Exact relationship between electric field and intensity

I often see $I(t)\propto |E(t)|^2$. What is the exact form ? Which constants are missing to make this an equality and why are they so often omitted ?

Answer

Intensity is related to the Poynting vector (https://en.wikipedia.org/wiki/Poynting_vector) by simply taking the magnitude. Note that $\vec{S}=\vec{E}\times \vec{H}$, and $|\vec{H}| = \sqrt{\frac{\epsilon_o}{\mu_o}}|\vec{E}|$ for an electromagnetic wave in vacuum. Thus,

$$|\vec{S}|=|\vec{E}\times\vec{H}|=|\vec{E}|\cdot |\vec{H}|=\sqrt{\frac{\epsilon_o}{\mu_o}}|\vec{E}|^2$$

where the second equality follows from using $|\vec{A}\times\vec{B}|=|\vec{A}||\vec{B}|\sin(\theta)$ and $\theta =90^\circ$ since magnetic and electric fields are perpendicular to the direction of propagation for waves.

Another way to write this would be

$$I=|\vec{S}|=\frac{|\vec{E}|^2}{Z_0} $$

Where $Z_0$ is the impedance of free space, with a value of about 377 ohms.

The constants are frequently omitted if we are doing a theoretical derivation since constants typically factor out of the entire problem and are not an interesting consideration. In experiment, we can typically perform some calibration for the constants. Constants are, however, useful for dimensional considerations and can be useful for checking that your final units are correct.