You illuminate a slit with a width of 77.3 um with a light of wavelength 721 nm and observe the resulting diffraction pattern on screen that is situated 2.55 m from the slit. What is the width, in centimeters_ of the pattern’s central maximum? width: cm
The Correct Answer and Explanation is:
Correct Answer:
The width of the central maximum is 4.76 cm.
Textbook-Style Explanation:
To determine the width of the central maximum in a single-slit diffraction pattern, the key is understanding how light behaves when it passes through a narrow slit and diffracts onto a distant screen.
The central maximum in a single-slit diffraction pattern spans from the first minimum on one side to the first minimum on the opposite side. The position yyy of the first minimum on either side of the central maximum is given by the diffraction formula: y=Lλay = \frac{L \lambda}{a}y=aLλ
Where:
- LLL = distance from slit to screen = 2.55 m
- λ\lambdaλ = wavelength of light = 721 nm = 721×10−9721 \times 10^{-9}721×10−9 m
- aaa = width of the slit = 77.3 μm = 77.3×10−677.3 \times 10^{-6}77.3×10−6 m
Substitute the values into the formula: y=2.55×721×10−977.3×10−6=1.83855×10−677.3×10−6=0.02378 my = \frac{2.55 \times 721 \times 10^{-9}}{77.3 \times 10^{-6}} = \frac{1.83855 \times 10^{-6}}{77.3 \times 10^{-6}} = 0.02378 \text{ m}y=77.3×10−62.55×721×10−9=77.3×10−61.83855×10−6=0.02378 m
This distance represents half the width of the central maximum (from center to one minimum). So, the full width of the central maximum is: Width=2y=2×0.02378=0.04756 m\text{Width} = 2y = 2 \times 0.02378 = 0.04756 \text{ m}Width=2y=2×0.02378=0.04756 m
Convert meters to centimeters: 0.04756 m=4.756 cm0.04756 \text{ m} = 4.756 \text{ cm}0.04756 m=4.756 cm
Rounded to two decimal places, the width of the central maximum is: 4.76 cm\boxed{4.76 \text{ cm}}4.76 cm
This outcome demonstrates how wave properties of light cause noticeable diffraction effects when passing through narrow openings. The broader the wavelength or the narrower the slit, the wider the resulting diffraction pattern.
