Physics

Problem

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

L_1 = 1.0\,\text{m}

and

L_2 = 3.0\,\text{m}

from the slit. For each screen, the experimenter measures the average spacing

\Delta y

between successive dark bands. The data are summarized: Screen 1:

\Delta y_1 = 4.0\,\text{mm}

Screen 2:

\Delta y_2 = 11.9\,\text{mm}

(a) Derive an expression for the spacing

\Delta y

between adjacent dark bands on a screen in terms of the unknown slit width

a

, the wavelength

\lambda

, and the screen distance

L

. State any approximations you introduce. (b) Using the expression from part (a), obtain the theoretical ratio

\dfrac{\Delta y_2}{\Delta y_1}

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

\Delta y

and discuss qualitatively how it could affect the comparison performed in part (c).

Intuitive explanation

Single-slit diffraction dark fringes are spaced proportionally to the distance from the slit: $\Delta y = \frac{\lambda L}{a}$ , where $a$ is the slit width. Since you have two measurements at different distances, you can set up a ratio to eliminate the unknown wavelength and solve for $a$ —then use either dataset to find $\lambda$ . The key is recognizing that the ratio of spacings directly reveals the slit width without needing to know the light's wavelength beforehand.

Intuitive explanation

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