Complex variables in compound expressions of electrodynamics

[andresordonez]

When dealing with electrodynamics it is usual to use complex variables for the electromagnetic field while taking into account that the electromagnetic field is real and that at the end one has to take the real part of the complex solution for the field. However, what happens to compound expressions like the energy
$$E = \frac{1}{2}(\epsilon \vec{E}^2 + \mu \vec{H}^2)$$

or the Poynting vector
$$\vec{S} = \vec{E}\times\vec{H}$$

?

For the energy I see two options, either I stick with the real part of the field
$$E = \frac{1}{2}(\epsilon \left[\Re(\vec{E})\right]^2 + \mu \left[\Re(\vec{H})\right]^2)$$
or I use a more general definition of the energy
$$E = \frac{1}{2}(\epsilon \left|\vec{E}\right|^2 + \mu \left|\vec{H}\right|^2)$$

Note these two definitions for the energy are different since if
$$z=x+iy$$
then
$$\Re(z)^2 = x^2$$
$$\left|z\right|^2 = x^2+y^2$$

For the Poynting vector there will be even more options since it is not as symmetric as the energy
$$\vec{S} = \Re(\vec{E})\times\Re(\vec{H})$$
$$\vec{S} = \vec{E} \times \vec{H}^*$$
$$\vec{S} = \vec{E}^* \times \vec{H}$$
$$\vec{S} = \left|\vec{E}\right| \times \left|\vec{H}\right|$$

My initial guess was that one should use only the real part of the field when calculating these compound energies (i.e. use the first options), but then I was reading Jackson's electrodynamics and I found an expression for the Poynting vector like in the second option, i.e.
$$\vec{S} = \vec{E} \times \vec{H}^*$$
and an expression for the energy like in the second option as well, i.e.
$$E = \frac{1}{2} \times (\epsilon \left|\vec{E}\right|^2 + \mu \left|\vec{H}\right|^2)$$

I suppose that the expressions given by Jackson are the right ones but I really don't understand why. For example why isn't the Poynting vector calculated as in the third option instead? or why should I take into account the imaginary part of the fields if they are just a convenience for the calculations? Are the imaginary parts more than just an instrument for the calculations? Is Jackson wrong?

I'd really be very grateful to anyone who could clarify this to me.

 

[Jano L.]

Hello andresordnez!
All physical quantities, like E or B, are real. The complex notation is just a (sometimes) useful way to write down quantities that vary _sinusoidally_ (or like $e^{-kt}\sin(\omega t)$) with time or spatial coordinate, because the equations of motion are mathematically easily solved with use of complex numbers. You can solve the equation for complex E and then take the real part as the factual quantity, provided the terms in equation are linear in E and the expression of equation itself does not contain non-real numbers (i.e., eq. like $\ddot E = -\omega^2 E is OK, but eq. \ddot E = i\omega^2 E.$ is not.

For quadratic expressions this strategy fails and you have to plug in the real quantities. For Poynting's vector, you have to use

$$\mathbf S = \mathrm{Re} \mathbf E \times \mathrm{Re} \mathbf B.$$

The expression Jackson gives can be used with the same rule maintained that the rule that the physical quantities are real parts of the mathematical complex quantities, i.e.

$$\mathrm{Re} (\mathbf E \times \mathbf B^*) = \mathrm{Re} \mathbf E \times \mathrm{Re} \mathbf B.$$


If this is too complicated, nevermind and use just the real quantities. It is always correct and it is not a big complication, you just have to know the formulae for $\sin(x+y), \cos(x+y)$, but these you have to know anyway. After some (quite long for me) time using the real quantities, you will develop intuition when to use complex quantities and how to do it.