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 \(k=2h\).
