Deriving Magnostatics equations from steady currents

[FG_313]

I've always heard that maxwell's equations contains essentially all of eletromagnetic theory. However, there's one thing I'm having trouble doing for myself: deriving the magnestatics equations from the maxwell's equations. Of course: it's clear that if you put ∂[t]E=∂[t]B=0 (partial derivative in relation to time), you get the magnostatic equations. What I'm wondering is: can we get that ∂[t]E=∂[t]B=0 by assuming steady currents (∂[t]ρ=0) and using the maxwell's equations? The motivation for this question is that the condition for being in magnostatics is steady currents, so the Maxwell's equations should reflect:
(Steady currents)⇒(Magnostatics equations). I believe I've proved the other direction of the implication, but not the one above, that is the most interesting for me: it would mean that, given the maxwell's equation, we can never have non-static situations coming from a "static" current distribution. Please contribute.

 

[vanhees71]

Well, you can solve the Maxwell equations for given sources $(\rho,\vec{j})$, in terms of the socalled Jefimenko equations (althought I never understood why these equations are named after Jefimenko, because they are known much longer in terms of the retarded potentials, to my knowledge first written down by Ludvig Lorenz in the 1860ies, but that's a minor historical detail)

https://en.wikipedia.org/wiki/Jefimenko's_equations

Now consider stationary sources $\partial_t \rho=0$, $\partial_t \vec{j}=0$ and see what you get!

 

[Dale]

Supporting the comments by @vanhees71 the retarded potentials are particularly easy to evaluate for static sources since the sources are the same at every retarded time, so you don't actually need to calculate the correct retarded time.

 

[ORF]

Hello

The formula is easy to derive, but difficult to apply to a real case...
https://en.wikipedia.org/wiki/Magne...cs_as_a_special_case_of_Maxwell.27s_equations

Greetings.