Refraction through a prism

Hold a triangular glass prism up to a beam of sunlight and something strange happens twice over. First, the whole beam bends — it does not pass straight through the way it would through a slab of window glass. Second, if the light is white, a fan of colours spills out the far side: red at one edge, violet at the other, the rest of the rainbow strung out between them. Isaac Newton settled a century-old argument about this in 1666 with nothing more elaborate than two prisms and a dark room, showing that white light is not "purified" or "coloured" by the glass — it is taken apart into components that were in it all along. The bending itself is the more disciplined part of the story, and it is what CBSE Class 12 asks you to master. A prism refracts light not once but twice — entering the first slanted face and leaving the second — and the two refractions do not simply add up to something easy to predict at a glance. As you tilt the incoming ray through different angles of incidence, the total bending, called the angle of deviation $\delta$ , traces out a curve with a single lowest point: the angle of minimum deviation, $\delta_m$ . That one special angle turns out to be locked to the prism's own geometry (its refracting angle $A$ ) and to the refractive index $n$ of the glass by a compact formula — one of the cleanest experimental routes to measuring $n$ in a school laboratory. This section builds that formula from the two refracting surfaces up, using only the geometry of triangles and Snell's law you already know from thin lenses, then puts it to work in two worked numericals and a set of the mark-losing traps examiners watch for every year.

Key takeaways

A prism has two refracting surfaces meeting at the angle of prism $A$ ; light bends at each surface, and the total bend from incident-ray direction to emergent-ray direction is the angle of deviation $\delta$ .
As the angle of incidence $i_1$ is varied, $\delta$ first falls, touches a single minimum $\delta_m$ , then rises again — a graphical fact, not a monotonic trend.
At minimum deviation the ray inside the glass runs symmetrically, parallel to the base, with $i_1=i_2$ and $r_1=r_2=A/2$ .
The prism formula links geometry to refractive index: $n=\dfrac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin\left(\frac{A}{2}\right)}$ , derived directly from $A=r_1+r_2$ and $\delta=i_1+i_2-A$ .
Measuring $A$ (fixed, geometric) and $\delta_m$ (found by rotating the prism on a spectrometer table) gives $n$ without needing any curved surface or lens.
A prism's $n$ (and so its $\delta_m$ ) depends weakly on wavelength — violet bends more than red — which is why a prism spreads white light into a spectrum (dispersion), while the prism formula itself is written and used one wavelength at a time.
Key idea: because a prism bends light at two inclined surfaces, the deviation dips to a single minimum $\delta_m$ where the internal ray runs symmetrically and $i_1=i_2=r+A/2$ -driven algebra collapses into $n=\sin\left(\frac{A+\delta_m}{2}\right)/\sin\left(\frac{A}{2}\right)$ , the standard way to measure a transparent material's refractive index.

The core insight

A prism's two inclined refracting surfaces bend a ray twice, the total deviation dips to one minimum $\delta_m$ where the internal ray runs symmetrically ( $i_1=i_2$ , $r_1=r_2=A/2$ ), and this symmetry collapses the geometry into the prism formula $n=\sin\left(\frac{A+\delta_m}{2}\right)/\sin\left(\frac{A}{2}\right)$ used to measure refractive index.

Where it fits in the syllabus

In Unit IX, "Ray Optics and Optical Instruments," refraction through a prism is the topic that bridges the lens work (thin lens formula, lens maker's formula, combination of thin lenses in contact) and the optical-instruments topic that follows (microscopes and telescopes, which use lens combinations to form images). CBSE's 2025-26 prescription is explicit that a full derivation of the prism formula is optional — the required core is the qualitative and graphical picture: how $\delta$ varies with $i_1$ , the existence and meaning of $\delta_m$ , and the ability to use the prism formula relating $A$ , $\delta_m$ , and $n$ . This article gives the short geometric derivation anyway, because it uses nothing beyond triangle angle sums and Snell's law already familiar from the earlier refraction topics, and because seeing where the formula comes from makes it far harder to misapply on an exam.

The physical picture

A prism is a transparent medium — glass, in almost every school example — bounded by two plane refracting surfaces inclined to each other at the angle of prism $A$ (also called the refracting angle). Unlike a rectangular glass slab, whose two faces are parallel so a ray shifts sideways but exits parallel to how it entered, a prism's two faces meet at an angle, so a ray genuinely changes direction on its way through. Follow one ray. It strikes the first face at angle of incidence $i_1$ (measured from the normal to that face) and refracts towards the normal, since it is entering the denser glass; call the refraction angle $r_1$ . It crosses the glass and strikes the second face from inside, at an internal angle $r_2$ , then refracts away from the normal as it leaves into air, emerging at angle $i_2$ . The overall angle between the original incident-ray direction and the final emergent-ray direction is the angle of deviation, $\delta$ . Two purely geometric facts tie these four angles together, and both come from nothing more than "angles in a triangle add to $180°$ " applied to the triangle formed by the ray path and the two refracting surfaces inside the prism:

A=r_1+r_2

\delta=(i_1-r_1)+(i_2-r_2)=i_1+i_2-A

where: $A$ is the fixed geometric angle of the prism, $r_1$ and $r_2$ are the internal refraction angles at the first and second surface, and $i_1$ , $i_2$ are the angles of incidence and emergence in air. The physical reading of the second equation is that the total deviation is simply how much bigger the sum of the two air-side angles is than the prism's own angle — each surface contributes its own share of bending, $(i_1-r_1)$ at entry and $(i_2-r_2)$ at exit, and the two add. If you now imagine slowly increasing $i_1$ from a very small (grazing-normal) value up towards $90°$ (grazing the face), and at each stage work out $i_2$ from Snell's law and hence $\delta$ from the equation above, you get a curve, not a straight line. $\delta$ starts large, falls as $i_1$ increases, reaches a single lowest value $\delta_m$ at one particular $i_1$ , and then rises again as $i_1$ approaches grazing incidence. This is the graphical result CBSE asks you to know qualitatively: deviation versus angle of incidence is a curve with one minimum, not a monotonic rise or fall.

Figure

A light ray entering the left face of a triangular prism bending toward the normal then travelling through the glass and bending away from the normal as it leaves the right face into air, with the angle of prism at the top vertex and the total deviation angle marked between the original and final ray directions

Refraction through a prism: minimum deviation and the prism formula

Derivation AWhy the prism formula only holds at minimum deviation

The goal is to show that at the single special angle of incidence where deviation is smallest, the messy four-angle relationship $A=r_1+r_2$ , $\delta=i_1+i_2-A$ collapses into one clean equation connecting only $A$ , $\delta_m$ and the refractive index $n$ — and to see exactly why that collapse needs the minimum-deviation condition and nothing less.

Step 1. Start from the two geometric relations already established: $A=r_1+r_2$ and $\delta=i_1+i_2-A$ . These hold for any angle of incidence $i_1$ , not just at minimum deviation — they are pure geometry of the triangle formed by the ray path and the two faces, true before we bring in the graph of $\delta$ against $i_1$ at all.

Step 2. Bring in the graphical fact from the physical picture above: as $i_1$ is varied, $\delta$ dips to a single minimum $\delta_m$ . At that turning point of the curve, it is a general property of any such symmetric refraction path that the ray retraces the same geometry if you reverse it — swap "in" for "out" and the picture is identical. This symmetry forces $i_1=i_2$ at the minimum (call this common value $i$ ), and correspondingly $r_1=r_2$ (call this common value $r$ ). Physically, the ray inside the glass runs parallel to the prism's base at exactly this angle of incidence.

Step 3. Substitute the symmetry into the two geometric relations. From $A=r_1+r_2=2r$ , we get $r=A/2$ . From $\delta=i_1+i_2-A=2i-A$ , evaluated at the minimum, we get $\delta_m=2i-A$ , so $i=\dfrac{A+\delta_m}{2}$ .

Step 4. Apply Snell's law at the first surface, $\sin i_1=n\sin r_1$ , which at minimum deviation reads $\sin i=n\sin r$ . Substitute the Step 3 results for $i$ and $r$ :

n=\frac{\sin i}{\sin r}=\frac{\sin\left(\dfrac{A+\delta_m}{2}\right)}{\sin\left(\dfrac{A}{2}\right)}

This is the prism formula. Notice it required the minimum-deviation symmetry at Step 2 — without $i_1=i_2$ , the general relations from Step 1 involve four separate angles and cannot be collapsed into one equation with only $A$ , $\delta_m$ and $n$ . This is exactly why the formula must not be used with an arbitrary, non-minimum pair of $i_1$ and $\delta$ . Check: for $A=60°$ and a typical crown-glass value $n=1.5$ , the formula gives $\sin\left(\frac{60+\delta_m}{2}\right)=1.5\sin(30°)=0.75$ , so $\frac{60+\delta_m}{2}=48.6°$ and $\delta_m\approx37.2°$ — a physically reasonable deviation, comfortably less than $A$ itself would suggest without any bending correction.

A related, familiar fact — needed only qualitatively at this level — is that a prism's refractive index depends slightly on the wavelength of light, larger for violet than for red. Since $n$ enters the prism formula, this means violet light has a slightly larger $\delta_m$ than red light for the same prism, so the two colours emerge along very slightly different directions. Sending white light through a prism therefore does not produce one bent white beam but a fan of colours — red bending least, violet bending most — the phenomenon of dispersion. The prism formula itself, and the worked examples below, always refer to a single wavelength (a single $n$ ) at a time; the wavelength-dependence is used here only to explain why prisms make rainbows, not as a separate quantitative topic.

Worked examples

Board level - refractive index from a measured minimum deviation

A prism has a refracting angle of $60°$ . When light passes through it at minimum deviation, the angle of minimum deviation is found to be $30°$ . Calculate the refractive index of the prism material.

Step 1. Write down what is given: $A=60°$ , $\delta_m=30°$ . The quantity to find is $n$ , and since both $A$ and $\delta_m$ are known, the prism formula applies directly — no need to solve for $i$ or $r$ separately.

Step 2. Substitute into the prism formula:

n=\frac{\sin\left(\dfrac{A+\delta_m}{2}\right)}{\sin\left(\dfrac{A}{2}\right)}=\frac{\sin\left(\dfrac{60°+30°}{2}\right)}{\sin\left(\dfrac{60°}{2}\right)}=\frac{\sin(45°)}{\sin(30°)}

Step 3. Evaluate the sines and divide: $\sin(45°)=0.707$ , $\sin(30°)=0.5$ , so

n=\frac{0.707}{0.5}=1.414

The refractive index of the prism material is approximately $\mathbf{1.41}$ , close to that of a typical glass or a mineral like fluorite. This value could then be cross-checked against tables of known standard materials to identify what the prism is made of.

Competitive level - minimum deviation from a known glass and prism angle

A prism made of glass with refractive index $1.5$ has a refracting angle of $40°$ . Find the angle of minimum deviation for this prism.

Step 1. Write down what is given: $n=1.5$ , $A=40°$ . This time $n$ and $A$ are known and $\delta_m$ is the unknown, so the prism formula must be rearranged to isolate $\delta_m$ rather than used directly.

Step 2. Rearrange and substitute:

\sin\left(\frac{A+\delta_m}{2}\right)=n\sin\left(\frac{A}{2}\right)=1.5\times\sin(20°)=1.5\times0.342=0.513

Step 3. Take the inverse sine, then solve for $\delta_m$ :

\frac{A+\delta_m}{2}=\sin^{-1}(0.513)\approx30.9°\ \implies\ A+\delta_m\approx61.8°

\delta_m\approx61.8°-40°=21.8°

The angle of minimum deviation is approximately $\mathbf{21.8°}$ . Once $n$ and $A$ are fixed by the material and how the prism is cut, $\delta_m$ is completely determined — a useful design check for choosing a prism for a particular optical purpose, such as a spectrometer prism table. Timing note: this is a two-line rearrangement once you recognise which two of the three quantities $A$ , $\delta_m$ , $n$ are given — spend your time deciding what to isolate, not re-deriving the formula. Under exam pressure, write the prism formula once at the top of your working, box which symbol is unknown, then substitute; this avoids the common slip of plugging $n$ and $A$ into the wrong side of the equation.

Common traps

Assuming deviation increases (or decreases) monotonically with angle of incidence — the correct graphical picture is a curve with a single minimum, $\delta_m$ , not a straight monotonic trend, and examiners often ask you to sketch or read this curve.
Forgetting that the prism formula applies specifically at minimum deviation; plugging in an arbitrary (non-minimum) angle of incidence and deviation into this formula gives an incorrect refractive index, because the derivation above needed the symmetry $i_1=i_2$ , $r_1=r_2$ to collapse to one equation.
Mixing up angle of prism $A$ with angle of incidence or angle of deviation — $A$ is a fixed geometric property of the prism itself (the angle between its two refracting faces), unrelated to how the light ray happens to be aimed at it.
Forgetting the symmetry condition at minimum deviation (ray inside the prism parallel to the base, angle of incidence equals angle of emergence) — this is what makes locating $\delta_m$ experimentally practical: you rotate the prism and watch for the point where the deviated ray reverses its direction of drift on the far screen.
Treating dispersion as part of the prism-formula numericals — the formula itself is written for one wavelength (one $n$ ) at a time; the spread of colours is a separate qualitative consequence, not something you compute with a single $\delta_m$ value.

Common trapthe prism formula is not a general refraction equation — it is a special-case result valid only at the one angle of incidence that gives minimum deviation

The single biggest mark-loser is using $n=\sin\left(\frac{A+\delta_m}{2}\right)/\sin\left(\frac{A}{2}\right)$ with a $\delta$ that is not the minimum deviation, or with an $i_1$ that is not the corresponding symmetric angle. At any other angle of incidence, $i_1\ne i_2$ and $r_1\ne r_2$ in general, so the neat substitution in the derivation above simply does not apply — you would need the full pair of relations $A=r_1+r_2$ and $\delta=i_1+i_2-A$ together with Snell's law at each surface separately, which is well beyond what this syllabus requires. Always check (or be told) that the given $\delta$ is specifically the minimum deviation before reaching for the prism formula.

You should be able to…

Describe qualitatively, and sketch, how deviation through a prism varies with angle of incidence, including the existence of a single minimum.
State the prism formula and correctly identify each symbol: $A$ (angle of prism), $\delta_m$ (angle of minimum deviation), $n$ (refractive index).
Reproduce, or at least follow, the short geometric derivation connecting $A=r_1+r_2$ and $\delta=i_1+i_2-A$ to the prism formula via the minimum-deviation symmetry $i_1=i_2$ , $r_1=r_2=A/2$ .
Use the prism formula to find $n$ given $A$ and $\delta_m$ , or to find $\delta_m$ given $A$ and $n$ , rearranging confidently in either direction.
Explain how the minimum-deviation condition can be located experimentally (rotating the prism, watching the deviated ray reverse drift) and used to determine a material's refractive index.
Explain, qualitatively, why a prism disperses white light into a spectrum, and why the prism formula itself is applied one wavelength at a time.

Ask the tutor: want to see the deviation-versus-incidence curve traced out point by point, or need a hand rearranging the prism formula for a tricky unknown — ask the AI tutor to work through it with you.

Why does the prism formula only work at minimum deviation, and not at just any angle I shine the light in?

Because the clean formula comes from a shortcut: at minimum deviation, and only there, the ray path is symmetric, meaning the angle of incidence equals the angle of emergence and the two internal refraction angles are equal too. That symmetry is what lets the general relations $A=r_1+r_2$ and $\delta=i_1+i_2-A$ collapse down to a single equation with just $A$ , $\delta_m$ and $n$ in it, as shown step by step in the derivation above. At any other angle of incidence, the four angles $i_1$ , $r_1$ , $r_2$ , $i_2$ are all different from each other, and you would need Snell's law at both surfaces separately to work anything out — there is no single-line formula for that general case at this level.

How do you actually measure the minimum deviation angle in a real lab?

On a spectrometer, you mount the prism on a rotating table, shine a fixed collimated beam onto one face, and watch the deviated ray through a telescope that can be rotated to follow it. As you slowly rotate the prism table, the deviated ray's direction sweeps across the field of view, slows down, momentarily reverses its apparent direction of drift, and then sweeps back the other way. The reversal point is the minimum deviation — you read off the telescope angle there and compare it with the straight-through (undeviated) direction to get $\delta_m$ directly, then plug it and the known $A$ into the prism formula to find $n$ .

If white light splits into colours through a prism, does that mean the prism formula gives a different answer for red and violet light?

Yes, in the sense that $n$ itself is slightly different for red and violet light in the same glass — real glass is very slightly more refractive for shorter (violet) wavelengths than for longer (red) ones. Since the prism formula has $n$ inside it, a slightly larger $n$ for violet means a slightly larger $\delta_m$ for violet, which is exactly why violet bends a touch more than red and the colours fan out. But for any single numerical problem, you use one fixed $n$ for one specific wavelength or colour — the prism formula itself does not model the colour-splitting directly, it just explains, qualitatively, why splitting happens at all.

Is the derivation of the prism formula required for my board exam, or is it optional like the syllabus note says?

The CBSE 2025-26 prescription explicitly marks the full derivation as optional, so you will not be penalised for not reproducing every step of it. What is expected is the qualitative and graphical understanding — that deviation has a single minimum, what the symmetry condition at that minimum looks like physically, and confident use of the prism formula itself to solve for $n$ , $A$ , or $\delta_m$ given the other two. Knowing the short derivation given here is still worth it, because it makes the "only at minimum deviation" restriction obvious rather than something you have to memorise as a rule with no reason behind it.