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It relates magnetic fields to electric currents that produce them. Using Ampere's law, one can determine the magnetic field associated with a given current or current associated with a given magnetic field, providing there is no time changing electric field present.In its historically original form, Ampere's Circuital Law relates the magnetic field to its electric current source. The law can be written in two forms, the "integral form" and the "differential form". The forms are equivalent, and related by the Kelvin–Stokes theorem. It can also be written in terms of either the B or H magnetic fields. Again, the two forms are equivalent (see the "proof" section below).

Ampere's circuital law is now known to be a correct law of physics in a magnetostatic situation: The system is static except possibly for continuous steady currents within closed loops. In all other cases the law is incorrect unless Maxwell's correction is included (see below).

In SI units (cgs units are later), the "integral form" of the original Ampere'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 through which the electric current passes through (again arbitrary but not closed - since no 3d 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 [2] [3] .

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 ):

Alternatively in terms of free current, the line integral of the magnetic H-field (in ampere per metre. Am –1 ) around closed curve C equals the free current I _{f, enc} through a surface S :