Theory: Poynting Vector, Intensity, and Radiation Pressure
1) What the Poynting vector means
The Poynting vector describes the rate of electromagnetic energy flow (power per unit area).
In SI units:
\[
\mathbf{S}=\frac{1}{\mu}\,(\mathbf{E}\times \mathbf{B})
\qquad\Rightarrow\qquad
|\mathbf{S}|=\frac{EB\sin\theta}{\mu}
\]
Here \(\theta\) is the angle between \(\mathbf{E}\) and \(\mathbf{B}\).
The direction of \(\mathbf{S}\) is given by the right-hand rule for \(\mathbf{E}\times\mathbf{B}\).
2) Plane waves (the standard wave case)
For a plane electromagnetic wave propagating in a uniform medium:
- \(\mathbf{E}\perp \mathbf{B}\perp \mathbf{k}\) (mutually perpendicular)
- \(\dfrac{E}{B}=c\), where \(c=\dfrac{1}{\sqrt{\mu\varepsilon}}\)
\[
c=\frac{1}{\sqrt{\mu\varepsilon}}
\]
In vacuum, \(\mu=\mu_0\) and \(\varepsilon=\varepsilon_0\).
In materials, you may use \(\mu=\mu_0\mu_r\) and \(\varepsilon=\varepsilon_0\varepsilon_r\).
3) Intensity as a time-average
For sinusoidal fields \(E(t)=E_0\cos(\cdot)\) and \(B(t)=B_0\cos(\cdot)\), the instantaneous magnitude scales like:
\[
S(t)=\frac{E(t)\,B(t)}{\mu}=\frac{E_0B_0}{\mu}\cos^2(\cdot)
\]
Since \(\langle \cos^2\rangle=\dfrac{1}{2}\), the average intensity is:
\[
I=\langle S\rangle=\frac{S_{\max}}{2}=\frac{E_0B_0}{2\mu}
\]
For a plane wave (where \(B_0=E_0/c\)), this becomes the widely used form:
\[
I=\frac{1}{2}c\varepsilon E_0^2
\]
4) Radiation pressure
Electromagnetic waves carry momentum. When light hits a surface, it exerts a pressure \(P\).
Using intensity \(I\):
\[
P=\frac{I}{c}\quad(\text{perfect absorber}),
\qquad
P=\frac{2I}{c}\quad(\text{perfect reflector})
\]
This tool allows you to compute pressure from either the average intensity \(\langle S\rangle\) or the peak value \(S_{\max}\).
5) Worked sample (from the calculator prompt)
Plane wave in vacuum with \(E_0=100\ \mathrm{V/m}\). Then \(B_0=\dfrac{E_0}{c}\approx \dfrac{100}{3\times 10^8}\ \mathrm{T}\).
\[
S_{\max}=\frac{E_0B_0}{\mu_0}\approx 796\ \mathrm{W/m^2},
\qquad
I=\frac{S_{\max}}{2}\approx 398\ \mathrm{W/m^2}
\]
Numbers may differ slightly depending on the value of \(c\) used.
6) “Solar constant” idea
The solar constant at Earth is roughly \(I\approx 1361\ \mathrm{W/m^2}\).
Using \(I=\dfrac{1}{2}c\varepsilon_0E_0^2\), you can estimate the corresponding field amplitude:
\[
E_0=\sqrt{\frac{2I}{c\varepsilon_0}}
\]
The calculator includes a one-click “Solar constant example” preset for this.
