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Hi.

According to classical electromagnetism (and common sense) the intensity of a beam of light entering a dielectric medium should remain constant. Hence the length of the poynting vector must remain constant.

But how do you derive mathematically the last point? Because if you just replace ##c## with ##v=c/n## and ##\epsilon_0## with ##\epsilon = \epsilon_0 n^2## and ##E## with ##E/n^2## you get into trouble when trying to transform the poynting vector.

Let's say you have light entering glass from vacuum with ##n = \sqrt{\epsilon/\epsilon_0}##. => Before: ##<S_{vac}> = \frac{c^2 \epsilon_0}{2} E_{vac} B_{vac}##. After: ##<S_{glass}> = \frac{(c^2/n^2) (\epsilon_0 n^2)}{2} (E/n^2) B = \frac{(c^2) (\epsilon_0}{2} (E_{vac}/n^2) B_{vac} \neq< S_{vac}>##

All thoughts on this are highly appreciated.

EDIT: fixed typo

According to classical electromagnetism (and common sense) the intensity of a beam of light entering a dielectric medium should remain constant. Hence the length of the poynting vector must remain constant.

But how do you derive mathematically the last point? Because if you just replace ##c## with ##v=c/n## and ##\epsilon_0## with ##\epsilon = \epsilon_0 n^2## and ##E## with ##E/n^2## you get into trouble when trying to transform the poynting vector.

Let's say you have light entering glass from vacuum with ##n = \sqrt{\epsilon/\epsilon_0}##. => Before: ##<S_{vac}> = \frac{c^2 \epsilon_0}{2} E_{vac} B_{vac}##. After: ##<S_{glass}> = \frac{(c^2/n^2) (\epsilon_0 n^2)}{2} (E/n^2) B = \frac{(c^2) (\epsilon_0}{2} (E_{vac}/n^2) B_{vac} \neq< S_{vac}>##

All thoughts on this are highly appreciated.

EDIT: fixed typo

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