Complex physical quantities

Question

I have a question regarding complex physical quantities. Why do we consider only the real part of a complex physical quantity? Why not the modulus? Since, for $z=a+bi$, we have $|z| = \sqrt{a^2+b^2}$, and so the imaginary part contributes to the modulus, it is not clear to me when to use the modulus or the real part.

For example, the electric (or magnetic) field. And/or the Poyting vector. Below I take texts from Jackson, Classical Electrodynamics:

"Because the diffusion equation is second order in the spatial derivatives and first order in the time, it is convenient to use complex notation, with the understanding that the physical fields are found by taking the real parts of the solutions." (Jackson, 3rd. Edition, page 220)

"Then (6.131) can be written as $$\frac{1}{2} \int_V \mathbf{J^*\!\cdot E}\,d^3x+2i\omega \int_V (w_e-w_m)\, d^3x +\oint_S \mathbf{S\cdot n}\,da \tag{6.134}$$

... It is a complex equation whose real part gives the conservation of energy for the time-averaged quantities and whose imaginary part relates to the reactive or stored energy and its alternating flow." (Jackson, 3rd. Edition, page 265)

"...With the convention that the physical electric and magnetic fields are obtained by taking the real parts of complex quantities, we write the plane wave fields as $$ \mathbf{E}(x,t)=\mathcal{E} e^{ik\mathbf{n\cdot x}-i\omega t} \ \ \ \ (7.8) \\ \mathbf{B}(x,t)=\mathcal{B} e^{ik\mathbf{n\cdot x}-i\omega t}$$" (Jackson, 3rd. Edition, page 296)

Answer 1

There are cases where we use the modulus, imaginary part, argument or some combinations to represent a physical quantity. It depends on how we arrived at using complex numbers in the first place.

Typically one of the first places where complex numbers are encountered when studying physics is when dealing with waves or oscillations as a method of simplifying the algebra, as we can replace trig functions with simpler exponentials. In this case the complex numbers were introduced by observing that the physical quantities we are interested in could be written as the real part of certain complex numbers, so it is the real part that retains the physical significance. We could, in principle, find a way to represent the same phyical quantity as the magnitude of a complex number, but it would likely result in a more complicated calculation.