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Why can't we use retarded times to make an expression for retarded fields instead of potentials? As far as I know it doesn't work, since the solutions produced ("retarded fields") don't satisfy Maxwell's equations, but I would like a more physical explanation, if that is possible. As a reference, one can look up Griffiths "Introduction to electrodynamics" p.423. What is it that makes potentials "special", since we know that the wave equations that potentials have to obey are derived from Maxwell's equations (that fields have to obey).

Thanks!

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The potentials have simple expression (10.19 in Griffiths) only because particular gauge was chosen, where the equations for potentials are simple inhomogeneous wave equations, and also because this is one of two simplest particular solutions of inhomogeneous wave equation in 3 spatial dimensions. In other words, the mathematically simplest case is considered.

In other gauges, the equations are not that pretty and the solutions are more complicated. Also different boundary conditions, in general, will cause the solution to be more complicated.

These retarded formulae cannot be derived for electric or magnetic field, because these fields obey different equations, where instead of $\rho,\mathbf j$, derivatives of these with respect to position and time appear. Since values of electric and magnetic fields have measurable consequences on the behaviour of the system, there is no gauge freedom and no way to alter the equations into different, more convenient form like it was done for potentials.

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