Maxwell's equations and time-harmonic solutions

• fog37
In summary, the two curl Maxwell's equations for a linear medium change when considering solutions that are space-dependent vector functions of the form ##\tilde{\bf{E}} (x,y,z) e^{-i \omega t}##. These solutions can represent either standing waves or traveling waves, depending on the values of the complex-valued functions ##\tilde{\bf{E}} (x,y,z)## and ##p(t)##. The physical fields are always real-valued, but the solutions can involve complex functions. Examples of standing and traveling waves in 1D are provided.
fog37
Hello,
For a linear medium, the two curl Maxwell's equations
$$\nabla \times \bf{E} = - \frac {\partial \bf{B}} {\partial t}$$
$$\nabla \times \bf{H} = \frac {\partial \bf{D}} {\partial t}$$
change to
$$\nabla \times \bf{E} = i \omega \bf{B}$$
$$\nabla \times \bf{H} = - i \omega\bf{D}$$

whose all solutions are space-dependent vector functions of the form ##\tilde{\bf{E}} (x,y,z) e^{-i \omega t}##. The last two equations actually have the complex-functions ## \tilde{\bf{E}} (x,y,z)## as solutions.
• Is it correct to assume that if the field solution ##\tilde{\bf{E}} (x,y,z)## is real-valued (zero imaginary part), the overall solution field ##\tilde{\bf{E}} (x,y,z) e^{-i \omega t}## would then represent a standing wave while if ## \tilde{\bf{E}} (x,y,z)## has both nonzero ##Re## and ##Im## parts the field solution would be a traveling wave?
• The spatial complex-valued function ## \tilde{\bf{E}} (x,y,z)## is truly the three scalar functions: $$\tilde{E}_{x} (x,y,z)$$ $$\tilde{E}_{y} (x,y,z)$$ $$\tilde{E}_{z} (x,y,z)$$ Is it possible for any of these scalar functions to be complex-valued while the others are real-valued? What kind of field would that be? Hybrid, i.e. traveling and stationary at the same time?

Last edited:
Usually the idea with the ansatz with the exponential function is that the physical fields are the real parts of these complex fields. The ##\tilde{\boldsymbol{E}}## and ##\tilde{\boldsymbol{B}}## fields are then general complex solutions of your equations.

You have a standing wave, wenn the field is a product of a space-dependent factor with a time dependent function. Your solutions are indeed of this kind if these fields are purely real or imaginary. Usually you get standing waves in a cavity, e.g., the interior of a hollow sphere (or more easy to calculate a cuboid). If you have a wave guide (e.g., a cylinder or a coaxial cable) you have both traveling waves along the axis of the wave guide and standing waves in any plane perpendicular to it.

fog37

A few points:
• I am familiar with the fact that, at the very end of all calculations, we must alway take the real part of the solution field to obtain the final answer since physical fields are always real-valued because measuring devices produce numerical answers with real values.
• In the case of a purely real-valued function ##f(x,t)## is separable, it always represents a standing wave because the spatial and temporal coordinates pertain to the two different functions. For example, in 1D, the real field ##E(x,t) = g(x) p(t)## is separable and does not represent a traveling field.
• However, if the functions ##g(x)## and ##p(t)## are both complex valued, the field could also be a traveling wave and not necessarily a standing wave. I would state that the field ##E(x,t) = g(x) p(t)## is a standing wave when one of the function ##g(x)## and ##p(t)## or both of them are real-valued.
Thanks!

vanhees71
exactly!

Great, thank you!

Here some (possibly correct) examples of 1D field functions representing standing and traveling waves:

Real-valued standing wave: ##E(x,y) = (2t^2) (3x)##
Complex valued standing wave: ## E(x,t) = (x^3-4) e^{i\omega t}##
Complex valued standing wave: ## E(x,t) = e^i(x^3-4) cos(\omega t)##
Complex-valued traveling wave: ##E(x,t) = (2x+ i 4x^3) (5t + i t^2)##
Real valued traveling wave: ##E(x,t)=3(x-t)^4##

1. What are Maxwell's equations?

Maxwell's equations are a set of four fundamental equations that describe the behavior of electric and magnetic fields in space. They were developed by James Clerk Maxwell in the 19th century and are an essential part of classical electromagnetism.

2. What is the significance of time-harmonic solutions in Maxwell's equations?

Time-harmonic solutions are a special type of solution to Maxwell's equations where the electric and magnetic fields vary sinusoidally with time. They are important because they allow us to simplify the equations and make them easier to solve, and they also have practical applications in fields such as telecommunications and signal processing.

3. How do Maxwell's equations relate to the behavior of electromagnetic waves?

Maxwell's equations describe the relationship between electric and magnetic fields, and they also predict the existence of electromagnetic waves. These waves are disturbances in the electric and magnetic fields that travel through space at the speed of light, and they are responsible for phenomena such as light, radio waves, and X-rays.

4. Can Maxwell's equations be applied to all types of materials?

Maxwell's equations can be applied to all types of materials, but the behavior of the electric and magnetic fields may vary depending on the properties of the material. For example, in conductors, the fields may be affected by the presence of free charges, while in dielectric materials, the fields may be affected by the polarization of the material.

5. How are Maxwell's equations used in practical applications?

Maxwell's equations have many practical applications, including the design and analysis of electronic circuits, the development of communication technologies, and the understanding of electromagnetic phenomena in nature. They are also used in fields such as optics, acoustics, and plasma physics.

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