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A question in my textbook involve finding the electric field amplitude at a point in space given the intensity of light. It uses the following equation to solve it: -

$I=\frac{1}{2}\epsilon_{0}|E_{rms}|^2c$

But where did this equation come from? I am unable to find an explanation for this anywhere.

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  • $\begingroup$ It came from observation and analysis. What is your question? The relationship between wave amplitude and power ( $ I \propto A^2$ ) holds for pretty much any kind of wave, not just E-M. $\endgroup$ – Carl Witthoft Feb 16 '16 at 13:24
  • $\begingroup$ It comes from the Poynting vector theorem which can be derived from the wave equation. Have a look on wikipedia and apply it for a plane TEM wave $\endgroup$ – Ronan Tarik Drevon Feb 16 '16 at 15:30
  • $\begingroup$ Check any college textbook on wave optics. $\endgroup$ – Ján Lalinský Feb 16 '16 at 21:49
  • $\begingroup$ @CarlWitthoft yeah everybody knows that, i was interested in its mathematical derivation. $\endgroup$ – Marcus Feb 17 '16 at 18:37
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As light is an electromagnetic wave, it is a combination of both electric field and magnetic field. So intensity of light is basically the power transmitted through electric and magnetic field divided by the cross section area of that light beam.

The energy density of the electric field is $\frac{1}{2}\epsilon_0 E^2$, and the energy density of the magnetic field is $\frac{1}{2}\frac{B^2}{\mu_0}$. The total energy density of an EM wave is then: $$\frac{1}{2}\left(\epsilon_0 E^2 + \frac{B^2}{\mu_0}\right)$$ The total energy transmitted per second per unit area is then: $$\frac{c}{2}\left(\epsilon_0 E^2 + \frac{B^2}{\mu_0}\right)\tag{1}$$ As we know $|\vec{E}|=c|\vec{B}|$ and $c^2=\frac{1}{\epsilon_0 \mu_0}$, so $(1)$ turns out to be: $$\epsilon_0 E^2_{\rm RMS}c$$ $$\frac{1}{2}\epsilon_0 E^2 c$$

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  • $\begingroup$ Posting images of derivations or any mathematical content in general is discouraged since the site supports mathjax, which both looks much better and makes the content searchable. $\endgroup$ – Kyle Oman Mar 4 at 8:07
  • $\begingroup$ I am new to this site. So i don't know much about how to write with mathjax. I will try next time $\endgroup$ – user224359 Mar 4 at 8:09
  • $\begingroup$ if you know how to markup math in latex, it's basically the same. Otherwise, you can read the post I linked and/or find some posts with math content and click 'edit' to see how it was done. Not terribly complicated for simple equations. $\endgroup$ – Kyle Oman Mar 4 at 8:11
  • $\begingroup$ And there's an example for you with your own post. $\endgroup$ – Kyle Oman Mar 4 at 8:21

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