The Coulomb gauge is chosen in such a way that, which corresponds to the case of magnetostatics. In terms of λ, this means that it must satisfy the equation
This choice of function results in the following formulation of Maxwell's equations:
Several features about Maxwell's equations in the Coulomb gauge are as follows. Firstly, solving for the electric potential is very easy, as the equation is a version of Poisson's equation. Secondly, solving for the magnetic vector potential is particularly difficult. This is the big disadvantage of this gauge. The third thing to note, and something which is not immediately obvious, is that the electric potential changes instantly everywhere in response to a change in conditions in one locality.
For instance, if a charge is moved in New York at 1 pm local time, then a hypothetical observer in Australia who could measure the electric potential directly would measure a change in the potential at 1 pm New York time. This seemingly goes against the prohibition in special relativity of sending information, signals, or anything faster than the speed of light. The solution to this apparent problem lies in the fact that, as previously stated, no observers measure the potentials, they measure the electric and magnetic fields. So, the combination of ∇φ and ∂A/∂t used in determining the electric field restores the speed limit imposed by special relativity for the electric field, making all observable quantities consistent with relativity.
Read more about this topic: Mathematical Descriptions Of The Electromagnetic Field, Potential Field Approach, Gauge Freedom
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... The Coulomb gauge (also known as the transverse gauge) is much used in quantum chemistry and condensed matter physics and is defined by the gauge condition (more precisely ... The Coulomb gauge has a number of properties The potentials can be expressed in terms of instantaneous values of the fields and densities (in SI units) where ρ(r, t) is the electric charge density, R =
... In the coulomb gauge, Maxwell's equations are although the solutions contrast the above, since A is a retarded potential yet φ changes instantly, given by This is presents an advantage and a disadvantage of the ... they can be expressed neatly in terms of fields Thus the potentials in the Coulomb gauge are unique and depend on the instantaneous values of fields in the ...