Question
A discrete spectral line occurs at an angle of 9.1° in the first order spectrum. If the grating has 3,660 lines/cm, what is the wavelength of this color of light?
Group of answer choices
4.3x10-7
5.3x10-7
5.1x10-7
4.7x10-7
Group of answer choices
4.3x10-7
5.3x10-7
5.1x10-7
4.7x10-7
Answers
GPT-4o mini
To calculate the wavelength of the light corresponding to a spectral line in a diffraction grating, we can use the grating equation:
\[
d \sin(\theta) = m \lambda
\]
where:
- \(d\) is the grating spacing (the distance between adjacent lines),
- \(\theta\) is the angle of the spectral line,
- \(m\) is the order of the spectrum,
- \(\lambda\) is the wavelength.
### Given Data:
- Lines per cm = 3660 lines/cm
- Angle \(\theta = 9.1°\)
- First order spectrum \(m = 1\)
### Steps to Solve:
1. **Convert the number of lines per cm to grating spacing \(d\):**
\[
d = \frac{1}{\text{lines per cm}} = \frac{1}{3660 \text{ lines/cm}} \approx 2.737 \times 10^{-4} \text{ cm} = 2.737 \times 10^{-6} \text{ m}
\]
2. **Convert the angle to radians (if necessary):**
For this calculation, we can use degrees since sine function available in calculators can take degree inputs.
3. **Compute \(\sin(\theta)\):**
\[
\sin(9.1°) \approx 0.158
\]
4. **Substitute the values into the grating equation:**
\[
d \sin(9.1°) = m \lambda
\]
\[
2.737 \times 10^{-6} \cdot 0.158 = 1 \cdot \lambda
\]
\[
\lambda \approx 2.737 \times 10^{-6} \cdot 0.158 \approx 4.319 \times 10^{-7} \text{ m} = 4.319 \times 10^{-7} \text{ m} = 4.3 \times 10^{-7} \text{ m}
\]
### Conclusion
The wavelength of the color of light corresponding to the discrete spectral line at an angle of 9.1° is approximately \(4.3 \times 10^{-7} \text{ m}\).
**Correct Answer: \(4.3 \times 10^{-7} m\)**
\[
d \sin(\theta) = m \lambda
\]
where:
- \(d\) is the grating spacing (the distance between adjacent lines),
- \(\theta\) is the angle of the spectral line,
- \(m\) is the order of the spectrum,
- \(\lambda\) is the wavelength.
### Given Data:
- Lines per cm = 3660 lines/cm
- Angle \(\theta = 9.1°\)
- First order spectrum \(m = 1\)
### Steps to Solve:
1. **Convert the number of lines per cm to grating spacing \(d\):**
\[
d = \frac{1}{\text{lines per cm}} = \frac{1}{3660 \text{ lines/cm}} \approx 2.737 \times 10^{-4} \text{ cm} = 2.737 \times 10^{-6} \text{ m}
\]
2. **Convert the angle to radians (if necessary):**
For this calculation, we can use degrees since sine function available in calculators can take degree inputs.
3. **Compute \(\sin(\theta)\):**
\[
\sin(9.1°) \approx 0.158
\]
4. **Substitute the values into the grating equation:**
\[
d \sin(9.1°) = m \lambda
\]
\[
2.737 \times 10^{-6} \cdot 0.158 = 1 \cdot \lambda
\]
\[
\lambda \approx 2.737 \times 10^{-6} \cdot 0.158 \approx 4.319 \times 10^{-7} \text{ m} = 4.319 \times 10^{-7} \text{ m} = 4.3 \times 10^{-7} \text{ m}
\]
### Conclusion
The wavelength of the color of light corresponding to the discrete spectral line at an angle of 9.1° is approximately \(4.3 \times 10^{-7} \text{ m}\).
**Correct Answer: \(4.3 \times 10^{-7} m\)**