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The term "Maxwell's equations" is often also used for equivalent alternative formulations. Versions of Maxwell's equations based on the electric and magnetic scalar potentials are preferred for explicitly solving the equations as a boundary value problem, analytical mechanics, or for use in quantum mechanics.
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 commonly used set of units is the called the SI, which defines two constants, the vacuum permittivity (ε 0) and the vacuum permeability (μ 0). These can be used to convert SI units to their corresponding Heaviside–Lorentz values, as detailed below. For example, SI charge is √ ε 0 L 3 M / T 2.
All but the last term of can be written as the tensor divergence of the Maxwell stress tensor, giving: = +, As in the Poynting's theorem, the second term on the right side of the above equation can be interpreted as the time derivative of the EM field's momentum density, while the first term is the time derivative of the momentum density for ...
That discussion is followed with several particular examples. A formulation of Maxwell's equations based upon division of charges and currents into "free" and "bound" charges and currents leads to introduction of the D- and P-fields: = +, where P is called the polarization density.
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations and the Lorentz force) in a form that is manifestly invariant under Lorentz transformations, in the formalism of special relativity using rectilinear inertial coordinate systems.
SI units for Maxwell's equations and the particle physicist's sign convention for the signature of Minkowski space (+ − − −), will be used throughout this article. Relationship with the classical fields
In fact, Maxwell's equations were crucial in the historical development of special relativity. However, in the usual formulation of Maxwell's equations, their consistency with special relativity is not obvious; it can only be proven by a laborious calculation. For example, consider a conductor moving in the field of a magnet. [8]