A narrow plane mirror of width w=3.0mm is used to reflect sunlight onto a screen located L=10m away. The Sun subtends an angle of about 0.53∘ as seen from Earth, but you may ignore the Sun’s finite size in the following analysis. Observers measure the width of the illuminated patch on the screen and find it noticeably larger than what simple geometry predicts.
(a) Treating the mirror as an ideal reflector in geometric optics, derive an expression for the width Wray of the bright patch at the screen in terms of w and L, and calculate its value.
(b) Treating the mirror as a slit that reflects rather than transmits, derive an expression for the angular position of the first dark fringe of the diffraction pattern. Hence determine the width Wwave of the central maximum at the screen, and compute its value for incident light of wavelength λ=550nm. Assume normal incidence.
(c) Compare the two calculated widths with the observers’ qualitative report. Which model better accounts for the observation, and why?
(d) From your analysis, formulate a quantitative condition involving w and λ that must be satisfied for geometric optics to give an adequate prediction of the spot size in this setup.