To find the angular separation between the second (m=2) and third (m=3) maximum for yellow light passing through a diffraction grating, we will use the diffraction grating formula. Here are the steps to solve this:
1. Identify the given parameters:
- Wavelength of the yellow light, [tex]\(\lambda\)[/tex] = 600 nm = [tex]\(600 \times 10^{-9} \)[/tex] meters
- Grating spacing, [tex]\(d = 2.35 \times 10^{-6} \)[/tex] meters
- Orders of maxima, [tex]\(m_2 = 2\)[/tex] and [tex]\(m_3 = 3\)[/tex]
2. Use the diffraction grating formula to find the angles of the maxima:
[tex]\[
d \sin(\theta) = m \lambda
\][/tex]
3. Find the angle for the second maximum ([tex]\(m = 2\)[/tex]):
[tex]\[
\sin(\theta_2) = \frac{m_2 \lambda}{d}
\][/tex]
Substituting the given values:
[tex]\[
\sin(\theta_2) = \frac{2 \times 600 \times 10^{-9}}{2.35 \times 10^{-6}}
\][/tex]
[tex]\[
\sin(\theta_2) \approx 0.510638
\][/tex]
Therefore, [tex]\(\theta_2 \approx \arcsin(0.510638) \approx 0.53593\)[/tex] radians.
4. Convert [tex]\(\theta_2\)[/tex] from radians to degrees:
[tex]\[
\theta_2 \approx 0.53593 \text{ radians} \approx 30.71 \text{ degrees}
\][/tex]
5. Find the angle for the third maximum ([tex]\(m = 3\)[/tex]):
[tex]\[
\sin(\theta_3) = \frac{m_3 \lambda}{d}
\][/tex]
Substituting the given values:
[tex]\[
\sin(\theta_3) = \frac{3 \times 600 \times 10^{-9}}{2.35 \times 10^{-6}}
\][/tex]
[tex]\[
\sin(\theta_3) \approx 0.765957
\][/tex]
Therefore, [tex]\(\theta_3 \approx \arcsin(0.765957) \approx 0.87253\)[/tex] radians.
6. Convert [tex]\(\theta_3\)[/tex] from radians to degrees:
[tex]\[
\theta_3 \approx 0.87253 \text{ radians} \approx 49.99 \text{ degrees}
\][/tex]
7. Calculate the angular separation between the second and third maxima:
[tex]\[
\Delta\theta = \theta_3 - \theta_2
\][/tex]
[tex]\[
\Delta\theta \approx 0.87253 - 0.53593 \text{ radians} \approx 0.33660 \text{ radians}
\][/tex]
8. Convert the angular separation from radians to degrees:
[tex]\[
\Delta\theta \approx 0.33660 \text{ radians} \approx 19.29 \text{ degrees}
\][/tex]
Therefore, the angular separation between the second and third maximum is approximately 19.29 degrees.
