Refraction through a Glass Prism and Dispersion - Tenth Grade Science
When a narrow beam of sunlight traverses a triangular glass prism, it not only bends away from its original path but also fans out into a vivid band of spectral colors ranging from red to violet. This phenomenon demonstrates that white light is a composite blend of multiple wavelengths, each traveling at distinct speeds within transparent optical media.
Geometric Structure of a Triangular Prism
A triangular glass prism is an optical element bounded by two triangular parallel bases and three rectangular lateral faces inclined toward one another.
TRIANGULAR PRISM GEOMETRY
A <-- Angle of Prism
/\
/ \
/ \
/ Refra\
/ cting \
/ Face \
/ \
B--------------C
Base
Key Geometric Definitions
The Refracting Faces are the two rectangular lateral surfaces through which light enters and emerges from the prism. The Refracting Edge is the line of intersection where the two refracting faces meet. The Angle of the Prism ($A$), or refracting angle, is the dihedral angle formed between the two inclined refracting surfaces. The Base of the prism is the bottom rectangular face opposite the refracting edge $A$.
Refraction of Light through a Glass Prism
Unlike a rectangular glass slab where refracting faces are parallel (causing incident and emergent rays to exit parallel to one another with only lateral displacement), the refracting faces of a prism are tilted at an angle. This inclined geometry causes incoming rays to undergo continuous deviation toward the base.
RAY TRACING THROUGH A GLASS PRISM
A
/\
/ \
/ \
/ \
Incident Ray / \ Emergent Ray
------------->PE / EF \ FS ----------->
\ i / \ / \ e /
\ / \ / \ /
\ / \ / \ /
\ / \/ \/
Q \ R
/ \ \ / \
/ \ \ / \
B--------------------------C
<--- Base --->
PE: Incident Ray i: Angle of Incidence
EF: Refracted Ray r1, r2: Angles of Refraction
FS: Emergent Ray e: Angle of Emergence
D: Angle of Deviation A: Angle of Prism
Step-by-Step Ray Analysis
Consider a monochromatic light ray $PE$ striking the first refracting face $AB$ at point $E$. At face $AB$, the ray transitions from optically rarer air into optically denser glass. In accordance with Snell's law, it bends toward the normal $NN'$, forming angle of incidence $i$ and first angle of refraction $r_1$. The refracted ray $EF$ travels through the glass prism and strikes the second refracting face $AC$ at point $F$. At this internal interface, the ray forms an angle of incidence $r_2$ with normal $MM'$. At face $AC$, the ray exits from optically denser glass into optically rarer air. It bends away from the normal, emerging along path $FS$ as the emergent ray with angle of emergence $e$.
The Angle of Deviation ($D$)
If the incident ray $PE$ is projected forward along its original line of sight, and the emergent ray $FS$ is extended backward into the prism, their intersection defines the Angle of Deviation ($D$). The angle of deviation represents the total net angular turn experienced by the light ray in traversing the prism.
Fundamental Geometric Derivation
From the cyclic quadrilateral formed by the refracting faces and surface normals, and the interior triangle $AEF$, fundamental geometric relations emerge:
$$r_1 + r_2 = A$$
Summing the exterior deviations at both faces yields the universal relation linking prism geometry, incidence, emergence, and net deviation:
$$A + D = i + e$$
This equation states that the sum of the prism angle and the angle of deviation is identically equal to the sum of the angle of incidence and the angle of emergence.
ANGLE OF DEVIATION vs ANGLE OF INCIDENCE
Deviation (D)
^
| \ /
| \ /
| \ /
| \ /
| \ /
Dm |-------'. .'
| '------------'
| |
0-----------------im------------> Angle of Incidence (i)
At minimum deviation (Dm): i = e and r1 = r2 = A / 2
Dispersion of White Light by a Glass Prism
In 1665, Sir Isaac Newton conducted a landmark experiment demonstrating that when a narrow pencil of sunlight passes through a glass prism, it splits into a continuous spectrum of seven distinct colors: Violet, Indigo, Blue, Green, Yellow, Orange, and Red (VIBGYOR).
DISPERSION OF WHITE LIGHT INTO SPECTRUM
A
/\
/ \
/ \ Red (Least deviated)
White Light / \=============> Orange
==================> / \============> Yellow
/ Glass \===========> Green
/ Prism \==========> Blue
/ \=========> Indigo
/ \========> Violet (Most deviated)
B------------------C
Base
Optical Origin of Dispersion
White light is not a singular monochromatic entity; it is a polychromatic superposition of all wavelengths across the visible spectrum (approximately 400 nm for violet to 700 nm for red).
In a vacuum, all electromagnetic waves travel at the identical speed ($c = 3.00 \times 10^8 \text{ m/s}$). However, inside a transparent material medium like glass, light interacts with atomic electron clouds. Because of this interaction, different wavelengths travel at different phase velocities: Red light has the longest visible wavelength ($\lambda_r \approx 700 \text{ nm}$) and travels fastest through glass ($v_r$ is maximum). Because $n = c / v$, glass exhibits the lowest refractive index for red light ($n_r$ is minimum). Violet light has the shortest visible wavelength ($\lambda_v \approx 400 \text{ nm}$) and travels slowest through glass ($v_v$ is minimum). Glass exhibits the highest refractive index for violet light ($n_v$ is maximum).
According to Snell's law, a higher refractive index produces greater ray bending. Therefore, violet light is refracted and deviated most toward the prism base, while red light is deviated least. The intermediate wavelengths experience intermediate amounts of bending, separating the composite white beam into the VIBGYOR band.
VISIBLE SPECTRUM CHARACTERISTICS
Color Wavelength (nm) Speed in Glass Refractive Index Deviation
----------------------------------------------------------------------------
Red ~700 nm Fastest Lowest Least
Orange ~620 nm . . .
Yellow ~580 nm . . .
Green ~530 nm Medium Medium Medium
Blue ~470 nm . . .
Indigo ~430 nm . . .
Violet ~400 nm Slowest Highest Most
Newton's Recombination Experiment
Before Newton's work, contemporary philosophers hypothesized that the glass prism actively stained or dyed pure white light into colors. Newton decisively refuted this hypothesis through his recombination experiment using two identical prisms.
NEWTON'S RECOMBINATION EXPERIMENT
Prism 1 (Upright) Prism 2 (Inverted)
A B'
/\ /\
/ \ Red / \
White Light / \-------------------->/ \ Recombined
============> / \ / \===> White Light
/ Prism \ Violet / Inverted\
/ 1 \------------->/ Prism \
B------------C A'------------C'
Experimental Setup and Observation
Newton placed a second identical glass prism in an inverted orientation relative to the first prism. When white light entered the first upright prism, it dispersed into the divergent VIBGYOR spectrum.
These separated colored rays then entered the inverted second prism. The second prism presented opposite refracting angles, bending each color back by an equal and opposite amount. The rays recombined at the exit face, emerging as a single collimated beam of pure white light. This experiment proved that prisms do not create colors; rather, colors are intrinsic constituents of white sunlight.
Natural Dispersion: The Formation of a Rainbow
A rainbow is a spectacular natural spectrum appearing in the sky after rainfall, produced by the dispersion, refraction, and internal reflection of sunlight within millions of tiny spherical raindrops suspended in the atmosphere.
FORMATION OF A RAINBOW IN A RAINDROP
Sunlight
=======> \
\ (Refraction & Dispersion)
.---*---.
.' \
/ Water \
| Droplet | (Total Internal Reflection)
\ /
'. *
'---*---'
/ \
/ \
v v
Red Violet (Refraction into air)
Observer sees Red at 42 deg, Violet at 40 deg
Three Optical Steps in Rainbow Formation
- Refraction and Dispersion upon Entry: Parallel sunlight strikes the upper curved surface of a suspended spherical raindrop. As light enters water from air, it refracts and disperses into its constituent colors, with violet bending most and red bending least.
- Internal Reflection at the Rear Surface: The dispersed rays travel through the raindrop and strike its back internal boundary at an angle exceeding the critical angle for water ($\theta_c \approx 48.6^\circ$). Consequently, the light undergoes total internal reflection, bouncing forward toward the lower front surface.
- Refraction and Final Angular Separation upon Exit: Upon striking the lower front surface, the rays exit the droplet back into air. As they pass from water to air, they refract once more, increasing their angular divergence.
Red light emerges at an angle of approximately $42^\circ$ relative to the incoming sunlight ray, while violet light emerges at approximately $40^\circ$. To an observer standing with their back to the Sun, light emerging from thousands of droplets across the sky enters the eye, creating a circular arc with red along the outer rim and violet along the inner edge.
Chapter Exercises and Solutions
Exercise 1: A ray of light traversing an equilateral glass prism exhibits an angle of incidence of $48^\circ$ and an angle of emergence of $48^\circ$. Calculate the angle of the prism and determine the resulting angle of deviation.
Solution 1: An equilateral prism has three equal internal angles, so its refracting angle is $A = 60^\circ$. The problem specifies that the angle of incidence is $i = 48^\circ$ and the angle of emergence is $e = 48^\circ$. The universal prism relation establishes that $A + D = i + e$, where $D$ represents the angle of deviation. Substituting the values gives $60^\circ + D = 48^\circ + 48^\circ = 96^\circ$. Solving for deviation gives $D = 96^\circ - 60^\circ = 36^\circ$. Because $i = e$, this configuration corresponds to the state of minimum deviation, and the ray turns through an angle of $36^\circ$.
Exercise 2: Explain why a thick rectangular glass slab produces no spectral dispersion of white light on a viewing screen, whereas a triangular glass prism produces a wide, distinct rainbow spectrum.
Solution 2: In a rectangular glass slab, the opposing refracting interfaces are strictly parallel to one another. When a polychromatic beam of white light enters the first interface at an angle, it disperses slightly inside the glass, with violet bending more than red. However, when these separated rays reach the second parallel interface, the geometrical angles of incidence inside the glass are identical to their refraction angles at entry. Upon exiting into air, each constituent color bends away from the normal by an angle equal to the original incident angle. As a result, all colors emerge completely parallel to one another. Because the parallel colored rays overlap, they fuse back into white light, showing only slight lateral displacement at the edges. In a triangular prism, the refracting interfaces are inclined at angle $A$. The rays cannot emerge parallel; the second interface bends the separated colors even further in the same angular direction, projecting a distinct spectral band.
Exercise 3: State the essential meteorological and geometric conditions required for an observer to view a natural rainbow in the sky, and explain why the sun is never visible in the same patch of sky as the rainbow arc.
Solution 3: For a rainbow to be visible, three simultaneous conditions must be met: there must be moisture in the form of water droplets suspended in the atmosphere (such as clearing rain or a waterfall mist), the Sun must be shining brightly behind the observer, and the observer's eyes must be positioned between the Sun and the water droplets. A rainbow cannot form in the same patch of sky as the Sun because the optical path requires sunlight to undergo total internal reflection inside raindrops, redirecting the light rays backward toward the source direction at characteristic angles of $40^\circ$ to $42^\circ$. Therefore, an observer must look away from the Sun toward the rain shower to receive these reflected and dispersed rays.