Physics equations/Ampere law

Ampère's law

The "integral form" of the original Ampère's circuital law is a line integral of the magnetic field around some closed curve C (arbitrary but must be closed). The curve C in turn bounds both a surface S which the electric current passes through (again arbitrary but not closed—since no three-dimensional volume is enclosed by S), and encloses the current. The mathematical statement of the law is a relation between the total amount of magnetic field around some path (line integral) due to the current which passes through that enclosed path (surface integral). It can be written in a number of forms.

In terms of total current, which includes both free and bound current, the line integral of the magnetic B-field (in tesla, T) around closed curve C is proportional to the total current Ienc passing through a surface S (enclosed by C):

${\displaystyle \oint _{C}\mathbf {B} \cdot \mathrm {d} {\boldsymbol {\ell }}=\mu _{0}\iint _{S}\mathbf {J} \cdot \mathrm {d} \mathbf {S} =\mu _{0}I_{\mathrm {enc} }}$

where J is the total current density (in ampere per square metre, Am−2). Also,

• ${\displaystyle \scriptstyle \oint _{C}}$ is the closed line integral around the closed curve C,
• ${\displaystyle \scriptstyle \iint _{S}}$ denotes a 2d surface integral over S enclosed by C
• • is the vector dot product,
• d is an infinitesimal element (a differential) of the curve C (i.e. a vector with magnitude equal to the length of the infinitesimal line element, and direction given by the tangent to the curve C)
• dS is the vector area of an infinitesimal element of surface S (that is, a vector with magnitude equal to the area of the infinitesimal surface element, and direction normal to surface S. The direction of the normal must correspond with the orientation of C by the right hand rule), see below for further explanation of the curve C and surface S.