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One difference between the Gaussian and SI systems is in the factor 4π in various formulas that relate the quantities that they define. With SI electromagnetic units, called rationalized, [3] [4] Maxwell's equations have no explicit factors of 4π in the formulae, whereas the inverse-square force laws – Coulomb's law and the Biot–Savart law – do have a factor of 4π attached to the r 2.
The maxwell is a non-SI unit. [8] 1 maxwell = 1 gauss × 2. That is, one maxwell is the total flux across a surface of one square centimetre perpendicular to a magnetic field of strength one gauss. The weber is the related SI unit of magnetic flux, which was defined in 1946. [9] 1 maxwell ≘ 10 −4 tesla × (10 −2 metre) 2 = 10 −8 weber
In partial differential equation form and a coherent system of units, Maxwell's microscopic equations can ... colloquially "in Gaussian units", [10] the Maxwell ...
In particular, in abstract index notation, the set of Maxwell's equations (in the Lorenz gauge) may be written (in Gaussian units) as follows: = = where is the d'Alembertian and is the four-current. The first equation is the Lorenz gauge condition while the second contains Maxwell's equations.
In three dimensions, the derivative has a special structure allowing the introduction of a cross product: = + = + from which it is easily seen that Gauss's law is the scalar part, the Ampère–Maxwell law is the vector part, Faraday's law is the pseudovector part, and Gauss's law for magnetism is the pseudoscalar part of the equation.
The Heaviside–Lorentz unit system, like the International System of Quantities upon which the SI system is based, but unlike the CGS-Gaussian system, is rationalized, with the result that there are no factors of 4π appearing explicitly in Maxwell's equations. [2]
The gauss is the unit of magnetic flux density B in the system of Gaussian units and is equal to Mx/cm 2 or g/Bi/s 2, while the oersted is the unit of H-field. One tesla (T) corresponds to 10 4 gauss, and one ampere (A) per metre corresponds to 4π × 10 −3 oersted .
This tensor simplifies and reduces Maxwell's equations as four vector calculus equations into two tensor field equations. In electrostatics and electrodynamics, Gauss's law and Ampère's circuital law are respectively: