# Physics equations/Ampere law

### Ampère's law[edit | edit source]

from https://en.wikipedia.org/w/index.php?title=Amp%C3%A8re%27s_circuital_law&oldid=578507291

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 *I*_{enc} passing through a surface *S* (enclosed by *C*):

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

- is the closed line integral around the closed curve
*C*, - denotes a 2d surface integral over
*S*enclosed by*C* - • is the vector dot product,
- d
**ℓ**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*) - d
**S**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*.