All Creatures Great and Small

E-field of light range from 10^(-14) Wm^-2 to 10^(12) Wm^-2 .  Superposition at the detector would mean the measured value is dependent on Intensity.  Since the the dipole comes around to the the same position $f$ times per second that means, time average over one second, the measured value is dependent on the frequency of the circular motion also.

We are looking for possibly one small value.  Let this small value be $\delta E_l$

According to Poynting,

From $S= E \times H$

$[S] = \cfrac{{E}^{2}_o}{2{Z}_{w}}$ Wm^-2

where ${Z}_{w}$ the wave impedence is ${Z}_{w}=\cfrac{{E}_{o}}{{H}_{o}}=\sqrt{\cfrac{{\mu}_{r}{\mu}_{o} }{{\varepsilon }_{r}{\varepsilon }_{o}}}$ = 376.7 $\Omega$ for free space. So, for a photon,

$[S_p] = \cfrac{{E}^{2}_{l}}{2{Z}_{w}}$

According to Planck,

${E}_{n}= h.f$

where $h$ is the Planck constant and $f$ the frequency of the light.  So the power of a helix of photon is given by

$W = c.N.{E}_{n}= c.N.h.f$ Wm^-2

where $N$ is the number of photons per unit volume (m^-3) and $c$ light speed.

Together with Poynting formulation where N = 1 m^-3,

$\cfrac{{{\delta E_l}^2}}{2{Z}_{w}} =c.h.f$,

since, $c = \cfrac{1}{\sqrt{{ \varepsilon  }_{ o }{ \mu  }_{ o }}}$ and ${Z}_{w}=\sqrt{\cfrac{{\mu}_{o} }{{\varepsilon }_{o}}}$

${\delta E_l}^2=\cfrac{2h}{{\varepsilon }_{o}}.f$

This is just a restatement of Planck relation.  And so,

${\delta E_l}=\sqrt{\cfrac{2h}{{\varepsilon }_{o}}.f}$

and correspondingly,

${\delta H_l}=\sqrt{\cfrac{2h}{{\mu}_{o}}.f}$

A plot of ${\delta E_l}$ over the THz range spectrum, 10^(5)*(2*6.62606957*10^(-12)*x/1.112650056)^(0.5) scaled  in the y axis by 10^(5).

Typical values are, at 430 THz is 7.156*10^(-5) Vm^-1 and at 720 THz, 9.26*10^(-5) Vm^-1. And the corresponding E-field are 9.49072*10^(-13) Wm^-2 and 1.22816*10^(-12) Wm^-2.  I have thought I can get rid of $f$ by multiplying it to ${\delta E_l}$, as it turn out Poynting energy conservation has already in cooperated $f$.  So, photon is not a simple dipole rotating at $f$, along a helix.  This dipole is being stretched at higher frequency.  In this treatment, ${\delta E_l}$ is just ${E}_{o}$ of a electromagnetic wave going on at light speed $c$ and $\omega$ = $2\pi f$.

If we just consider the distance $a$ between the charges in a dipole,  ${\delta E_l}=\cfrac{1}{4\pi\varepsilon_o}\cfrac{q.a}{z^3}$

$\cfrac{a_1}{a_2} = \sqrt{\cfrac{{f}_{1}}{{f}_{2}}}$,    or

$\cfrac{a^2_1}{a^2_2} = \cfrac{{f}_{1}}{{f}_{2}}$

since according to Planck, frequency is proportional to energy via Planck constant, $h$,

 ${E}_{n} = h.f=\cfrac{k}{2}a^2$,    where $\cfrac{k}{2}$ mimic a spring.

This suggest that the dipole can be model as a spring of stretch length $a=\sqrt{f}$ and spring constant $k=2h$.