Step 1: Relate the Poynting vector to momentum. The average Poynting vector \(S_{av}\) quantifies the average energy flux (energy per unit area per unit time) of an electromagnetic wave. Momentum flux, defined as momentum per unit area per unit time, is given by \(\frac{S_{av}}{c}\). This momentum flux equates to the radiation pressure experienced by the wave upon complete absorption.
Step 2: Consider total reflection. During total reflection, the wave's momentum is reversed.- The incoming momentum flux per unit area per unit time is \(\frac{S_{av}}{c}\).- The outgoing momentum flux per unit area per unit time is \(-\frac{S_{av}}{c}\).The change in momentum per unit area per unit time is calculated as the difference between the final and initial momentum fluxes:\[ \Delta (\text{momentum flux}) = \left(-\frac{S_{av}}{c}\right) - \left(\frac{S_{av}}{c}\right) = -\frac{2S_{av}}{c} \]
Step 3: Determine surface pressure. Radiation pressure on the surface, representing force per unit area, equals the magnitude of the momentum change per unit area per unit time transferred to the surface.\[ P_{rad} = \left| \Delta (\text{momentum flux}) \right| = \frac{2S_{av}}{c} \]