Jonathan T. answered • 10/29/23
10+ Years of Experience from Hundreds of Colleges and Universities!
To find the width of the central maximum in the diffraction pattern, you can use the formula for the angular width of the central maximum in a single-slit diffraction:
\[θ = \frac{λ}{w}\]
Where:
- \(θ\) is the angular width of the central maximum (in radians).
- \(λ\) is the wavelength of the light (in meters).
- \(w\) is the width of the slit (in meters).
Given:
- Slit width (\(w\)) = 76.9 micrometers = \(76.9 \times 10^{-6}\) meters
- Wavelength (\(λ\)) = 715 nm = \(715 \times 10^{-9}\) meters
Now, calculate \(θ\):
\[θ = \frac{715 \times 10^{-9} \, \text{m}}{76.9 \times 10^{-6} \, \text{m}}\]
Simplify:
\[θ = \frac{715}{76.9} \times 10^{-3} \, \text{radians}\]
Now, calculate \(θ\):
\[θ ≈ 9.296 \times 10^{-3} \, \text{radians}\]
To find the width of the central maximum on the screen (in centimeters), you can use the following formula:
\[W = 2L \cdot \tan\left(\frac{θ}{2}\right)\]
Where:
- \(W\) is the width of the central maximum on the screen (in meters).
- \(L\) is the distance from the slit to the screen (in meters).
Given:
- \(L\) = 2.69 meters
Now, calculate \(W\) in meters:
\[W = 2 \cdot 2.69 \cdot \tan\left(\frac{9.296 \times 10^{-3}}{2}\right)\]
Calculate \(W\):
\[W ≈ 2 \cdot 2.69 \cdot \tan(4.648 \times 10^{-3})\]
\[W ≈ 2 \cdot 2.69 \cdot 4.647 \times 10^{-3}\]
\[W ≈ 2 \cdot 2.69 \cdot 4.647 \times 10^{-3} \, \text{m}\]
Now, convert meters to centimeters (1 meter = 100 centimeters):
\[W ≈ 2 \cdot 2.69 \cdot 4.647 \times 10^{-3} \times 100 \, \text{cm}\]
\[W ≈ 24.96 \, \text{cm}\]
So, the width of the central maximum in the diffraction pattern is approximately 24.96 centimeters.
