Biot-Savart law and Ampere's law

Intuition [Beginner]

Static charges produce electric fields. Moving charges produce something else: a magnetic field $\mathbf{B}$. The experimental fact is that a current-carrying wire deflects a nearby compass needle — the same needle that points toward geographic north in Earth's field rotates when you switch on the current. The needle responds to $\mathbf{B}$; the current creates it.

The Lorentz force law tells you what a magnetic field does to a charge $q$ moving with velocity $\mathbf{v}$:

$$\mathbf{F}=q\mathbf{v}\times \mathbf{B}.$$

The cross product means the force is perpendicular to both the velocity and the field. A charge moving along $\mathbf{B}$ feels no force; a charge moving across $\mathbf{B}$ curves sideways. Magnetic forces do no work (force perpendicular to displacement), so they change direction but not speed.

The Biot-Savart law tells you the $\mathbf{B}$ field produced by a small piece of current-carrying wire. For a short segment of wire carrying current $I$, the field at a point a distance $r$ away points perpendicular to both the wire and the line from wire to point, and its magnitude drops as $1/r^2$. Adding up the contributions from every segment of wire gives the total field. For a long straight wire, the result is a field that circles the wire, with magnitude ${\mu }_{0}I/(2\pi r)$ where ${\mu }_0$ is the permeability of free space.

Ampere's law is the magnetic analogue of Gauss's law. Where Gauss's law relates the total electric flux through a closed surface to the enclosed charge, Ampere's law relates the total $\mathbf{B}$ field tangent to a closed loop to the current passing through the loop. The line integral of $\mathbf{B}$ around any closed loop equals ${\mu }_0$ times the total current threading that loop. This is most useful when symmetry pins down the direction of $\mathbf{B}$: for the long straight wire, a solenoid, and a toroid, Ampere's law gives the field in a few lines.

Visual [Beginner]

A long straight wire carrying current $I$ out of the page produces circular magnetic field lines centred on the wire, lying in the plane of the page. The field lines get farther apart as you move outward — the field weakens as $1/r$. A compass placed near the wire orients tangent to the nearest circle. Reverse the current direction and every arrow on every circle reverses.

Inside a solenoid (a long cylindrical coil of wire), the field lines run straight and parallel down the bore, tightly packed and uniform — a region of nearly constant $\mathbf{B}$. Outside the solenoid the field lines loop back around, but for a long solenoid the exterior field is weak compared to the interior.

The right-hand rule determines direction: point your right thumb along the current; your fingers curl in the direction of $\mathbf{B}$.

Worked example [Beginner]

Two long parallel wires carry current $I=10$ A in the same direction, separated by $d=0.05$ m. Find the magnetic force per unit length between them.

Step 1. Field of wire 1 at the location of wire 2. Wire 1 creates a field of magnitude $B_1={\mu }_{0}I/(2\pi d)$ at the position of wire 2. Substitute:

$$B_1=\frac{(4\pi \times 10^{-7})(10)}{2\pi (0.05)}=\frac{4\times 10^{-6}}{0.1}=4\times 10^{-5}\text{T}.$$

Step 2. Force on wire 2. A length $\ell$ of wire 2, carrying current $I$ in a field $B_1$, experiences force $F=I\ell B_1$ (the cross-product magnitude, since the field is perpendicular to the wire). Force per unit length:

$$\frac{F}{\ell }=I{B}_1=(10)(4\times 10^{-5})=4\times 10^{-4}\text{N/m}.$$

Step 3. Direction. Two parallel currents in the same direction attract. (If they were opposed, the force would be repulsive.) The result is an attractive force of $4\times 10^{-4}$ N/m — small, but measurable, and the principle behind the operational definition of the ampere.

Check your understanding [Beginner]

Formal definition [Intermediate+]

A steady current density $\mathbf{J}({\mathbf{r}}^{\prime })$ produces a static magnetic field. The fundamental law is the Biot-Savart law:

$$\mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int \frac{\mathbf{J}({\mathbf{r}}^{\prime })\times (\mathbf{r}-{\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}{d}^3{r}^{\prime }.$$

For a thin wire carrying current $I$ along a curve $\mathcal{C}$, this reduces to

$$\mathbf{B}(\mathbf{r})=\frac{{\mu }_{0}I}{4\pi }{\oint }_{\mathcal{C}}\frac{d{\mathbf{\ell }}^{\prime }\times (\mathbf{r}-{\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}.$$

The Biot-Savart law is the magnetic analogue of Coulomb's law for $\mathbf{E}$: both are $1/r^2$ laws, both superpose linearly, and both can be derived from the appropriate Maxwell equation plus a Green's-function inversion. The cross product in the Biot-Savart law reflects the vector character of current (a vector source) compared to the scalar character of charge.

Ampere's law (integral form). For any closed loop $\Gamma$ bounding a surface $S$,

$$\boxed{{\oint }_{\Gamma }\mathbf{B}\cdot d\mathbf{\ell }={\mu }_0{\int }_S\mathbf{J}\cdot d\mathbf{a}={\mu }_0{I}_{\mathrm{enc}}.}$$

The left side is the circulation of $\mathbf{B}$ around $\Gamma$; the right side is ${\mu }_0$ times the total current $I_{\mathrm{enc}}$ threading $S$. The sign convention: the direction of $d\mathbf{\ell }$ and the orientation of $d\mathbf{a}$ are linked by the right-hand rule.

Ampere's law (differential form). Applying Stokes' theorem to the integral form,

$$\boxed{\nabla \times \mathbf{B}={\mu }_0\mathbf{J}.}$$

This is the magnetostatic Maxwell equation. The curl on the left is non-zero only where current is present — currents are the sources of curl $\mathbf{B}$, just as charges are the sources of $\nabla \cdot \mathbf{E}$. (The full time-dependent Maxwell equation adds the displacement current ${\mu }_0{ϵ}_0\partial \mathbf{E}/\partial t$ to the right side; see unit 10.04.01 pending.)

The vector potential. Since $\nabla \cdot \mathbf{B}=0$ (no magnetic monopoles), there exists a vector field $\mathbf{A}$ such that

$$\mathbf{B}=\nabla \times \mathbf{A}.$$

The vector potential $\mathbf{A}$ is the magnetic analogue of the electrostatic scalar potential $V$ ($\mathbf{E}=-\nabla V$). It is not unique: replacing $\mathbf{A}\to \mathbf{A}+\nabla \chi$ for any smooth function $\chi$ leaves $\mathbf{B}$ unchanged. This gauge freedom is fixed by imposing a gauge condition — the standard magnetostatic choice is the Coulomb gauge $\nabla \cdot \mathbf{A}=0$, developed in the Master tier.

In the Coulomb gauge, the vector potential satisfies a Poisson equation ${\nabla }^2\mathbf{A}=-{\mu }_0\mathbf{J}$, solved by

$$\mathbf{A}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int \frac{\mathbf{J}({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{r}^{\prime }.$$

Magnetic force on a current loop. A current-carrying wire in an external field $\mathbf{B}$ experiences a force. For a wire segment carrying current $I$ along $d\mathbf{\ell }$, the force element is $d\mathbf{F}=Id\mathbf{\ell }\times \mathbf{B}$. A closed loop in a uniform field experiences zero net force but a torque

$$\mathbf{\tau }=\mathbf{m}\times \mathbf{B},$$

where $\mathbf{m}=I\mathbf{a}$ is the magnetic dipole moment and $\mathbf{a}$ is the oriented area vector of the loop. This is the magnetic analogue of $\mathbf{\tau }=\mathbf{p}\times \mathbf{E}$ for an electric dipole.

Counterexamples to common slips

• Ampere's law in the form $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$ is valid only in magnetostatics (steady currents, $\partial \mathbf{E}/\partial t=0$). Time-varying fields require the displacement-current correction $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}+{\mu }_0{ϵ}_0\partial \mathbf{E}/\partial t$. Applying the magnetostatic form to a charging capacitor gives a contradiction — the displacement current resolves it.
• The Biot-Savart law gives the field of a steady current. It is not the correct Green's function for time-dependent currents; the retarded-potential formulation of the full Maxwell equations 10.04.01 pending replaces instantaneous propagation with retardation at speed $c$.
• The vector potential $\mathbf{A}$ is gauge-dependent and not directly measurable in classical electrodynamics. Its line integral $\oint \mathbf{A}\cdot d\mathbf{\ell }={\Phi }_B$ (the magnetic flux through the loop) is gauge-invariant. In quantum mechanics $\mathbf{A}$ acquires direct physical significance through the Aharonov-Bohm effect (Master tier).
• A magnetic dipole moment $\mathbf{m}$ is not a pair of magnetic charges (a "north" and "south" pole). All known magnetic dipoles are current loops (or spin, which is quantum-mechanical). No magnetic monopole has been observed; $\nabla \cdot \mathbf{B}=0$ holds in all known situations.

Key theorem with proof [Intermediate+]

Theorem (Ampere's law from the Biot-Savart law). The Biot-Savart law

$$\mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int \frac{\mathbf{J}({\mathbf{r}}^{\prime })\times (\mathbf{r}-{\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3}{d}^3{r}^{\prime }$$

implies $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$ for steady currents satisfying $\nabla \cdot \mathbf{J}=0$.

Proof. Write the Biot-Savart kernel using the identity $(\mathbf{r}-{\mathbf{r}}^{\prime })/|\mathbf{r}-{\mathbf{r}}^{\prime }{|}^3=-\nabla (1/|\mathbf{r}-{\mathbf{r}}^{\prime }|)$. Then

$$\mathbf{B}(\mathbf{r})=-\frac{{\mu }_0}{4\pi }\int \mathbf{J}({\mathbf{r}}^{\prime })\times \nabla \frac{1}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{r}^{\prime }=\frac{{\mu }_0}{4\pi }\nabla \times \int \frac{\mathbf{J}({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{r}^{\prime },$$

where the curl passes through the integral (it acts on $\mathbf{r}$, not ${\mathbf{r}}^{\prime }$) and the sign flip comes from $\mathbf{J}\times (-\nabla f)=\nabla f\times \mathbf{J}=\nabla \times (\mathbf{J}f)$ with $\mathbf{J}({\mathbf{r}}^{\prime })$ treated as constant under $\nabla$. Recognising the integral as the vector potential $\mathbf{A}$ in the Coulomb gauge, $\mathbf{B}=\nabla \times \mathbf{A}$.

Now take the curl of $\mathbf{B}$:

$$\nabla \times \mathbf{B}=\nabla \times (\nabla \times \mathbf{A})=\nabla (\nabla \cdot \mathbf{A})-{\nabla }^2\mathbf{A}.$$

In the Coulomb gauge, $\nabla \cdot \mathbf{A}=0$. The vector Poisson equation ${\nabla }^2\mathbf{A}=-{\mu }_0\mathbf{J}$ follows from the Green's-function solution for $\mathbf{A}$ and the identity ${\nabla }^2(1/|\mathbf{r}-{\mathbf{r}}^{\prime }|)=-4\pi {\delta }^3(\mathbf{r}-{\mathbf{r}}^{\prime })$. Therefore

$$\nabla \times \mathbf{B}=0+{\mu }_0\mathbf{J}={\mu }_0\mathbf{J}.■$$

The steady-current condition $\nabla \cdot \mathbf{J}=0$ enters at the step where the Coulomb gauge is imposed: the vector potential $\mathbf{A}=({\mu }_0/4\pi )\int \mathbf{J}({\mathbf{r}}^{\prime })/|\mathbf{r}-{\mathbf{r}}^{\prime }|d^3{r}^{\prime }$ satisfies $\nabla \cdot \mathbf{A}=0$ if and only if ${\nabla }^{\prime }\cdot \mathbf{J}({\mathbf{r}}^{\prime })=0$ (integrate by parts; the boundary term vanishes for localised currents).

Worked example: field of a solenoid at intermediate level

A solenoid of length $L$ with $n$ turns per unit length carries current $I$. Use Ampere's law to find $\mathbf{B}$ inside and outside.

Choose a rectangular Amperean loop with one long side inside the solenoid (parallel to the axis, length $\ell$) and the opposite long side outside. The two short sides are perpendicular to the axis. By symmetry, $\mathbf{B}$ inside is axial and uniform; outside, the field of an infinite solenoid is zero (field lines that leave one end return through the other, and the exterior region is field-free in the ideal limit).

Ampere's law:

$$\oint \mathbf{B}\cdot d\mathbf{\ell }=B_{\mathrm{in}}\ell ={\mu }_0(n\ell )I.$$

The enclosed current is $n\ell$ turns times $I$. Solving:

$$B_{\mathrm{in}}={\mu }_{0}nI.$$

The field is uniform, axial, and independent of the solenoid's cross-section. Outside, the same Amperean loop with both long sides exterior gives zero enclosed current for a loop that does not thread the solenoid, so $B_{\mathrm{out}}=0$ in the infinite-length limit.

Exercises [Intermediate+]

Biot-Savart as convolution and the Helmholtz decomposition [Master]

The Biot-Savart law is a convolution of the current density with a vector Green's function:

$$\mathbf{B}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int \mathbf{J}({\mathbf{r}}^{\prime })\times \frac{\hat{\mathbf{R}}}{R^2}{d}^3{r}^{\prime }=(\mathbf{G}\ast \mathbf{J})(\mathbf{r}),$$

where $\mathbf{R}=\mathbf{r}-{\mathbf{r}}^{\prime }$, $R=|\mathbf{R}|$, and $\mathbf{G}(\mathbf{R})=({\mu }_0/4\pi )\hat{\mathbf{R}}\times /R^2$ is the magnetic Green's kernel. The cross-product structure distinguishes this from the scalar Coulomb Green's function $1/(4\pi {ϵ}_{0}R)$.

The Helmholtz decomposition states that any sufficiently smooth vector field $\mathbf{F}$ vanishing at infinity can be decomposed as $\mathbf{F}=-\nabla \Phi +\nabla \times \mathbf{A}$ where $\Phi$ is a scalar potential and $\mathbf{A}$ is a vector potential. For the magnetic field: $\nabla \cdot \mathbf{B}=0$ eliminates the scalar part ($\Phi =0$), leaving $\mathbf{B}=\nabla \times \mathbf{A}$ as the unique decomposition. The Biot-Savart law provides the explicit convolution formula for $\mathbf{A}$:

$$\mathbf{A}(\mathbf{r})=\frac{{\mu }_0}{4\pi }\int \frac{\mathbf{J}({\mathbf{r}}^{\prime })}{|\mathbf{r}-{\mathbf{r}}^{\prime }|}{d}^3{r}^{\prime }.$$

The curl of this gives Biot-Savart for $\mathbf{B}$; the divergence of this (using $\nabla \cdot \mathbf{J}=0$) is zero — the Coulomb gauge is automatic.

Gauge freedom and the Coulomb gauge [Master]

The vector potential is defined only up to a gradient: $\mathbf{A}\to \mathbf{A}+\nabla \chi$ leaves $\mathbf{B}=\nabla \times \mathbf{A}$ unchanged for any smooth $\chi$. This is gauge freedom, and it reflects a redundancy in the description — the physical field $\mathbf{B}$ has fewer degrees of freedom than the mathematical object $\mathbf{A}$.

The Coulomb gauge fixes this redundancy by imposing $\nabla \cdot \mathbf{A}=0$. In this gauge the static equations become:

$${\nabla }^2\mathbf{A}=-{\mu }_0\mathbf{J},\nabla \cdot \mathbf{A}=0.$$

The Poisson equation for each component of $\mathbf{A}$ is solved by the integral formula above. The Coulomb gauge condition $\nabla \cdot \mathbf{A}=0$ holds automatically when $\nabla \cdot \mathbf{J}=0$ (steady currents) because the divergence passes through the integral and ${\nabla }^2(1/R)=-4\pi {\delta }^3(\mathbf{R})$ gives $\nabla \cdot \mathbf{A}=({\mu }_0/4\pi )\int \mathbf{J}\cdot \nabla (1/R)d^3{r}^{\prime }=-({\mu }_0/4\pi )\int ({\nabla }^{\prime }\cdot \mathbf{J})/R{d}^3{r}^{\prime }=0$.

Other gauge choices exist. The Lorenz gauge $\nabla \cdot \mathbf{A}+{\mu }_0{ϵ}_0\partial V/\partial t=0$ is the natural choice for time-dependent problems because it decouples the wave equations for $\mathbf{A}$ and $V$. The Coulomb gauge is natural for magnetostatics because it renders $\mathbf{A}$ instantaneous (no retardation) and the Poisson equation is straightforward.

The Aharonov-Bohm effect (preview) [Master]

In classical electrodynamics, $\mathbf{A}$ is a mathematical convenience — only $\mathbf{B}=\nabla \times \mathbf{A}$ is physical. In quantum mechanics this is no longer true. The Aharonov-Bohm effect (1959) demonstrates that a charged particle can be affected by $\mathbf{A}$ even in a region where $\mathbf{B}=0$.

Consider a long solenoid carrying current, producing a confined magnetic field inside and $\mathbf{B}=0$ outside. The vector potential outside is $\mathbf{A}=({\Phi }_B/2\pi s)\hat{\mathbf{\varphi }}$ where ${\Phi }_B$ is the flux through the solenoid. A quantum-mechanical electron beam split into two paths encircling the solenoid acquires a phase difference

$$\Delta \varphi =\frac{e}{\hslash }\oint \mathbf{A}\cdot d\mathbf{\ell }=\frac{e{\Phi }_B}{\hslash },$$

even though $\mathbf{B}=0$ along both paths. The interference pattern shifts by an amount depending on the enclosed flux — a direct measurement of the line integral of $\mathbf{A}$, which equals the flux of $\mathbf{B}$ through the enclosed area by Stokes' theorem. This confirms that $\mathbf{A}$ (or rather, the holonomy of the connection $\mathbf{A}$) has physical significance beyond $\mathbf{B}$.

The Aharonov-Bohm effect is the entry point to the gauge-theory viewpoint on electromagnetism: $\mathbf{A}$ is a connection on a $U(1)$ principal bundle over space, $\mathbf{B}$ is its curvature, and gauge transformations are bundle automorphisms. The holonomy (Wilson loop) $\oint \mathbf{A}\cdot d\mathbf{\ell }$ is the gauge-invariant observable associated with a closed curve.

Multipole expansion for magnetic fields [Master]

For a localised current distribution, the vector potential admits a multipole expansion analogous to the electrostatic case. In the Coulomb gauge, the leading term at large distance is

$$\mathbf{A}(\mathbf{r})\approx \frac{{\mu }_0}{4\pi }\frac{\mathbf{m}\times \hat{\mathbf{r}}}{r^2},$$

where the magnetic dipole moment is

$$\mathbf{m}=\frac{1}{2}\int {\mathbf{r}}^{\prime }\times \mathbf{J}({\mathbf{r}}^{\prime })d^3{r}^{\prime }.$$

There is no magnetic monopole term ($\nabla \cdot \mathbf{B}=0$ forbids it). The dipole field is

$${\mathbf{B}}_{\mathrm{dip}}(\mathbf{r})=\frac{{\mu }_0}{4\pi r^3}[3(\mathbf{m}\cdot \hat{\mathbf{r}})\hat{\mathbf{r}}-\mathbf{m}].$$

This has the same angular structure as the electric dipole field ${\mathbf{E}}_{\mathrm{dip}}$ with $\mathbf{p}\to \mathbf{m}$ and $1/{ϵ}_0\to {\mu }_0$. Higher-order terms (magnetic quadrupole, octupole) fall as $1/r^3$, $1/r^4$, etc. For most purposes the dipole approximation suffices at distances large compared to the source size.

The absence of a magnetic monopole term in the multipole expansion is a direct consequence of $\nabla \cdot \mathbf{B}=0$. If magnetic monopoles existed (Dirac monopoles, charge $g$), the leading term would be $\mathbf{B}=({\mu }_0/4\pi )(g\hat{\mathbf{r}}/r^2)$ and the quantisation condition $ge=n\hslash /2$ would follow from requiring single-valuedness of the wave function.

Historical notes [Master]

The magnetic effect of electric currents was discovered by Hans Christian Oersted in 1820, who observed that a compass needle deflects near a current-carrying wire. This was the first experimental link between electricity and magnetism.

Jean-Baptiste Biot and Felix Savart (1820) measured the $1/r$ dependence of the field around a long wire and formulated the $1/r^2$ law for the contribution of a current element. Andre-Marie Ampere (1820-1826) established the force law between current-carrying wires, developed the mathematical theory of the interaction of current elements, and formulated the circuital law bearing his name. Ampere's experimental and theoretical work was so thorough that Maxwell called him the "Newton of electricity."

The vector potential was introduced by Franz Ernst Neumann (1845) and developed by William Thomson (Lord Kelvin) and James Clerk Maxwell. The gauge-theory interpretation dates to Weyl (1918, scale invariance) and London (1927, phase invariance), with the modern $U(1)$ bundle formulation due to Wu and Yang (1975).

Connections [Master]

To differential geometry. The vector potential $\mathbf{A}$ is a connection 1-form on a $U(1)$ principal bundle over ${\mathbb{R}}^3$. The magnetic field $\mathbf{B}$ is the curvature 2-form $F=dA$. Gauge transformations are bundle automorphisms. The Aharonov-Bohm phase is the holonomy of the connection around a closed curve. This is the simplest example of a Yang-Mills theory and the template for the gauge-theory description of the other fundamental forces.

To Faraday's law 10.03.01 pending. The magnetostatic Maxwell equation $\nabla \times \mathbf{B}={\mu }_0\mathbf{J}$ becomes one of the four time-dependent Maxwell equations when the displacement current is added (unit 10.04.01 pending). The vector potential $\mathbf{A}$ will reappear as the dynamical variable in the Lagrangian and Hamiltonian formulations of electrodynamics, and its time derivative is related to the electric field by $\mathbf{E}=-\partial \mathbf{A}/\partial t-\nabla V$.

To quantum mechanics. The minimal-coupling Hamiltonian $H=|\mathbf{p}-e\mathbf{A}{|}^2/(2m)+eV$ is the starting point for the quantum theory of charged particles in electromagnetic fields. Landau levels, the quantum Hall effect, and flux quantisation in superconductors all derive from the interplay between $\mathbf{A}$ and the wave function's phase.

To biophysics 17.09.02 pending. Ionic current loops in neurons produce magnetic fields detectable by magnetoencephalography (MEG). The Biot-Savart law gives the field from a known current distribution; inverse problems (inferring current from measured $\mathbf{B}$) are the core computational challenge of MEG source localisation.

Bibliography [Master]

• Ampere, A.-M. (1826). Theorie des phenomenes electro-dynamiques, uniquement deduite de l'experience. Megnetiques et Electrodynamiques.
• Biot, J.-B. and Savart, F. (1820). "Note sur la magnetisme de la pile de Volta." Annales de chimie et de physique, 15, 222–223.
• Griffiths, D. J. (2017). Introduction to Electrodynamics, 4th ed. Cambridge University Press. Ch. 5.
• Jackson, J. D. (1999). Classical Electrodynamics, 3rd ed. Wiley. Ch. 5.
• Oersted, H. C. (1820). "Experiments on the effect of a current of electricity on the magnetic needle." Annals of Philosophy, 16, 273–276.
• Tong, D. "Electromagnetism." Lecture notes, Cambridge. §2.
• Zangwill, A. (2013). Modern Electrodynamics. Cambridge University Press. Ch. 9.
• Aharonov, Y. and Bohm, D. (1959). "Significance of electromagnetic potentials in the quantum theory." Physical Review, 115(3), 485–491.
• Wu, T. T. and Yang, C. N. (1975). "Concept of nonintegrable phase factors and global formulation of gauge fields." Physical Review D, 12(12), 3845–3857.