1. Title and aim
Refractive index of a glass slab using a travelling microscope. Determine the refractive index of a plane-parallel glass slab from microscope measurements of real and apparent depth, then compare individual ratios with a straight-line estimate.
2. Abstract
Five observations were generated from the ideal teaching model over the settings listed below. A straight-line fit gave an estimate of refractive index of 1.5000 dimensionless, differing by 0.002% from the model value. The reported scatter comes from scale rounding and any stated model approximation, not measured laboratory noise.
3. Apparatus and fixed settings
- Travelling microscope: Focus successively on a bare mark, its image through glass and a marker on the upper surface; read the same vertical carriage scale each time.
- Plane-parallel glass slabs: Provide different real thicknesses of the same transparent material, with model refractive index 1.500 relative to air.
- Fine base mark and surface marker: Identify the lower object and the upper surface without confusing their focus positions.
- Levelled support and scale: Keep the viewing axis close to the surface normal and express all vertical differences in consistent units.
- Plane parallel ideal glass; air index taken as 1; fixed glass index n = 1.500.
- Paraxial normal viewing; focus-based apparent depth rounded to 0.01 mm.
- Aligned microscope datum: R₁ = 0; R₂ = real depth; R₃ = R₂ − apparent depth.
4. Theory and graph choice
A mark below glass appears closer to the upper surface than it really is. Use this apparent rise to connect refraction with a measurable depth. Vary slab thickness, collect simulated apparent-depth readings, and infer refractive index from their graph. The virtual apparatus uses one glass material and one fixed wavelength; changing thickness does not change its index.
The calculation fits y = mx + b in SI units. The graph may display scaled axes for readability; those display factors are removed before calculating the physical result. The gradient has units dimensionless.
5. Method
- Level the microscope and identify an upward-increasing vertical scale. Focus the crosshairs first, then focus the bare base mark and label its carriage height R1.
- Place a clean slab over the mark without moving the base. Select real thickness t in the simulator; the simplified virtual datum is R1 = 0 and R2 = t.
- In the physical method focus the apparent mark through glass to obtain R3, then a fine marker on the upper surface to obtain R2. Approach final focus from the same carriage direction. The virtual reading returns their apparent-depth difference directly.
- Measure and record five well-spaced thicknesses of the same model material. For example use 3, 6, 9, 12 and 15 mm. Repeat focus judgements in a real apparatus.
- Calculate n = t/a for one row. Convert both depths to metres for the stored fit, then fit a = st + b without forcing the intercept to zero.
- Calculate n = 1/s, inspect residuals and intercept, and state the resolution, fixed-material assumptions and omitted uncertainties. Label all virtual readings simulated.
6. Observations and calculation table
All observations use fixed model settings. Values are rounded only at the instrument reading stage; further calculations use full precision. Repeated physical trials are not claimed.
| No. | Real slab depth t (mm) | Apparent depth a (mm) | x (SI) | y (SI) | Fitted y (SI) | Residual (SI) |
|---|---|---|---|---|---|---|
| 1 | 3.0 | 2.00 | 0.00300 | 0.00200 | 0.00200 | -5.8434e-07 |
| 2 | 5.5 | 3.67 | 0.00550 | 0.00367 | 0.00367 | 2.7122e-06 |
| 3 | 8.0 | 5.33 | 0.00800 | 0.00533 | 0.00533 | -3.9912e-06 |
| 4 | 11.5 | 7.67 | 0.01150 | 0.00767 | 0.00767 | 2.6240e-06 |
| 5 | 15.0 | 10.00 | 0.01500 | 0.01000 | 0.01000 | -7.6075e-07 |
Download the sample observations (CSV) ↓
7. Worked calculation for one observation
8. Graph, gradient and final calculation
Dots show the sample observations. The line is the ordinary least-squares fit.
Σ(x − x̄)² = 9.0700e-05
Σ(x − x̄)(y − ȳ) = 6.0468e-05
m = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² = 0.66668137
b = ȳ − mx̄ = 5.4024e-07
n = 1/m = 1/(0.66668137) = 1.499967 dimensionless
9. Uncertainty and comparison
Percentage difference = |estimate − model value| / |model value| × 100 = 0.0022%. Percentage difference is not experimental uncertainty.
For n = 5 observations, Σresidual² = 3.1092e-11; s² = Σresidual²/(n − 2). Gradient standard error = √[s²/Σ(x − x̄)²] = 3.3803e-04. Intercept standard error = √[s²(1/n + x̄²/Σ(x − x̄)²)] = 3.2440e-06.
Propagating only the relevant fit coefficient gives a fit-only standard error of 7.6054e-04 dimensionless. This is a one-standard-error quantity under the regression assumptions. Deterministic rounded model data do not supply a complete physical uncertainty budget.
Microscope focusing, scale calibration, surface parallelism and parallax affect each depth. Real and apparent depth share the R₂ reading, so their errors are correlated. A vertical equal-weight fit ignores this covariance. The model uses paraxial viewing through a plane parallel transparent slab.
10. Result and discussion
Model comparison value: 1.5000 dimensionless
The observations follow the stated model relationship. Check whether the intercept and residual pattern suggest a systematic offset or a missing approximation. Agreement with the model does not independently verify its input constants.
11. Precautions and limitations
- Keep the viewing axis close to normal to the clean, parallel surfaces.
- Keep the mark and slab position fixed while obtaining the three focus heights.
- Approach each final focus from the same direction and avoid forcing the carriage.
- Use a fine surface marker and distinguish apparent depth from apparent rise.
The lesson assumes a uniform plane-parallel slab, paraxial viewing, fixed illumination and air index one. Apparent-depth values are deterministic simulated differences rounded to 0.01 mm; independent microscope focusing, vernier reading, backlash and repeated-trial scatter are not modelled. A plotted fit does not include uncertainty in real depth or covariance between shared focus readings. The reference glass-to-air index is n = 1.500. Real thickness t ranges from 3 to 15 mm. The generator uses a = t/n, rounded to 0.01 mm. Under the virtual height datum R1 = 0 and R2 = t, the apparent focus height would be R3 = t − a. These are illustrative heights, not three independently measured scale readings. The fitted x and y are real and apparent depth in metres; n is the reciprocal of the fitted gradient.
12. Conclusion
The worked example illustrates how to obtain refractive index from measurements, graph variables and a fitted coefficient. The estimate is 1.5000 dimensionless. A real experiment must additionally evaluate the apparatus-specific uncertainties and corrections described above.