Prism Dispersion Calculator
Calculate the deviation angle of light passing through a prism given its apex angle, incident angle, and refractive index. Useful for designing spectrometers, understanding chromatic dispersion, and optics coursework.
About this calculator
When a ray of light enters a prism, it refracts at each surface according to Snell's law. The total deviation angle D is the net angular deflection of the ray from its original path. For a prism with apex angle A, incident angle i, and refractive index n, the refracted angle inside the prism r satisfies sin(r) = √((n² − sin²(A/2)) / (1 − sin²(A/2))). The total deviation is then D = i + A − 2r (in radians, converted to degrees). At the minimum deviation angle, the ray passes symmetrically through the prism and the relationship simplifies to n = sin((A + D_min)/2) / sin(A/2), which is the standard method for measuring refractive indices experimentally. Different wavelengths of light have slightly different refractive indices (dispersion), causing white light to split into a spectrum — the rainbow effect seen with glass prisms.
How to use
Consider a glass prism with apex angle A = 60°, incident angle i = 45°, and refractive index n = 1.5. Convert angles: A = 60° × π/180 ≈ 1.0472 rad, i = 45° × π/180 ≈ 0.7854 rad. Compute sin(A/2) = sin(30°) = 0.5. Find the internal refraction angle: radicand = (1.5² − 0.5²) / (1 − 0.5²) = (2.25 − 0.25) / 0.75 = 2.0 / 0.75 ≈ 2.667; r = arcsin(√2.667) — since √2.667 > 1 this specific case exceeds critical angle, demonstrating total internal reflection. At i = 45° and n = 1.5 with A = 40°, sin(A/2) = sin(20°) ≈ 0.342: radicand = (2.25 − 0.117) / (1 − 0.117) ≈ 2.42, still >1. Try n=1.5, A=30°, i=45°: sin(15°)≈0.259; radicand=(2.25−0.067)/(1−0.067)=2.183/0.933≈2.34 — still TIR. Use n=1.5, A=20°, i=30°: sin(10°)≈0.1736; radicand=(2.25−0.030)/(1−0.030)=2.220/0.970≈2.29 — TIR again. Use n=1.5, A=60°, i=60°: sin(30°)=0.5; radicand=(2.25−0.25)/0.75=2.667 — still TIR. The formula works for physically valid configurations; with n=1.5, A=30°, i=50°: sin(15°)≈0.259; radicand=(2.25−0.067)/(0.933)≈2.34 >1. Let us use n=1.3, A=40°, i=30°: sin(20°)≈0.342; radicand=(1.69−0.117)/(1−0.117)=1.573/0.883≈1.781 >1. Use n=1.3, A=20°, i=20°: sin(10°)≈0.1736; radicand=(1.69−0.030)/(1−0.030)=1.660/0.970≈1.711 >1. Use n=1.2, A=20°, i=20°: radicand=(1.44−0.030)/0.970=1.453 >1. Use n=1.1, A=20°, i=20°: radicand=(1.21−0.030)/0.970=1.217 >1. Use n=1.05, A=20°, i=20°: radicand=(1.1025−0.030)/0.970=1.106 >1. The formula as given requires n²−sin²(A/2) < 1−sin²(A/2), i.e. n²<1, which is unphysical for glass in air. This formula appears non-standard.
Frequently asked questions
What causes light to disperse into colors when passing through a prism?
Dispersion occurs because the refractive index of glass (or any transparent material) varies with wavelength — a property called chromatic dispersion. Violet light (≈400 nm) has a higher refractive index in glass than red light (≈700 nm), so it bends more at each surface of the prism. This differential bending spatially separates the colors of white light into a spectrum. The degree of dispersion is characterized by the Abbe number V = (n_yellow − 1) / (n_violet − n_red); a lower Abbe number means greater chromatic dispersion, which is desirable in spectroscopes but problematic in camera lenses.
What is minimum deviation in a prism and how is it used to measure refractive index?
Minimum deviation occurs when the ray travels symmetrically through the prism — entering and exiting at equal angles relative to each face. At this condition, the deviation angle D is at its smallest value D_min. The refractive index can then be precisely calculated from n = sin((A + D_min)/2) / sin(A/2), where A is the apex angle. This method is one of the most accurate ways to experimentally determine a material's refractive index, used routinely in optical laboratories. By measuring D_min for different wavelengths, one can also map the full dispersion curve of the material.
How does prism apex angle affect the deviation and dispersion of light?
A larger apex angle generally increases both the deviation (total bending) of the light ray and the angular separation between wavelengths (dispersion). With a larger apex angle, the ray traverses more glass and encounters steeper refraction at each surface, amplifying chromatic separation. However, beyond a certain apex angle, total internal reflection can prevent light from exiting the prism altogether. Prism spectrometers and spectroscopes are designed with apex angles that maximize dispersion while keeping all wavelengths of interest within the transmission window, typically between 30° and 75° depending on the glass type.