To determine the angular separation between the sodium D-lines in the first order spectrum when sodium light is incident normally on a grating, we can use the principles of diffraction and the grating equation. Let's break down the problem step by step.
Understanding the Grating Parameters
First, we need to gather the necessary information:
- Grating width: 4 × 10-3 m
- Total number of lines on the grating: 2000
Calculating the Grating Spacing
The grating spacing, often denoted as "d," is the distance between adjacent lines on the grating. We can find this by dividing the width of the grating by the number of lines:
d = Width of grating / Number of lines
d = (4 × 10-3 m) / 2000 = 2 × 10-6 m
The Sodium D-lines
Sodium light primarily emits two closely spaced wavelengths known as the sodium D-lines, which are:
- λ1 = 589.0 nm (D2 line)
- λ2 = 589.6 nm (D1 line)
To proceed, we need to express these wavelengths in meters:
- λ1 = 589.0 × 10-9 m
- λ2 = 589.6 × 10-9 m
Using the Grating Equation
The grating equation is given by:
d sin(θ) = mλ
where:
- d: grating spacing
- θ: angle of diffraction
- m: order of diffraction (in this case, m = 1 for first order)
- λ: wavelength of light
Calculating Angles for Each D-line
We will now find the angles for both wavelengths at the first order (m = 1).
For λ1 (589.0 nm)
Using the grating equation:
sin(θ1) = (mλ1) / d
sin(θ1) = (1 × 589.0 × 10-9) / (2 × 10-6)
sin(θ1) = 0.2945
Now, calculating θ1:
θ1 = sin-1(0.2945) ≈ 17.2°
For λ2 (589.6 nm)
Similarly, we calculate:
sin(θ2) = (mλ2) / d
sin(θ2) = (1 × 589.6 × 10-9) / (2 × 10-6)
sin(θ2) = 0.2948
Now, calculating θ2:
θ2 = sin-1(0.2948) ≈ 17.3°
Finding the Angular Separation
The angular separation between the two D-lines in the first order spectrum is simply the difference between the two angles:
Δθ = θ2 - θ1
Δθ ≈ 17.3° - 17.2° = 0.1°
Final Result
The angular separation between the sodium D-lines in the first order spectrum is approximately 0.1°.
