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For a steady filamentary electric current $I$ in vacuum, the Biot-Savart law isThe line element points with conventional electric current, and the cross product fixes the magnetic flux density direction by the right-hand rule.

As an application, an infinitely long straight wire has circular symmetry about its axis. At perpendicular distance $s$,Its direction follows the right-hand rule. The following part applies the same law to a circular current loop.

Parameterize the loop by ${\mathbf{r}}^{\prime }=(a\cos \varphi ,a\sin \varphi ,0)$ and choose positive electric current counterclockwise as seen from $+z$. At the observation point $\mathbf{r}=(0,0,z)$,The numerator of the Biot-Savart law isThe distance $\sqrt{a^2+z^2}$ is constant round the loop. The transverse terms integrate to zero, while the axial term integrates to $2\pi a^2$. HenceFor use below, define the undisturbed magnetic field intensity $H_0(z)=B_0(z)/{\mu }_0=a^{2}I/[2(a^2+z^2{)}^{3/2}]$. This distinguishes the applied field from the internal field after adding the magnetic sample.

Because the magnetic field obeys $\nabla \cdot \mathbf{B}=0$, it can be represented locally as the curl of a magnetic vector potential:This automatically enforces the absence of magnetic monopoles. It is also a global representation throughout ordinary all-space problems without monopole singularities. It is not unique: the gauge transformationleaves $\mathbf{B}$ unchanged because the curl of a gradient vanishes. In a time-dependent problem,makes the same transformation leave both physical fields unchanged and automatically satisfies Faraday's law.

In vacuum magnetostatics, choose the Coulomb gauge $\nabla \cdot \mathbf{A}=0$. Substitution into Ampère's law and the identity $\nabla \times (\nabla \times \mathbf{A})=\nabla (\nabla \cdot \mathbf{A})-{\nabla }^2\mathbf{A}$ givesfor a localized steady electric current density and a potential vanishing at infinity. Taking the curl recovers the Biot-Savart law. The magnetic vector potential converts a constrained vector field problem into componentwise Poisson equations once a gauge is chosen.

For a uniform magnetic field, $\mathbf{A}=\frac{1}{2}\mathbf{B}\times \mathbf{r}$ is a convenient choice. For a localized magnetic dipole moment $\mathbf{m}$, the leading far-field potential isAnother useful relation is Stokes theorem, which relates a closed-path integral to the magnetic flux through a spanning surface:The units of $\mathbf{A}$ are $\mathrm{T}\mathrm{m}$, equivalently $\mathrm{Wb}{\mathrm{m}}^{-1}$.

For a steady thin-wire current in vacuum, the Biot-Savart law isThe line element follows conventional current direction. More generally replace $Id{\mathbf{\ell }}^{\prime }$ with $\mathbf{J}({\mathbf{r}}^{\prime })d^3{r}^{\prime }$. The cross product gives the circulating field direction and superposition allows contributions from separate wire segments to be added.

Integration gives $B={\mu }_{0}I/(2\pi r)$ around a long straight wire. On the axis of a circular loop of radius a, $B_z={\mu }_{0}I{a}^2/[2(a^2+z^2{)}^{3/2}]$, including the centre value ${\mu }_{0}I/(2a)$. Coil turns multiply the field when their geometries coincide; distributed coils and solenoids follow by integration. This is a magnetostatic or quasistatic calculation: rapidly changing currents require retarded electrodynamics and appropriate displacement-current effects, not an instantaneous wire formula. ^[ref]

For steady currents in vacuum, the Biot-Savart law sums the contributions of current elements:It is useful for known wire geometries; a circular loop of radius $a$ gives $B={\mu }_{0}I/(2a)$ at its centre.

Ampère's circuital law in magnetostatics isWith sufficient symmetry it finds fields directly: a long straight wire gives $B={\mu }_{0}I/(2\pi r)$, while a long solenoid has interior $B\simeq {\mu }_{0}nI$ and a small exterior field away from its ends. Without symmetry, a circulation value alone does not determine the local field. For time-dependent electric fields, use the Ampère-Maxwell equation with the displacement-current term ${\mu }_0{ϵ}_{0}d{\Phi }_E/dt$ as well as conduction current.

For a steady thin-wire electric current, the Biot-Savart law iswhere $\mathbf{R}$ points from the source element to the field point and $d\mathbf{\ell }$ follows positive electric current. On a circular loop's axis the transverse contributions cancel by symmetry. The axial numerator is $a^{2}d\phi$ and the distance is $\sqrt{a^2+z^2}$, givingIts graph is positive and even, with maximum ${\mu }_{0}I/(2a)$ at the centre and a $|z{|}^{-3}$ far-field decay. Reversing the electric current reverses its sign.

For two equally oriented $N$-turn coils centred at $\pm D$, superposition givesConsequentlyThis is an even differentiable function, so every odd derivative at zero vanishes, including the first and third. To check the second derivative explicitly, let $f(s)=(a^2+s^2{)}^{-3/2}$. ThenThus $2D=a$ makes the quadratic variation vanish, which is the Helmholtz coil condition. The central value and leading axial variation are

Near the centre, the coils' magnetic field lines are nearly straight and parallel to the axis. On the axis rotational symmetry makes $B_x=B_y=0$ exactly; in the central plane the radial components from the two coils cancel. More quantitatively, in the current-free central volume, axisymmetry and $\nabla \cdot \mathbf{B}=0$ implyThe suppressed axial derivatives therefore also suppress transverse magnetic fields near the centre. The smallness applies locally compared with the coil radius; it is not a claim of an exactly uniform field throughout the apparatus.

For steady currents in vacuum, the Biot-Savart law gives the magnetic field by summing oriented current elements. With $\mathbf{R}=\mathbf{r}-{\mathbf{r}}^{\prime }$ from the source to the observation point,For a thin wire it becomes $\mathbf{B}=({\mu }_{0}I/4\pi )\int d{\mathbf{l}}^{\prime }\times \hat{\mathbf{R}}/R^2$. The cross product fixes the direction by the right-hand rule. Fields from different steady circuits add by superposition; the entire current path, including return wires where relevant, belongs in the calculation.

An infinite straight wire gives circular field lines of magnitude $\boxed{B={\mu }_{0}I/(2\pi s)}$ at perpendicular distance $s$. This follows by integrating the transverse component of each element along the wire. A circular loop of radius $a$ has cancelling transverse components on its axis, leavingMultiple coincident turns multiply the field by their number. Far from the loop this is the field of a magnetic dipole of moment $\mathbf{\mu }=I\pi a^2\hat{\mathbf{n}}$. Integrating over many loops gives the approximately uniform interior field $B\simeq {\mu }_{0}nI$ of a long solenoid, with $n$ turns per unit length. Finite coils have fringing fields, and appropriately spaced coil pairs can produce a more uniform measurement region.

The law is useful for predicting fields of wires and coils where direct symmetry is insufficient, designing electromagnets and estimating stray fields. It is a magnetostatic law: steady charge flow satisfies charge continuity. An instantaneous application to rapidly changing currents neglects propagation and radiation; such problems require the time-dependent Maxwell's equations and retarded fields. A quasistatic approximation can be valid when circuit dimensions are small compared with the light-travel distance during the variation timescale.

Choose the electric current counterclockwise as viewed from $+z$; reversing it reverses the magnetic field. Parameterize the circular wire byThe cross product in the Biot-Savart law isConsequently the transverse terms integrate to zero over $0\le \varphi \le 2\pi$, while the axial term is constant:ThusThe direction is $+z$ on both halves of the axis for this electric current orientation, including $z<0$; it is not “away from the plane” on both sides. The right-hand rule gives the same direction.

Read the original diagram as a clockwise arc carrying electric current from its upper endpoint to its lower endpoint, with tangent semi-infinite leads extending to the right. Their crossing is insulated, so there is no extra junction. Choose $+z$ out of the page. At the centre the arc produces a negative axial magnetic field, and each straight lead a positive one, by the right-hand rule.

For the arc, every element is at distance $R$, with $d\mathbf{l}$ perpendicular to its displacement to the centre. The Biot-Savart law therefore givesFor each straight lead, write $s\ge 0$ for distance along the tangent measured from its endpoint. Its perpendicular distance from the centre is $R$, soThe integral equals $1/R$ because an antiderivative is $s/[R\sqrt{R^2+s^2}]$. Adding both leads and the arc,Using the full infinite-wire magnetic field for either lead would incorrectly double its contribution.

In magnetostatics, Ampère's circuital law isIt relates magnetic circulation around a closed curve to the steady current through a spanning surface. It is especially useful when symmetry makes the tangential field constant on a chosen path: an infinite straight wire gives $B={\mu }_{0}I/(2\pi r)$, while an ideal long solenoid gives approximately $B={\mu }_{0}nI$ inside.

The Biot-Savart law gives the field contribution from the actual current geometry:The cross product determines the direction by the right-hand rule. Integrating around a circular loop of radius $a$ gives $B={\mu }_{0}I/(2a)$ at its centre. A finite wire segment can also be integrated directly, even though it lacks enough symmetry for an Ampère-loop shortcut. The full steady-current circuit includes a return path; a segment calculation specifies that segment's contribution.

For an observation point in the wire's plane, both $d\mathbf{s}$ and $\mathbf{r}-\mathbf{s}$ lie in that plane. Their cross product is normal to it, and the Biot-Savart law integrates only such normal vectors. Thus the field is perpendicular to the plane at every point off the wire.

At the centre, all four sides contribute in the same normal direction by the right-hand rule. A single side is distance $a$ from the centre, so its contribution isMultiplying by four giveswhere $\mathbf{n}$ is the oriented normal fixed by the current.

A vortex filament is the ideal thin-core limit of a vortex tube, concentrating a finite circulation on a curve. In an unbounded incompressible fluid with velocity tending to zero at infinity, its induced velocity field follows the vorticity version of the Biot-Savart law. The ideal field is singular on the filament; finite-core models are needed there. A closed circular filament is a thin vortex ring.

A vortex ring is a vortex tube bent into a closed loop, often approximately a circle. Its vorticity runs around the loop, while the induced velocity field passes through the centre and returns outside. For a thin ring of radius $r$ and circulation $\Gamma$, the Biot-Savart law for incompressible vorticity gives an axial velocity $u_z(z)=\Gamma r^2/[2(r^2+z^2{)}^{3/2}]$, measured relative to the ring centre, with its sign set by the circulation orientation. A vortex ring is not a loop of fluid whose velocity everywhere merely follows the circular vortex line.