Are the equations of the Poynting vector and energy density of an electromagnetic wave only for the real waves?

Question

So, my book says that the Poynting Vector associated to an electromagnetic wave in matter with permeability $\mu$ is $\mathbf{S} = \frac{1}{\mu} \mathbf{E} \times \mathbf{B}$. The thing is, I am unsure whether I can compute this vector with the complex fields "associated" to my original waves, and then get the real part, or if this formula is only valid when using the original real fields.

My exercise in particular gives me the wave $E(y,z,t)=E_0cos(ay+bz-wt) \hat{x}$, which has associated complex wave $$E(y,z,t)=E_0e^{i(ay+bz-wt)} \hat{x}$$ The magnetic field (which I got from Maxwell's third equation) is:

$$B(y,z,t)=\dfrac{E_0}{w}(b\hat{y}-a\hat{z}) e^{i(ay+bz-wt)} $$ Now, if I compute the poynting vector with the first method(complex waves and then extract real part) I get: $\mathbf{S}=\dfrac{E_0^2}{\mu w}(b\hat{z}+a\hat{y})cos(2(ay+bz-wt))$

Whereas if I use directly the real fields I get: $\mathbf{S}=\dfrac{E_0^2}{\mu w}(b\hat{z}+a\hat{y})cos^2(ay+bz-wt)$.

Obviously they cannot both be right, since one has $cos(2\alpha)$ and the other $cos^2(\alpha)$.I also have a similar confusion with the formula for the energy density $\frac{1}{2}\epsilon_0 E^2 + \frac{1}{2\mu_0} B^2$. Should I use the modules of the complex waves, or the modules of the real waves? Thanks in advance.

Answer 1

With a slight exaggeration one could say that the real purpose of complex amplitudes in EM is to simplify calculations of averages, not to simplify the solutions of linear differential equations and when you multiply two phasors of the same frequency you have to remember that goal. Let us just do this for $V$ and $I$, so we do not have to deal with vectors.

You have two sinusoids $V(t)=V_0 cos(\omega t +\phi)$ and $I(t)=I_0 cos (\omega t +\psi)$, voltage and current, and you want to calculate the corresponding power (energy rate) they represent: $$P(t)=V(t)I(t)\\ =\frac{1}{2} V_0 I_0 cos(\phi-\psi) + \frac{1}{2}V_0 I_0 cos(2\omega t +\phi+\psi). $$

You notice that this contains a static term $ P_0= \frac{1}{2} V_0 I_0 cos(\phi-\psi)$ and a term that fluctuates at $2\omega$ rate: $p(t)=\frac{1}{2} V_0 I_0 cos(2\omega t +\phi+\psi)$. Especially at RF and beyond the double frequency term is usually irrelevant because we are usually interested in delivered average power and its average is zero already over a single cycle.

Now if we do the power calculation not with complex sinusoids but with complex amplitudes then we want to get only the static term $ P_0= \frac{1}{2}V_0 I_0 cos(\phi-\psi)$, and that is very simple because we have the complex representation $\tilde V = V_0 e^{\mathfrak j \phi}$ and $\tilde I = I_0 e^{\mathfrak j \psi}$ from which $P_0 = \frac{1}{2} \Re{[\tilde V \tilde I^*]}$. It is sometimes customary to use rms values instead of peak amplitudes and then bury the $\frac{1}{2}$ factor in the formula so that $V_{rms}=\frac{1}{\sqrt2}V_0$, $I_{rms}=\frac{1}{\sqrt2} I_0$ and write $ P_0 = \Re{[\tilde V_{rms} \tilde I_{rms}^*]}$. Of course it would be equally valid to write $ P_0 = \Re{[\tilde V_{rms}^* \tilde I_{rms}]}.$

Answer 2

Complex number expressions for fields in EM theory are just a different mathematical representation of fields to make solving differential equations on paper easier. The actual physical field can be described by either the real or the imaginary component of those complex expressions, depending solely on our choice; usualy the real component is meant to represent the actual field.

The Poynting energy expressions are derived and meant to be used with the actual physical fields. It is not correct to put auxiliary complex representations of fields into them, you would get wrong results.