We formulate an existence theorem that states that, given localized scalar and vector time-dependent sources satisfying the continuity equation, there exist two retarded fields that satisfy a set of four field equations. If the theorem is applied to the usual electromagnetic charge and current densities, the retarded fields are identified with the electric and magnetic fields and the associated field equations with Maxwell’s equations. This application of the theorem suggests that charge conservation can be considered to be the fundamental assumption underlying Maxwell’s equations.

## REFERENCES

The evaluation at the retarded time must be interpreted unambiguously. The quantity $[\u2207\u2032\u2219J]$ means that we first calculate the divergence of the current density at the present time $\u2207\u2032\u2219J(x\u2032,t)$ and in the resulting expression $f1(x\u2032,t)$ we replace $t$ by the retarded time $t\u2032=t\u2212R\u2215c$, that is, $f(x\u2032,t\u2032)$. Analogously, the quantity $[\u2202\rho \u2215\u2202t]$ means that we first calculate the time derivative of the charge density at the present time $\u2202\rho (x\u2032,t)\u2215\u2202t$ and in the resulting expression $f2(x\u2032,t)$ we replace $t$ by the retarded time $t\u2032=t\u2212R\u2215c$, that is, $f2(x\u2032,t\u2032)$. We also note that $[\u2202\rho \u2215\u2202t]=[\u2202\rho \u2215\u2202t\u2032]$, which follows from the result $\u2202\rho (x\u2032,t\u2032)\u2215\u2202t=(\u2202\rho (x\u2032,t\u2032)\u2215\u2202t\u2032)(\u2202t\u2032\u2215\u2202t)=\u2202\rho (x\u2032,t\u2032)\u2215\u2202t\u2032$.

This nice interpretation of Eq. (5) was suggested by an anonymous referee.

We could also consider sources that vanish sufficiently at infinity, that is, sources of order $O\u2223x\u2223\u22122\u2212\delta $, where $\delta >0$ as $\u2223x\u2223$ goes to infinity, $x$ being the field point.

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