To determine the color of the light whose second-order bright band forms an angle of [tex]\(13.5^\circ\)[/tex] in a diffraction grating with 175 lines per mm, you can follow these steps:
1. Calculate the distance between lines on the grating:
[tex]\[
d_{\text{mm}} = \frac{1}{175} \text{ mm}
\][/tex]
This yields:
[tex]\[
d_{\text{mm}} = 0.005714285714285714 \text{ mm}
\][/tex]
2. Convert the distance from millimeters to meters:
[tex]\[
d_{\text{m}} = d_{\text{mm}} \times 10^{-3}
\][/tex]
This yields:
[tex]\[
d_{\text{m}} = 0.005714285714285714 \times 10^{-3} \text{ m}
\][/tex]
[tex]\[
d_{\text{m}} = 5.7142857142857145 \times 10^{-6} \text{ m}
\][/tex]
3. Convert the angle from degrees to radians:
[tex]\[
\theta_{\text{rad}} = \theta_{\text{deg}} \times \frac{\pi}{180}
\][/tex]
For [tex]\(\theta_{\text{deg}} = 13.5^\circ\)[/tex]:
[tex]\[
\theta_{\text{rad}} = 13.5 \times \frac{\pi}{180}
\][/tex]
[tex]\[
\theta_{\text{rad}} = 0.23561944901923448 \text{ radians}
\][/tex]
4. Use the diffraction grating formula to calculate the wavelength:
The general formula for diffraction grating is:
[tex]\[
d \sin(\theta) = n \lambda
\][/tex]
For the second-order ([tex]\(n = 2\)[/tex]):
[tex]\[
\lambda = \frac{d \sin(\theta)}{n}
\][/tex]
Substituting the given values:
[tex]\[
\lambda = \frac{5.7142857142857145 \times 10^{-6} \times \sin(0.23561944901923448)}{2}
\][/tex]
[tex]\[
\lambda = 6.669867538740154 \times 10^{-7} \text{ m}
\][/tex]
5. Convert the wavelength from meters to nanometers:
[tex]\[
\lambda_{\text{nm}} = \lambda \times 10^9
\][/tex]
[tex]\[
\lambda_{\text{nm}} = 6.669867538740154 \times 10^{-7} \times 10^9
\][/tex]
[tex]\[
\lambda_{\text{nm}} = 666.9867538740154 \text{ nm}
\][/tex]
6. Determine the color based on the wavelength:
Using the given wavelength-color relationship:
- Red: 650 nm
- Yellow: 570 nm
- Green: 510 nm
- Violet: 400 nm
The calculated wavelength of 666.9867538740154 nm falls closest to the "Red" range.
Therefore, the most likely color of the light is red.
