To Determine the Resolving Power of a Plane Diffraction Grating
1 Aim
2 Apparatus Used
3 Diagram
(Insert experimental setup diagram here showing spectrometer, grating, and light source arrangement)
Figure: Experimental setup for determining resolving power of plane diffraction grating
4 Theory
A diffraction grating is an optical component with a periodic structure that splits and diffracts light into several beams traveling in different directions. The resolving power of a grating is its ability to separate two spectral lines of nearly equal wavelengths.
When monochromatic light falls normally on a plane diffraction grating, it gets diffracted according to the grating equation. For sodium light, which consists of a doublet with wavelengths λ₁ = 5890 Å and λ₂ = 5896 Å, the resolving power can be determined by observing the separation of these two lines in different orders of diffraction.
The theoretical resolving power of a grating is given by the product of the order of diffraction (m) and the total number of lines (N) illuminated on the grating. Higher orders of diffraction provide better resolution, allowing us to distinguish between closely spaced spectral lines.
The angular dispersion of the grating increases with the order of diffraction, making it easier to resolve the sodium doublet in higher orders. This experiment demonstrates the fundamental relationship between the physical parameters of the grating and its optical resolving capability.
5 Formula
where d = grating element, θ = angle of diffraction, m = order, λ = wavelength
where m = order of diffraction, N = total number of lines illuminated
where $\bar{\lambda}$ = mean wavelength, Δλ = difference in wavelengths
where W = width of grating illuminated, d = grating element
6 Procedure
7 Observation Table
Given Data:
- Grating specification: _______ lines per inch
- Grating element (d): _______ cm
- Width of grating illuminated (W): _______ cm
- Sodium wavelengths: λ₁ = 5890 Å, λ₂ = 5896 Å
| Order of Diffraction (m) | Left Side | Right Side | Mean Angle θ | Doublet Resolved? | Calculated λ (Å) | ||
|---|---|---|---|---|---|---|---|
| Position 1 | Position 2 | Position 1 | Position 2 | ||||
| 1 | |||||||
| 2 | |||||||
| 3 | |||||||
8 Calculations
Step 1: Calculate grating element if not given:
$$d = \frac{2.54}{N_{lines/inch}} \text{ cm}$$Step 2: Verify grating equation:
$$\lambda = \frac{d \sin \theta}{m}$$Step 3: Calculate theoretical resolving power:
$$R_t = mN = m \times \frac{W}{d}$$Step 4: Calculate practical resolving power:
$$R_p = \frac{\bar{\lambda}}{\Delta\lambda} = \frac{5893}{6} = 982.17$$Step 5: Compare theoretical and practical values:
$$\text{Percentage difference} = \frac{|R_t - R_p|}{R_t} \times 100\%$$Calculation Space:
Show detailed calculations for each order of diffraction here...
9 Result
Experimental Results:
1. Minimum order at which sodium doublet is resolved: _____ order
2. Resolving power values:
- Theoretical resolving power (Rt): _____
- Practical resolving power (Rp): 982.17
- Percentage difference: _____%
3. Verification of grating equation:
Calculated wavelength: _____ Å (Expected: ~5893 Å)
4. Conclusion:
The resolving power of the plane diffraction grating increases linearly with the order of diffraction, confirming the theoretical relationship R = mN. The sodium doublet becomes clearly resolved from the _____ order onwards.
10 Precautions
- Ensure the spectrometer is properly leveled using the spirit level before starting measurements.
- The grating surface should be clean and free from dust. Handle it carefully to avoid scratches.
- Mount the grating perpendicular to the collimator axis with grating lines vertical.
- Use minimum slit width that still provides clear visibility of spectral lines to improve resolution.
- Allow the sodium lamp to warm up completely before taking measurements for stable light output.
- Take readings from both sides of the direct beam to minimize systematic errors.
- Avoid parallax error while reading the vernier scale. Keep the eye in line with the scale.
- Do not force the telescope or prism table movements. Use gentle, smooth motions.
- Record observations immediately to avoid confusion and ensure accuracy.
- Check the alignment frequently, especially after handling the instrument.