Determine the refractive index of the material of a given glass prism by measuring the angle of the prism and the angle of minimum deviation (angle of minimum deviation method).
To determine the refractive index of the material of a given glass prism using a spectrometer, by measuring the angle of the prism and the angle of minimum deviation.
When a ray of monochromatic light is incident on one refracting face of a prism, it bends towards the normal on entering the denser medium (glass), travels through the prism, and bends away from the normal as it emerges from the second refracting face into air. This double refraction — governed at each surface by Snell’s law — causes the emergent ray to deviate from the direction of the original incident ray by a total angle called the angle of deviation (D).
As the angle of incidence on the first face is gradually increased from a small value, the angle of deviation D first decreases, reaches a minimum value, and then increases again as the angle of incidence is increased further. This minimum value of the deviation is called the angle of minimum deviation, Dm. It can be shown that minimum deviation occurs precisely when the ray passes symmetrically through the prism, that is, when the angle of incidence equals the angle of emergence, and the refracted ray inside the prism travels parallel to its base.
When light from the collimator falls on the apex of the prism and illuminates both refracting faces, the telescope catches the reflected ray from each face in turn. If θ is the total angle turned by the telescope between the two reflected images, then:
2A = θ ⇒ A = θ / 2
where A is the refracting angle of the prism.
Once the prism angle A and the angle of minimum deviation Dm are known, the refractive index n of the prism material is given by:
n = sin[(A + Dm) / 2] / sin(A / 2)
where:
This formula is derived directly by applying Snell’s law (n = sin i / sin r) at both refracting surfaces of the prism, together with the geometrical relation A = r1 + r2 connecting the two internal angles of refraction to the prism angle. At minimum deviation, symmetry makes i1 = i2 and r1 = r2 = A/2, which simplifies the general refraction equations to the compact single-variable formula above. Because the refractive index of glass depends on wavelength (dispersion), Dm — and hence n — is measured using a monochromatic source such as a sodium lamp to obtain a single, sharply defined value.
Figure: Spectrometer schematic showing the collimator producing a parallel beam through the slit, the prism mounted on the prism table with refracting angle A, refraction of the ray at both faces, the telescope receiving the emergent ray, and the angle of deviation D measured from the original ray direction (dashed).
| S.No. | Face | Vernier V1 (°) | Vernier V2 (°) | 2A (°) | A (°) |
|---|---|---|---|---|---|
| 1 | Face 1 reading | 115° 20′ | 295° 15′ | 120° 10′ | 60° 05′ |
| 2 | Face 2 reading | 355° 30′ | 175° 25′ |
Mean angle of the prism, A ≈ 60° (standard equilateral prism)
| S.No. | Direct reading (°) | Minimum deviation reading (°) | Dm = Difference (°) |
|---|---|---|---|
| 1 | 0° 00′ | 39° 40′ | 39.67 |
| 2 | 0° 00′ | 39° 35′ | 39.58 |
| 3 | 0° 00′ | 39° 45′ | 39.75 |
Mean Dm = (39.67 + 39.58 + 39.75) / 3 ≈ 39.67° ≈ 39.6°
Given:
Using the refractive index formula:
n = sin[(A + Dm) / 2] / sin(A / 2)
n = sin[(60° + 39.6°) / 2] / sin(60° / 2)
n = sin(49.8°) / sin(30°)
n = 0.7627 / 0.5000
n ≈ 1.52
This value is consistent with the typical refractive index of crown glass (n ≈ 1.50–1.52) for the sodium D-line.
The refractive index of the material of the given glass prism:
n = 1.52 (typical crown glass)
For the measured prism angle:
A = 60°
The angle of minimum deviation (Dm) is the smallest angle through which an incident ray is bent after passing through a prism, occurring when the ray travels symmetrically through the prism, that is, when the angle of incidence equals the angle of emergence. It is significant because at this unique position the refractive index of the prism material can be calculated directly using the simple formula n = sin[(A+Dm)/2] / sin(A/2), without needing to know the exact path of the ray inside the prism.
The spectrometer must be leveled so that the collimator, prism table, and telescope all lie in the same horizontal plane, and their rotation axes are exactly vertical and coincident. If the instrument is not level, the light ray will not travel in the horizontal plane containing the vernier scales, causing systematic errors in every angle measured and making it impossible to obtain a true angle of minimum deviation.
A sodium vapour lamp emits nearly monochromatic yellow light (the sodium D-lines at about 589 nm), which gives a single sharp, well-defined spectral line to observe through the telescope. Since the refractive index of glass varies with wavelength (dispersion), a monochromatic source ensures that only one well-defined angle of minimum deviation is measured, avoiding the blurred, overlapping images that a white light source would produce.
The angle of the prism A is measured by the reflection method: light from the collimator is made to fall on the apex of the prism so that it illuminates both refracting faces, and the telescope is turned in turn to catch the light reflected from each face. The angle between these two telescope positions equals 2A, so A is obtained by taking half of the angle turned through by the telescope.
The refractive index n of the prism material is related to the prism angle A and the angle of minimum deviation Dm by n = sin[(A+Dm)/2] / sin(A/2). A larger angle of minimum deviation for a given prism angle indicates a higher refractive index, since the material bends light more strongly, which is consistent with Snell’s law applied at both refracting surfaces of the prism.
