A single slit of fixed, unknown width is illuminated by monochromatic light. The resulting diffraction pattern is projected onto two separate screens placed normal to the beam at distances L1=1.0m and L2=3.0m from the slit. For each screen, the experimenter measures the average spacing Δy between successive dark bands. The data are summarized:
Screen 1: Δy1=4.0mm
Screen 2: Δy2=11.9mm
(a) Derive an expression for the spacing Δy between adjacent dark bands on a screen in terms of the unknown slit width a, the wavelength λ, and the screen distance L. State any approximations you introduce.
(b) Using the expression from part (a), obtain the theoretical ratio Δy1Δy2 expected for the two screen distances given.
(c) Compare the theoretical ratio with the experimental value determined from the data. Evaluate whether the data are consistent with theory within reasonable experimental uncertainty.
(d) Identify one plausible systematic source of error in the measurement of Δy and discuss qualitatively how it could affect the comparison performed in part (c).