Ampere's law

In physics, or more in particular in electrodynamics, Ampère's law relates the strength of a magnetic field to the electric current that causes it. The law was first formulated by André-Marie Ampère around 1825. Later (ca 1865) it was augmented by James Clerk Maxwell, who added displacement current to it. This extended form is one of the four Maxwell's laws that form the axiomatic basis of electrodynamics.

Formulation

We consider a closed curve C around an electric current i. Then Ampère's law reads

$\oint _{C}\mathbf {B} \cdot d\mathbf {s} =ki,$

where $\scriptstyle k=\mu _{0}$ for SI units and $\scriptstyle k=4\pi /c$ for Gaussian units. Here μ₀ is the magnetic constant (also known as vacuum permeability), and c is the speed of light. The vector field B is known as the magnetic induction.

Relation with Biot-Savart's law

Ampère's law can be seen as one of the postulates of the theory of electromagnetism and as such cannot be derived from earlier results. However, alternatively one may take the generalized form of the Biot-Savart law as a starting point of the theory

$\mathbf {B} (\mathbf {r} )={\frac {k}{4\pi }}{\boldsymbol {\nabla }}\times \iiint _{V}{\frac {\mathbf {J} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r'} |}}d\mathbf {r} ',$

where J(r') is the current density at position r' (i is the surface integral of J). It is shown in this article that from the generalized Biot-Savart law follows that

${\boldsymbol {\nabla }}\times \mathbf {B} (\mathbf {r} )=k\mathbf {J} (\mathbf {r} ),$

where $\scriptstyle k=\mu _{0}$ for SI units and $\scriptstyle k=4\pi /c$ for Gaussian units.

Integrating both sides over a surface S and noting that the infinitesimal element dS is a vector with length the surface of the element and direction the normal to the element, gives

$\iint _{S}{\Big (}{\boldsymbol {\nabla }}\times \mathbf {B} (\mathbf {r} ){\Big )}\cdot d\mathbf {S} =k\iint _{S}\mathbf {J} (\mathbf {r} )\cdot d\mathbf {S} =ki.$

Applying the Stokes theorem that reads for any vector field A,

$\iint _{S}{\Big (}{\boldsymbol {\nabla }}\times \mathbf {A} (\mathbf {r} ){\Big )}\cdot d\mathbf {S} =\oint _{C}\mathbf {A} \cdot d\mathbf {s} ,$

where S is the surface bordered by the closed curve C, we find

$\oint _{C}\mathbf {B} \cdot d\mathbf {s} =ki,$

which is indeed Ampère's law.