Maxwell's original circuital law
The original form of Maxwell's circuital law, which he derived in his 1855 paper "On Faraday's Lines of Force" based on an analogy to hydrodynamics, relates magnetic fields to electric currents that produce them. It determines the magnetic field associated with a given current, or the current associated with a given magnetic field.
The original circuital law is only a correct law of physics in a magnetostatic situation, where the system is static except possibly for continuous steady currents within closed loops. For systems with electric fields that change over time, the original law (as given in this section) must be modified to include a term known as Maxwell's correction (see below).
The original circuital law can be written in several different forms, which are all ultimately equivalent:
- An "integral form" and a "differential form". The forms are exactly equivalent, and related by the Kelvin–Stokes theorem.(see the "proof" section below)
- Forms using SI units, and those using cgs units. Other units are possible, but rare. This section will use SI units, with cgs units discussed later.
- Forms using either H magnetic fields. These two forms use the total current density and free current density, respectively. The B and H fields are related by the constitutive equation: B = μ0H where μ0 is the magnetic constant.
The integral form of the original 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).
In terms of total current, (which is the sum of both free current and bound current) the line integral of the B-field (in teslas, T) around closed curve C is proportional to the total current Ienc passing through a surface S (enclosed by C). In terms of free current, the line integral of the H-field (in amperes per metre, A·m−1) around closed curve C equals the free current If,enc through a surface S.
Forms of the original circuital law written in SI units
|Using B-field and total current
|Using H-field and free current
- J is the total current density (in amperes per square metre, A·m−2),
- Jf is the free current density only,
- ∮C is the closed line integral around the closed curve C,
- ∬S denotes a 2-D surface integral over S enclosed by C,
- · is the vector dot product,
- dl 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.
- ∇ × is the curl operator.
Ambiguities and sign conventions
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 ∮C 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.