# About curious form of Maxwell's Equations for a monochromatic field [closed]

In a review paper of Whispering-gallery waves from A.N. Oraevsky, he writes the source-free monochromatic Maxwell's Equations as

$\nabla\times E = ikH$

$\nabla\times H = -ikE$

and he defines $k = (\omega/c)\sqrt{\mu\varepsilon}$ where $\mu$ and $\varepsilon$ are the magnetic permeability and electric permitivity respectively of the medium, $\omega$ is the frequency (rad/s) and $c$ is the speed of light in vacuum.

I am puzzled with the form Maxwell's Equations are written, since they should be

$\nabla\times E = i\omega\mu H$

$\nabla\times H = -i\omega\varepsilon E$

for a time dependence $e^{-i\omega t}$. His definition does not agree with cgs units neither. What is this different form of writing Maxwell Equations, and how can they be related to those written by me above?.

• I've never seen that notation before, and it's definitely incorrect because both $k=\omega \mu$ and $k = \omega \epsilon$ have to be true at the same time. I suspect that the author has made a mistake. Commented Jan 14, 2016 at 21:11
• Maxwell's equations for the curls are $\nabla\times E = -\partial_t B$ and $\nabla\times B = \mu_0 (J + \epsilon_0\partial_t E)$. I neither understand your nor Oraesky's form of them, there is definitely some context lacking here. Commented Jan 15, 2016 at 0:35
• @ACuriousMind your equations are the same as mine. Since there is not sources $J=0$, $\partial_t \rightarrow -i\omega$ (harmonic monochromatic time-dependence), and of course $B=\mu H$. (Yours are in vacuum $\mu_0$, $\varepsilon_0$, mine are in a general medium $\mu$, $\varepsilon$) Commented Jan 15, 2016 at 11:59

The only explanation I found to this is the following:

From Maxwell's Equations in Lorentz-Heaviside units:

$$\nabla\times E = (i\omega\mu/c)H$$

$$\nabla\times H = (-i\omega\varepsilon/c)E$$

where now $$\mu$$ and $$\varepsilon$$ refer to the common relative permeability and permitivity respectively. Defining a sort of new $$H' = \sqrt{\mu/\varepsilon} H$$ then,

$$\nabla\times E = (i\omega\sqrt{\mu\varepsilon}/c)H'$$

$$\nabla\times H' = (-i\omega\sqrt{\mu\varepsilon}/c)E$$

which agrees with Oraevsky's equations. However, this requires to assume that $$\nabla\times ( \sqrt{\varepsilon/\mu} H') = \sqrt{\varepsilon/\mu}\nabla\times H$$ which is only valid for homogeneous media, and it does not seem to be the case for the context of the paper.