In physics, Ampère's law, discovered by André-Marie Ampère, relates the circulating magnetic field in a closed loop to the electric current passing through the loop. It is the magnetic equivalent of Faraday's law of induction,

Original Ampère's law

---

In its original form, Ampère's law relates the magnetic field B to its source, the current density J:

$\backslash oint\_C\; \backslash frac\{\backslash mathbf\{B\}\}\{\backslash mu\_0\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; \backslash int\backslash !\backslash !\backslash !\backslash !\backslash int\_S\; \backslash mathbf\{J\}\; \backslash cdot\; d\; \backslash mathbf\{A\}\; =\; I\_\{\backslash mathrm\{enc\}\}$

where

$\backslash oint\_C$ is the closed line integral around contour (closed curve) $C$.

$\backslash mathbf\{B\}$ is the magnetic field in Tesla,

$d\backslash mathbf\{l\}$ is an infinitesimal element (differential) of the contour $C$,

$\backslash mathbf\{J\}$ is the current density (in amperes per square meter) through the surface S enclosed by contour C

$d\; \backslash mathbf\{A\}\; \backslash !\backslash$ is a differential vector element of surface area A, with infinitesimally small magnitude and direction normal to surface S

$I\_\{\backslash mathrm\{enc\}\}\; \backslash !\backslash$ is the current enclosed by the curve $C$, or strictly, the current that penetrates surface $S$,

$\backslash mu\_0\; =\; 4\; \backslash pi\; \backslash times\; 10^\{-7\}$ is the permeability of free space (in henries per meter),

Equivalently, the original equation in differential form is

$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}$

The magnetic field H in linear media, is related to the magnetic flux density B (in teslas) by

$\backslash mathbf\{B\}\; \backslash \; =\; \backslash \; \backslash mu\; \backslash mathbf\{H\}$

Corrected Ampère's law: the Ampère-Maxwell equation

---

James Clerk Maxwell noticed a logical inconsistency when applying Ampère's law to a charging or discharging capacitor. If surface $S$ passes between the plates of the capacitor, and not through any wires, then $\backslash mathbf\{J\}\; =\; 0$ even though $\backslash oint\_C\; \backslash mathbf\{H\}\; \backslash cdot\; d\backslash mathbf\{l\}\backslash ne\; 0$. He concluded that this law had to be incomplete. To resolve the problem, he came up with the concept of displacement current and made a generalized version of Ampère's law which was incorporated into Maxwell's equations.

The generalized law, as corrected by Maxwell, takes the following integral form:

$\backslash oint\_C\; \backslash mathbf\{H\}\; \backslash cdot\; d\backslash mathbf\{l\}\; =\; \backslash iint\_S\; \backslash mathbf\{J\}\; \backslash cdot\; d\; \backslash mathbf\{A\}\; +$

{d \over dt} \iint_S \mathbf{D} \cdot d \mathbf{A}

where in linear media

$\backslash mathbf\{D\}\; \backslash \; =\; \backslash \; \backslash varepsilon\; \backslash mathbf\{E\}$

is the displacement current density (in amperes per square meter).

This Ampère-Maxwell law can also be stated in differential form:

$\backslash nabla\; \backslash times\; \backslash mathbf\{H\}\; =\; \backslash mathbf\{J\}\; +\; \backslash frac\{\backslash partial\; \backslash mathbf\{D\}\}\{\backslash partial\; t\}$

where the second term arises from the displacement current.

With the addition of the displacement current, Maxwell was able to postulate (correctly) that light was a form of electromagnetic wave. See Electromagnetic wave equation for a discussion on this important discovery.

See also

---

• Maxwell's equations
• Biot-Savart law
• Faraday's law of induction
• Gauss's law
• Electric current
• Vector calculus
• Stokes' theorem

References

---

External links

---

• MISN-0-138 Ampere's Law (PDF file) by Kirby Morgan for Project PHYSNET.
• MISN-0-145 The Ampere-Maxwell Equation; Displacement Current (PDF file) by J.S. Kovacs for Project PHYSNET.
• The Ampère's Law Song (PDF file) by Walter Fox Smith.

Electrostatics | Eponymous laws | Introductory physics

Ampèresches Gesetz | Ley de Ampère | Théorème d'Ampère | 앙페르의 법칙 | Legge di Ampere | חוק אמפר | Wet van Ampère | アンペールの法則 | Prawo Ampère'a | Lei de Ampère | Ampèren laki | Ampères lag | Định luật Ampere | ஆம்ப்பியர் விதி