I've studied and understand the differential and integral forms of Maxwell's equations, and understand the basic math and logic behind phasor notation for circuit analysis. Still, I'm confused as to the significance of Maxwell's equations in phasor form. They seem only to be functions of frequency, which from scattered research I gather is helpful in analyzing fields that only vary based on sinusoidal expressions with respect to time, What is the added intuition and/or computational value of the phasor form vs. the differential and integral forms (Ex: specific applications like in circuit analysis, wave propagation, etc.)? Or if any of my initial assumptions are wrong and I'm just rambling right now, what would be the correct interpretation?
2 Answers
It would be easier to define where the complex amplitude notation is not appropriate than listing innumerable applications where it is appropriate and used extensively.
First remember that Maxwell's differential equations are linear in the vector variables $\mathbf {E,D,H,B}$. As long as the constitutive (material) relationship between $\mathbf {E,D}$ and between $\mathbf {H,B}$ is linear complex amplitudes more than just dominate, they rule. Cases where it cannot be used are materials with strong nonlinear relationship between these variables, e.g., saturation, hysteresis, etc., the kind you find in the magnetostatics of ferromagnetism.
Between these extremes of linearity and nonlinearity of saturation/hysteresis you find various attempts to extend the use of complex phasors by locally linearizing the material response functions, if possible. Even when this response function is differentiable local linearization can fail for several reasons, among which are limitations for narrow bandwidth, reduced dynamic range, genuine (not just numerical) instabilities, etc. There are very few generally valid nonlinear results, it is a jungle out there.
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$\begingroup$ Oh wow, I didn't realize how widespread the applications of this were: I guess what I'm wondering then is what cases we use the phasor/amplitude form and which cases we use others, and why. $\endgroup$– LambdaCommented Apr 13, 2023 at 23:56
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$\begingroup$ do you mean the practical difference between using a complex number $z$ as $x=Re^{\mathfrak j \phi}$ or as $z=x+\mathfrak j y$? $\endgroup$ Commented May 18, 2023 at 13:40
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$\begingroup$ Moreso the problems in which we apply the integral form of equations like Gauss's Law (e.g. calculating the electric field of symmetric charge distributions) vs. the phasor form. $\endgroup$– LambdaCommented May 18, 2023 at 23:04
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$\begingroup$ I think you use the term "phasor" in a sense I am not familiar with. Would you elaborate how is a "phasor" related to Gauss's law? $\endgroup$ Commented May 19, 2023 at 1:04
A electromagnetic wave oscillates and it can be represented a sum of sines or cosines.However doing math with sines and cosines is really hard because the trigonometric functions are not linear.However you can do a trick.You can write the components of a EM field in their complex form using Euler's identity $e^{ia}=cos(a)+isin(a)$ perform all the hard math there,then take the real part of what you have found and you get your result which would be much MUCH harder if you stuck with sines and cosines.
