What happens to the wave impedance of free space at ω=0?

[Shinji83]

For a plane wave traveling in free space we know from Maxwell's equations that:

Z= E/H=√μ₀/ε₀ = 377Ω

The meaning of the wave impedance is that if we have an electric field oscillating with amplitude E₀ in a medium (in this case the vacuum) a magnetic field will be induced with amplitude E₀/Z.
Now this impedance is frequency independent so my doubt is what happens when ω=0?
In a static situation I can create an electrostatic field without having a magnetic field at all.
But the intrinsic impedance tells me that a static(?) magnetic field will be induced anyway.
How's that possible?

Also in metals the wave impedance becomes:
√jωμ/σ+jωε

So for ω=0 the impedance goes to zero as well, that means that if a static electric field existed in the metal there should be an infinite magnetic field. That can't happen and infact no static electric field can exist in a metal in a electrostatic situation (assuming that the metal is immersed in a static electric field).
But if we apply a constant voltage across the metal we can force a static electric field inside the metal. There will be a finite current if σ is finite with an associated finite static magnetic field inside the conductor.
But the impedance formula tells me that even if σ isn't infinite (ideal metal), wave impedance should be zero anyway in DC but that isn't possible as it would mean an infinite magnetic field.
So why doesn't the formula apply anymore? Maybe because to the applied voltage corresponds a superficial net density of charges on the metal that we must take care of in the Maxwell equation so that expression for the impedance is not correct anymore?

 

[mfb]

But the intrinsic impedance tells me that a static(?) magnetic field will be induced anyway.

It does not tell you that. The equation is true for planar electromagnetic waves only. It is not true for static fields, standing waves, constant current in metals and various other setups. Why should it?

 

[Shinji83]

It does not tell you that. The equation is true for planar electromagnetic waves only. It is not true for static fields, standing waves, constant current in metals and various other setups. Why should it?

Thank you for your reply first of all.
My doubt comes from the fact that when the impedance formula is obtained from Maxwell's equations in the phasor/fourier domain for a plane wave solution, ω=0 is a possible frequency.
A sinusoid with zero frequency exists, it's a constant value. A plane wave with zero frequency can still be interpreted as a constant signal/perturbation which propagates with velocity c in vacuum (so both ω and k are zero). There is nothing wrong in assuming ω=0 in the complex factor e^jω.
So why the formula shouldn't apply for sinusoidal plane waves with ω=0?
Yes it diverges into a static field and it doesn't work anymore (hence my doubt) but there's nothing in the derivation that tells me that it's special case.

 

[mfb]

So why the formula shouldn't apply for sinusoidal plane waves with ω=0?

Try to derive it for waves with ω=0. You'll hit a problem somewhere.
Static fields do not have to satisfy the usual wave equations. All their time-derivatives are zero, which gives more freedom for everything that is multiplied by them.