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. The direction of integration along C matches i in the sense of a right-handed screw. Drive such a screw in the direction of positive i (which is from + to − voltage) then the direction of rotation of the screw is the direction of integration along C. Equivalently: lay your right hand around C and integrate from wrist to fingers, then i runs in the direction of your stretched thumb.

Relation with Maxwell's equation

Ampère's law follows from the special case of one of Maxwell's equations for zero displacement currents:

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

where $\scriptstyle {\boldsymbol {\nabla }}\times \mathbf {B}$ is the curl of the vector field B. Further $\scriptstyle k=\mu _{0}$ 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 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.

See also

• Biot-Savart's law
• Ampere's equation