Question
A discrete spectral line occurs at an angle of 11.2° 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
5.1x10-7
5.3x10-7
4.7x10-7
4.3x10-7
Group of answer choices
5.1x10-7
5.3x10-7
4.7x10-7
4.3x10-7
Answers
GPT-4o mini
To find the wavelength of light corresponding to the given angle and number of lines per centimeter in a diffraction grating, we can use the diffraction grating formula:
\[
d \sin(\theta) = m \lambda
\]
where:
- \(d\) is the distance between adjacent grating lines (grating spacing),
- \(\theta\) is the angle of diffraction,
- \(m\) is the order of the spectrum (in this case, \(m = 1\) for the first order),
- \(\lambda\) is the wavelength of the light.
1. **Calculate the grating spacing \(d\)**:
The number of lines per centimeter is given as 3,660 lines/cm. Therefore, the distance between two adjacent lines can be calculated as:
\[
d = \frac{1 \text{ cm}}{3660 \text{ lines}} = \frac{1 \times 10^{-2} \text{ m}}{3660} \approx 2.737 \times 10^{-6} \text{ m}
\]
2. **Convert the angle from degrees to radians**:
First, we need to convert the angle from degrees to radians or simply use it in sine directly.
\(\theta = 11.2^\circ\)
We can use:
\[
\sin(11.2^\circ) \approx 0.195
\]
3. **Substituting values in the grating formula**:
Now we can substitute the values into the diffraction grating equation:
\[
d \sin(\theta) = \lambda
\]
So we have:
\[
\lambda = d \sin(11.2^\circ)
\]
Plugging values in:
\[
\lambda = 2.737 \times 10^{-6} \text{ m} \times 0.195
\]
\[
\lambda \approx 5.34 \times 10^{-7} \text{ m}
\]
We can convert this to nanometers by multiplying by \(10^9\):
\[
\lambda \approx 534 \text{ nm}
\]
Thus, given the answer choices, the closest option is:
**5.3 x \(10^{-7}\) m.**
\[
d \sin(\theta) = m \lambda
\]
where:
- \(d\) is the distance between adjacent grating lines (grating spacing),
- \(\theta\) is the angle of diffraction,
- \(m\) is the order of the spectrum (in this case, \(m = 1\) for the first order),
- \(\lambda\) is the wavelength of the light.
1. **Calculate the grating spacing \(d\)**:
The number of lines per centimeter is given as 3,660 lines/cm. Therefore, the distance between two adjacent lines can be calculated as:
\[
d = \frac{1 \text{ cm}}{3660 \text{ lines}} = \frac{1 \times 10^{-2} \text{ m}}{3660} \approx 2.737 \times 10^{-6} \text{ m}
\]
2. **Convert the angle from degrees to radians**:
First, we need to convert the angle from degrees to radians or simply use it in sine directly.
\(\theta = 11.2^\circ\)
We can use:
\[
\sin(11.2^\circ) \approx 0.195
\]
3. **Substituting values in the grating formula**:
Now we can substitute the values into the diffraction grating equation:
\[
d \sin(\theta) = \lambda
\]
So we have:
\[
\lambda = d \sin(11.2^\circ)
\]
Plugging values in:
\[
\lambda = 2.737 \times 10^{-6} \text{ m} \times 0.195
\]
\[
\lambda \approx 5.34 \times 10^{-7} \text{ m}
\]
We can convert this to nanometers by multiplying by \(10^9\):
\[
\lambda \approx 534 \text{ nm}
\]
Thus, given the answer choices, the closest option is:
**5.3 x \(10^{-7}\) m.**