# Wave impedance of free space

1. Feb 12, 2016

### Shinji83

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

Z= E/H=√μ00 = 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?

Last edited: Feb 12, 2016
2. Feb 13, 2016

### Staff: Mentor

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?

3. Feb 13, 2016

### Shinji83

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.
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.

4. Feb 13, 2016

### Staff: Mentor

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.