its charged particles, is formulated within this operator framework and sits inside the Standard Model of particle physics. Interactions in QED are ordinarily evaluated perturbatively: complex algebraic series expansions are organized and visualized with Feynman diagrams in which charged particle lines exchange virtual photons.
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QED’s predictions agree with experiment to extraordinarily high precision (currently at the level of about 10^-12, with the dominant limitation coming from experimental error), making it one of the best-confirmed physical theories. Many pedagogical and applied treatments nevertheless present classical-field equations; these are approximations to the full theory that are adequate for most macroscopic and everyday situations but omit genuinely quantum effects that QED is required to explain.
Earth’s magnetic field
The Earth’s magnetic field at the surface is well approximated by a dipole field whose axis is tilted roughly 11° with respect to the planet’s rotation axis; in the simple dipole picture the externally observed field resembles that of a giant bar magnet centered in the Earth, with field lines emerging and re-entering in a pattern offset from the geographic poles. This tilt accounts for the systematic displacement between geomagnetic and geographic coordinates and explains why the magnetic poles do not coincide with the rotational poles.
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The field is not produced by a permanent magnet in the crust but by a self‑sustaining geodynamo in the fluid outer core: convective motion of a liquid iron–nickel alloy generates electrical currents, and those currents produce magnetic fields that feed back on and maintain the flow, allowing the geomagnetic field to persist over geological timescales. A magnetic compass aligns with the ambient geomagnetic field; because opposite magnetic poles attract, the compass end labeled “north” points toward the Earth’s magnetic south pole (commonly called the North Magnetic Pole), a historical naming convention that can be misleading in physical terms.
The geomagnetic field is time dependent: both its intensity and the positions of the magnetic poles evolve (secular variation), and on longer timescales the dipole polarity has reversed repeatedly—the last full reversal occurred about 780,000 years ago.
Rotating magnetic fields
A rotating magnetic field is the operative principle by which alternating-current (AC) machines convert electromagnetic energy into mechanical rotation: a magnetic field that turns in space exerts a torque on a rotor so that the rotor continually reorients to align with the external field, producing usable shaft motion. In practical devices this torque arises from the interaction between the rotor’s magnetic elements (permanent magnets or current‑carrying conductors) and the time‑varying field produced by the stator.
The stator field is generated by spatially distributed coils driven with time‑shifted alternating currents. In the simplest laboratory demonstration, two orthogonally placed coils fed with sinusoidal currents 90° out of phase produce a vector sum whose direction rotates smoothly in the plane. Commercial machines employ three‑phase systems: three equal‑magnitude currents separated by 120° in phase energize coils at 120° geometric spacing on the stator, yielding a nearly uniform rotating field. That combination of equal amplitudes and 120° phase shifts is central to the smooth torque production of three‑phase motors and underpins the widespread adoption of three‑phase power distribution.
Rotors respond to the rotating stator field in two principal ways. Synchronous machines have rotor windings supplied with direct current (or otherwise magnetically excited) so that the rotor’s magnetic axis locks to and turns exactly in step with the stator field. Induction machines instead use short‑circuited rotor conductors (for example, a squirrel‑cage); the rotating stator field induces currents in those conductors, and the resulting electromagnetic forces—describable by the Lorentz force law acting on current elements in a magnetic field—produce torque and rotation without direct electrical connection to the rotor excitation.
The rotating magnetic field was identified independently in the late 19th century by Galileo Ferraris, who published his results to the Royal Academy of Turin in 1888, and by Nikola Tesla, who obtained U.S. Patent No. 381,968 for related electromagnetic motor technology. Their work established the foundation for modern AC machines.
Hall effect
When a current-carrying conductor is subjected to a magnetic field perpendicular to the direction of current, the charge carriers experience a lateral Lorentz force that drives them toward one side of the conductor. This charge separation generates a transverse electric field and an associated potential difference across the conductor, the Hall voltage. The redistribution of charge proceeds until the transverse electric force on the carriers exactly counterbalances the magnetic Lorentz force; at this steady state the transverse field (and thus the Hall voltage) scales predictably with the applied magnetic flux density and the current geometry.
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Because the sign of the Hall voltage depends on the sign of the predominant charge carriers, Hall measurements yield both the magnitude and the polarity of the magnetic field and reveal whether electrons or holes dominate electrical conduction. The measured Hall coefficient, which relates the transverse electric field to the product of current density and magnetic field, also provides a direct means of estimating carrier concentration. These properties make the Hall effect a practical diagnostic and sensing technique: it is widely used to measure external magnetic fields and to characterize semiconductor materials (for example, distinguishing n-type from p-type conductivity) under controlled current and field geometries.
In linear magnetic materials the flux density B and the field intensity H are proportional, B = μH, where μ (the magnetic permeability) is a material constant. This constitutive relation for magnetism mirrors the microscopic electrical law J = σE (current density proportional to electric field via conductivity σ), indicating a parallel between a material’s magnetic response and electrical conduction.
At the macroscopic, circuit level this correspondence yields a magnetic analogue of Ohm’s law: Φ = F / Rm. Here Φ = ∫ B · dA is the magnetic flux through a chosen surface, F = ∫ H · dℓ is the magnetomotive force around the magnetic path, and Rm (reluctance) quantifies the opposition to flux in the circuit in the same way resistance opposes current. Reluctance therefore plays the role of electrical resistance in magnetic networks.
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Because of these direct analogies, complex magnetic-field configurations can be treated with standard circuit methods. Magnetomotive forces, reluctances, and fluxes can be combined and analyzed like voltages, resistances, and currents, allowing the use of series/parallel reduction and network-analysis techniques to calculate flux distributions in magnetic circuits.
Largest-magnitude magnetic fields
This section is flagged for updating; the most recent recorded edits are dated October 2018 with a parenthetical review note (July 2021). The numerical claims below therefore reflect the documented record as of those dates and should be checked against more recent measurements before citation.
The largest reported magnetic field produced over a macroscopic volume outside a conventional laboratory environment is a 2.8 kT (2.8 × 10^3 T) pulse generated by VNIIEF in Sarov, Russia, in 1998. The highest macroscopic-volume field produced within a laboratory setting reported in the cited update is a 1.2 kT (1.2 × 10^3 T) field achieved by researchers at the University of Tokyo in 2018. By contrast, particle-physics experiments produce far greater peak fields but confined to microscopic regions: heavy-ion collisions at facilities such as RHIC generate transient magnetic fields estimated on the order of 10^14 T within extremely small volumes and durations. The strongest naturally occurring magnetic fields are found on magnetars (a class of neutron star), whose surface fields are reported in the range ~0.1–100 GT (≈10^8–10^11 T), greatly exceeding any sustained terrestrial or laboratory fields.
Common formulae for steady magnetic fields
This section collects closed‑form expressions for the magnetic induction B produced by simple steady current distributions. Symbols: μ0 is the permeability of free space, I the current (or total current), x or r the relevant radial/axial distance, R a characteristic radius, N the number of turns, n the turns per unit length, and m the magnetic dipole moment. Field senses follow the right‑hand rule unless otherwise indicated.
Finite straight segment
The magnetic field at a point located a perpendicular distance x from a finite straight conductor carrying current I is
B = (μ0 I / (4π x)) (cos θ1 + cos θ2),
where θ1 and θ2 are the angles between the lines from the point to the segment ends and the line extending the conductor. The field is azimuthal about the conductor.
Infinite straight wire and cylindrical conductor
For an infinitely long straight wire the familiar cylindrical (Biot–Savart/Ampère) result is
B = μ0 I / (2π x),
with x the radial distance from the wire axis and the field tangentially encircling the wire. A cylindrical conductor of radius R carrying a uniform current density reproduces this outer result for x ≥ R, while inside (x ≤ R) the field grows linearly with radius:
B = μ0 I x / (2π R^2),
so the interior field vanishes at the axis and reaches the external 1/x envelope at the surface. In both cases the field is azimuthal about the axis.
Circular loop on axis
Along the symmetry axis of a circular loop of radius R carrying current I, the axial induction at distance x from the loop center is
B = μ0 I R^2 / [2 (x^2 + R^2)^{3/2}],
directed along the loop axis (sign set by the current sense).
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Finite and infinite solenoids (axial fields)
A finite solenoid with n turns per unit length and current I produces on its axis
B = (μ0 n I / 2) (cos θ1 + cos θ2),
where θ1 and θ2 are the polar angles subtended by the near and far ends as seen from the field point; the field lies along the solenoid axis. In the idealized infinite (very long) solenoid the axial interior field becomes uniform,
B = μ0 n I,
and the external field is effectively zero.
Toroid (circular toroidal coil)
Within the core of a torus of mean radius R carrying N uniformly distributed turns with current I, the azimuthal field is
B = μ0 N I / (2π R),
circulating around the torus and confined to the toroidal region in the idealized limit.
Magnetic dipole far fields (axial and equatorial)
At distances large compared with the source size, a current loop behaves as a dipole moment m. On the equatorial plane (perpendicular to m) the leading far‑field is
B = − μ0 m / (4π r^3),
where the minus sign indicates the direction opposite to m; along the axis the far‑field is
B = μ0 m / (2π |x|^3),
showing the characteristic 1/r^3 decay and the different directional signs on and off the axis.
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Arc and regular polygon at center
The magnetic induction at the center of a circular arc of radius R and central angle θ (radians) is
B = μ0 θ I / (4π R),
directed perpendicular to the arc plane. For a regular N‑sided polygon of side length a carrying current I, the central field magnitude can be written in closed form as
B = (μ0 N I / (π a)) sin(π/N) tan(π/N),
again perpendicular to the polygon plane, with sense determined by the current direction.
These expressions summarize common analytic results used to build and approximate more complex magnetic field geometries.
History
René Descartes’ 1644 engraving stands among the earliest attempts to render magnetism visually, depicting the Earth as exerting an attractive influence on lodestones and thereby treating magnetic action as a planetary-scale phenomenon. By combining the globe and samples of naturally magnetic rock in a single graphic, Descartes used spatial representation to make an otherwise invisible force intelligible, effectively mapping how a terrestrial agency might account for the observed attraction of lodestones.
Accompanying the image, Descartes proposed a mechanical account in which magnetism results from the circulation of minute helical particles—his “threaded parts”—flowing through complementary “threaded pores” in magnetic materials. Although superseded by later electromagnetic theory, this particulate-circulation model is significant in the history of geophysics: it frames magnetism as arising from internal material processes and as extending spatially from the Earth to localized magnetic bodies, marking an early conceptualization of what would become the study of geomagnetism.
In 1269 Petrus Peregrinus of Maricourt produced the first systematic empirical map of a magnetic field by laying iron needles on a spherical magnet and using their orientations to trace continuous lines of magnetic direction. He noted that these lines converged at two antipodal loci on the sphere, which he termed “poles” in explicit analogy to terrestrial geography, and from his observations concluded that every magnet possesses two distinct ends—north and south—that persist even when the magnet is subdivided, an early empirical statement of magnetic dipolarity. Building on Peregrinus’s needle-mapping technique, William Gilbert of Colchester revisited these experiments and in 1600 published De Magnete, in which he argued that the Earth itself behaves like a giant magnet. Gilbert’s work thus extended laboratory-scale dipole observations to a planetary context and helped consolidate magnetism as a scientific field by combining systematic experimentation with the conceptual insight that planetary bodies may host global magnetic fields analogous to those mapped on smaller magnets.
Mathematical development
The mathematical development of classical magnetism traces a progression from a pole-based inverse-square picture to a unified, dynamical field theory. Early conjectures by John Michell (1750) proposed an inverse-square interaction for magnetic poles; experimental confirmation followed in Coulomb’s torsion-balance measurements (1785), which also emphasized the inseparability of north and south poles. Building on these empirical foundations, Poisson (1824) produced the first systematic mathematical representation by treating magnetized bodies as assemblies of infinitesimal north–south pole pairs and by defining an H-field produced by such discrete poles.
The year 1820 marked a decisive shift from static pole models to current-centered explanations. Ørsted’s demonstration that a steady electric current produces a circular magnetic effect around a conductor, together with Ampère’s observation that parallel currents exert mutual forces (attraction for like directions, repulsion for opposite), and the Biot–Savart measurements relating the force on a small magnet to the inverse perpendicular distance from a long straight current, collectively indicated that magnetism is intimately tied to electric currents. Laplace reached a corresponding differential formulation independently, though he did not publish it.
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Ampère synthesized these experimental facts by abandoning the pole-as-fundamental view in favor of continuous, closed current loops as the source of magnetic phenomena. He established the formal equivalence between steady currents and magnets, derived an expression for the force between current elements (Ampère’s force law), and formulated a law for the magnetic field generated by steady currents (Ampère’s law), results consistent with the Biot–Savart empirical relationship. Ampère’s work effectively inaugurated electrodynamics as the study of the interaction between currents and magnetic effects.
Electromagnetic induction introduced time dependence into the theory. Faraday (1831) showed that a changing magnetic flux produces a circulating electric field—a qualitative law that linked temporal variation of magnetic fields to induced electromotive forces. Neumann subsequently demonstrated that the electromotive effects observed when conductors move in magnetic fields can be derived from Ampère’s force law and, in so doing, formulated the magnetic vector potential as a convenient mathematical quantity that reproduces Faraday’s experimentally motivated picture.
By mid-century the need to distinguish different field concepts within matter became clear. William Thomson (Lord Kelvin) (1850) separated the pole-based field concept, denoted H, from the induction or flux density B associated with currents and changing fields, and introduced the notion of permeability to quantify how material media relate H and B.
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This conceptual and mathematical consolidation culminated in Maxwell’s synthesis (1861–1865). Maxwell combined laws for static and dynamic electricity and magnetism into a single set of field equations and demonstrated that these equations admit wave solutions propagating at the speed of light, thereby identifying light with an electromagnetic disturbance. Heinrich Hertz’s laboratory experiments in 1887–1888 provided the first direct experimental confirmation of Maxwell’s dynamical predictions, closing the historical arc from inverse‑square pole hypotheses to a complete classical field theory of electromagnetism.
The late nineteenth-century realization of the rotating magnetic field provided the practical foundation for modern induction motors. Nikola Tesla developed an alternating‑current polyphase induction motor in 1887 that exploited a rotating magnetic field to produce torque on a rotor; he received a patent for this design in May 1888. Independently, Galileo Ferraris had pursued both experimental and theoretical studies of rotating fields from 1885 and presented his results to the Royal Academy of Sciences in Turin in March 1888, shortly before Tesla’s patent.
Physically, the rotating field—generated by spatially and temporally phased polyphase currents—establishes a synchronous magnetic pattern in the stator that drags the rotor into motion without requiring an electrical connection to the rotor windings, a mechanism central to the operation of induction motors.
During the twentieth century, developments in fundamental theory placed classical electrodynamics within broader physical frameworks. Einstein’s 1905 special relativity showed that electric and magnetic fields are not separate entities but interrelated components of a single electromagnetic field whose decomposition into E and B depends on the observer’s inertial frame. Subsequent integration with quantum mechanics produced quantum electrodynamics (QED), the quantum field theory that quantizes electromagnetic energy and identifies photons as the discrete quanta mediating electromagnetic interactions.