# Interpretation of real and imaginary parts of the Poynting vector in a vacuum

It is known that the flux of the Poynting Vector through a certain surface represents the total electromagnetic power flowing through it.

Then, its real part is the active power, while its imaginary part is the reactive power. Now let's consider an EM wave in vacuum: what do active power and reactive power mean?

From a circuit point of view, I'd say that the active power is that is dissipated in resistances, while reactive power is that is stored in inductances and/or capacitances. But in vacuum there are not materials with those properties. So what is their physical meaning?

Real and imaginary parts of the electromagnetic fields are simplifications constructed on top of real-valued Maxwell's Equations with arbitrary time-dependence. So the question about what imaginary part of the Poynting vector is, is a bit moot, IMHO. You defined it, so you can choose it to be whatever you want to be.

Also, how do you end up with imaginary Poynting vector ($$\mathbf{S}$$) in free-space? Whenever I see Poynting vector in time-harmonic treatment you get something like:

$$\mathbf{S} = \frac{1}{2} \Re\left(\mathbf{E}^\dagger\times\mathbf{H}\right)$$

For complex-valued electric ($$\mathbf{E}$$) and magnetic ($$\mathbf{H}$$) fields. In frees-space electric and magnetic fields are in phase, so even $$\mathbf{E}^\dagger\times\mathbf{H}$$ will be real-valued (i.e. $$\Re$$ is superfluous).

• I always saw a definition of the Poynting vector S like this: S = 1/2 (E x H*) Commented Oct 8, 2019 at 4:44
• @Kindka-Byo. Well that's how we end up with the moot point (you defined something, so you 'chose' the phyiscal meaning). I would define Poynting vector as the flow of energy. Energy would be defined as the quantity conserved due to translational time-invariance of the Maxwell's Equations. With this scheme you always end up with instantaneous real-valued Poynting vector. The complex-valued fields can then be introduced through complex-harmonic time-dependence of fields. However, the complex-harmonic stuff only makes sense for quantities linear in fields.
– Cryo
Commented Oct 8, 2019 at 11:12
• For quantitites quadratic in time-harmonic fields, you have to be very careful. In fact it is better to take a step back to real valued fields (i.e. if $\mathbf{\tilde{E}}$ is complex time-harmonic, then $\mathbf{E}=\mathbf{\tilde{E}}\exp\left(i\omega t\right) + \mathbf{\tilde{E}}^\dagger\exp\left(-i\omega t\right)$ is real-valued; $\omega$ is the angular frequency), and clarify what you intend to seek.
– Cryo
Commented Oct 8, 2019 at 11:14