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The Lorenz gauge hence contradicted Maxwell's original derivation of the EM wave equation by introducing a retardation effect to the Coulomb force and bringing it inside the EM wave equation alongside the time varying electric field, which was introduced in Lorenz's paper "On the identity of the vibrations of light with electrical currents".
The Coulomb gauge (also known as the transverse gauge) is used in quantum chemistry and condensed matter physics and is defined by the gauge condition (more precisely, gauge fixing condition) (,) =. It is particularly useful for "semi-classical" calculations in quantum mechanics, in which the vector potential is quantized but the Coulomb ...
Position vectors r and r′ used in the calculation. The starting point is Maxwell's equations in the potential formulation using the Lorenz gauge: =, = where φ(r, t) is the electric potential and A(r, t) is the magnetic vector potential, for an arbitrary source of charge density ρ(r, t) and current density J(r, t), and is the D'Alembert operator. [2]
There is gauge freedom in A in that of the three forms in this decomposition, only the coexact form has any effect on the electromagnetic tensor F = d A {\displaystyle F=dA} . Exact forms are closed, as are harmonic forms over an appropriate domain, so d d α = 0 {\displaystyle dd\alpha =0} and d γ = 0 {\displaystyle d\gamma =0} , always.
The gauge-fixed potentials still have a gauge freedom under all gauge transformations that leave the gauge fixing equations invariant. Inspection of the potential equations suggests two natural choices. In the Coulomb gauge, we impose ∇ ⋅ A = 0, which is mostly used in the case of magneto statics when we can neglect the c −2 ∂ 2 A/∂t ...
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.
By combining the Lorentz force law above with the definition of electric current, the following equation results, in the case of a straight stationary wire in a homogeneous field: [30] =, where ℓ is a vector whose magnitude is the length of the wire, and whose direction is along the wire, aligned with the direction of the conventional current I.
The Lorentz reciprocity theorem is simply a reflection of the fact that the linear operator ^ relating and at a fixed frequency (in linear media): = ^ where ^ [()] is usually a symmetric operator under the "inner product" (,) = for vector fields and . [8] (Technically, this unconjugated form is not a true inner product because it is not ...