Ampère's original circuital law

Ampère's original circuital law

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, Ampère'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).


Ampère'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).

An electric current produces a magnetic field.

Integral form

In SI units (cgs units are later), 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):

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

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 If, enc through a surface S:

where Jf is the free current density only. Furthermore 

• is the closed line integral around the closed curve C,
• S denotes a 2d surface integral over S enclosed by C 
• •is the vector dot product,
• dℓ is an infinitesimal element (a differential[disambiguation needed]) 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.

The B and H fields are related by the constitutive equation

where μ0 is the magnetic constant.

There are a number of ambiguities in the above definitions that require clarification and a choice of convention.

First, three of these terms are associated with sign ambiguities: the line integral 

 could go around the loop in either direction (clockwise or counterclockwise); the vector area dS could point in either of the two directions normal to the surface; and Ienc is the net current passing through the surface S, meaning the current passing through in one direction, minus the current in the other direction—but either direction could be chosen as positive. These ambiguities are resolved by the right-hand rule: With the palm of the right-hand toward the area of integration, and the index-finger pointing along the direction of line-integration, the outstretched thumb points in the direction that must be chosen for the vector area dS. Also the current passing in the same direction as dS must be counted as positive. The right hand grip rule can also be used to determine the signs.

Second, there are infinitely many possible surfaces S that have the curve C as their border. (Imagine a soap film on a wire loop, which can be deformed by moving the wire). Which of those surfaces is to be chosen? If the loop does not lie in a single plane, for example, there is no one obvious choice. The answer is that it does not matter; it can be proven that any surface with boundary C can be chosen.

Differential form

By the Stokes' theorem, this equation can also be written in a "differential form". Again, this equation only applies in the case where the electric field is constant in time, meaning the currents are steady (time-independent, else the magnetic field would change with time); see below for the more general form. In SI units, the equation states for total current:

and for free current

where ∇× is the curl operator.