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Why does simply substituting retarded time in the Biot-Savart Law and Coulomb's law not give correct relations for electric and magnetic field? How and why are fields different from potentials where substituting time with retarded time give correct results for the scalar and vector potentials?

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  • $\begingroup$ my handwaving answer is that potentials are more fundamental than fields since they show up directly in the Hamiltonian, they have fewer independent elements (3+1 vs. 3+3) so they are the more natural to "retard".. $\endgroup$ – hyportnex Feb 5 at 14:22
  • $\begingroup$ @hyportnex: Have you heard of the Coulomb gauge where the scalar potential is not retarded? $\endgroup$ – Vladimir Kalitvianski Feb 5 at 14:49
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Scalar and vector potentials are ambiguous, so what you think is "correct" may be changed by gauge terms with a quite weird time-space dependence.

The electromagnetic fields $\vec{E}$ and $\vec{B}$, on the other hand, are unambiguous and coupled to each other (remember, for example, the displacement current $\partial\vec{E}/\partial t$, which is absent in a static solution), so they should be derived from the exact time-dependent equations rather than by inserting the retarded time in some static solutions.

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  • $\begingroup$ Are implying that in the Coulomb gauge you only need a scalar potential to describe E and B? But if not then you are is just misleading the OP with your comment. $\endgroup$ – hyportnex Feb 5 at 15:04
  • $\begingroup$ No, on the contrary, it is your handwaving answer, which is misleading. $\endgroup$ – Vladimir Kalitvianski Feb 5 at 15:15
  • $\begingroup$ maybe he will tell us if my comment has mislead him or not, if yes I will delete it. $\endgroup$ – hyportnex Feb 5 at 15:20

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