Mathematical Descriptions of The Electromagnetic Field - Potential Field Approach - Gauge Freedom - Coulomb Gauge

Coulomb Gauge

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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