Richard observes that a single slit that is2,022 nm wide forms a diffraction pattern when illuminated by monochromatic light of673 -nm wavelength. At an angle of 10° from the central maximum, what is the ratio of the intensity to the intensity of the central maximum? Give your answer with two decimal places please.
Richard observes that a single slit that is2,022 nm wide forms a diffraction pattern when illuminated by monochromatic light of673 -nm wavelength. At an angle of 10° from the central maximum, what is the ratio of the intensity to the intensity of the central maximum? Give your answer with two decimal places please.
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![### Diffraction Pattern Analysis
**Problem Statement:**
Richard observes that a single slit that is 2,022 nm wide forms a diffraction pattern when illuminated by monochromatic light of 673 nm wavelength. At an angle of 10° from the central maximum, what is the ratio of the intensity to the intensity of the central maximum? Give your answer with two decimal places, please.
**Solution:**
To determine the ratio of the intensity at an angle (θ) to the intensity at the central maximum for a single slit diffraction pattern, we use the diffraction intensity formula:
\[ I(\theta) = I_0 \left(\frac{\sin(\beta/2)}{\beta/2}\right)^2 \]
where:
- \( \beta = \frac{2 \pi a \sin(\theta)}{\lambda} \)
- \( a \) is the width of the slit
- \( \lambda \) is the wavelength of the light
- \( \theta \) is the angle from the central maximum
- \( I_0 \) is the central maximum intensity.
Given:
- \( a = 2022 \) nm
- \( \lambda = 673 \) nm
- \( \theta = 10° \)
We first calculate \( \beta \):
\[ \beta = \frac{2 \pi \times 2022 \times \sin(10°)}{673} \]
\[ \sin(10°) \approx 0.1736 \]
\[ \beta = \frac{2 \pi \times 2022 \times 0.1736}{673} \]
\[ \beta \approx \frac{2201.584 \times 0.1736}{673} \]
\[ \beta \approx 0.5699 \]
Now plug \( \beta \) into the intensity formula:
\[ I(\theta) = I_0 \left(\frac{\sin(0.5699/2)}{0.5699/2}\right)^2 \]
\[ I(\theta) = I_0 \left(\frac{\sin(0.28495)}{0.28495}\right)^2 \]
\[ \sin(0.28495) \approx 0.2810 \]
\[ I(\theta) = I_0 \left(\frac{0.2810}{0.28495}\right)^2 \]
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Transcribed Image Text:### Diffraction Pattern Analysis
**Problem Statement:**
Richard observes that a single slit that is 2,022 nm wide forms a diffraction pattern when illuminated by monochromatic light of 673 nm wavelength. At an angle of 10° from the central maximum, what is the ratio of the intensity to the intensity of the central maximum? Give your answer with two decimal places, please.
**Solution:**
To determine the ratio of the intensity at an angle (θ) to the intensity at the central maximum for a single slit diffraction pattern, we use the diffraction intensity formula:
\[ I(\theta) = I_0 \left(\frac{\sin(\beta/2)}{\beta/2}\right)^2 \]
where:
- \( \beta = \frac{2 \pi a \sin(\theta)}{\lambda} \)
- \( a \) is the width of the slit
- \( \lambda \) is the wavelength of the light
- \( \theta \) is the angle from the central maximum
- \( I_0 \) is the central maximum intensity.
Given:
- \( a = 2022 \) nm
- \( \lambda = 673 \) nm
- \( \theta = 10° \)
We first calculate \( \beta \):
\[ \beta = \frac{2 \pi \times 2022 \times \sin(10°)}{673} \]
\[ \sin(10°) \approx 0.1736 \]
\[ \beta = \frac{2 \pi \times 2022 \times 0.1736}{673} \]
\[ \beta \approx \frac{2201.584 \times 0.1736}{673} \]
\[ \beta \approx 0.5699 \]
Now plug \( \beta \) into the intensity formula:
\[ I(\theta) = I_0 \left(\frac{\sin(0.5699/2)}{0.5699/2}\right)^2 \]
\[ I(\theta) = I_0 \left(\frac{\sin(0.28495)}{0.28495}\right)^2 \]
\[ \sin(0.28495) \approx 0.2810 \]
\[ I(\theta) = I_0 \left(\frac{0.2810}{0.28495}\right)^2 \]
\[ \
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