Dispersion through Prisms

What is Dispersion?

When white light passes through a prism, it splits into seven colours — that’s dispersion. The prism does this because the refractive index of glass depends on the wavelength of light. Red light bends the least, violet bends the most, and the rest fan out in between. The phenomenon is the working principle behind spectrometers, the rainbow, and the iconic Pink Floyd album cover.

For board exams and JEE/NEET, dispersion sits at the intersection of geometric optics and wave optics. We need refractive index, deviation through a prism, the angle of minimum deviation, and how dispersive power varies between glasses. Let’s get the formulas straight first, then work through the standard problem types.

A prism has two refracting surfaces meeting at the apex, with angle $A$ between them. Light entering one face refracts, traverses the glass, and refracts again at the second face. The total bending is the angle of deviation $\delta$ .

Key Terms & Definitions

Refracting angle of prism ( $A$ ): the angle at the apex between the two refracting faces.

Angle of incidence ( $i$ ): angle the incoming ray makes with the normal at the first face.

Angle of emergence ( $e$ ): angle the outgoing ray makes with the normal at the second face.

Angle of deviation ( $\delta$ ): angle between the original direction of the incoming ray and the final direction of the outgoing ray. Given by $\delta = i + e - A$ .

Minimum deviation ( $\delta_m$ ): the smallest deviation, occurring when $i = e$ (symmetric path through the prism).

Dispersive power ( $\omega$ ): measure of how much a prism spreads colours, defined as $\omega = (\mu_v - \mu_r)/(\mu - 1)$ where $\mu_v$ , $\mu_r$ are violet and red refractive indices and $\mu$ is the mean.

Methods & Concepts

Prism formula at minimum deviation

$\mu = \frac{\sin\left(\frac{A + \delta_m}{2}\right)}{\sin(A/2)}$

This is the most-tested formula in the chapter. It assumes the path through the prism is symmetric. To recognise minimum deviation, look for “symmetric ray”, " $i = e$ ", or " $r_1 = r_2 = A/2$ ".

Thin prism approximation

For small prism angles ( $A < 10^\circ$ ), the deviation simplifies to

$\delta = (\mu - 1)A$

Two thin prisms in contact give net deviation $\delta_{\text{net}} = (\mu_1 - 1)A_1 + (\mu_2 - 1)A_2$ . Setting this to zero gives achromatic combination (no net deviation but residual dispersion). Setting net dispersion to zero gives direct vision (no net dispersion but residual deviation).

Angular dispersion

The angular spread between violet and red after passage through a thin prism is

$\theta = (\mu_v - \mu_r)A$

Combining gives dispersive power $\omega = \theta/\delta = (\mu_v - \mu_r)/(\mu - 1)$ .

Solved Examples

Example 1 (CBSE — Easy)

A glass prism of refracting angle $60^\circ$ has refractive index $1.5$ . Find the angle of minimum deviation.

Use the prism formula. $\sin((A+\delta_m)/2) = \mu \sin(A/2) = 1.5 \times \sin 30^\circ = 0.75$ .

$(A+\delta_m)/2 = \sin^{-1}(0.75) \approx 48.6^\circ$ .

$\delta_m = 2(48.6) - 60 = 37.2^\circ$ .

Example 2 (JEE Main — Medium)

A thin prism of angle $5^\circ$ produces deviation $2.5^\circ$ . Find the refractive index.

$\delta = (\mu - 1)A \Rightarrow 2.5 = (\mu - 1)(5) \Rightarrow \mu - 1 = 0.5 \Rightarrow \mu = 1.5$ .

Example 3 (JEE Advanced — Hard)

Two thin prisms of angles $A_1 = 4^\circ$ ( $\mu = 1.5$ ) and $A_2 = ?$ ( $\mu = 1.6$ ) are combined for direct vision (zero net deviation). Find $A_2$ .

For zero net deviation: $(\mu_1 - 1)A_1 = (\mu_2 - 1)A_2 \Rightarrow 0.5(4) = 0.6\,A_2 \Rightarrow A_2 = 10/3 \approx 3.33^\circ$ .

The two prisms must be oriented oppositely (apex of one against base of the other).

Exam-Specific Tips

For CBSE 12 boards, the prism formula at minimum deviation is a guaranteed 3-mark derivation question. Practise the geometric proof using the quadrilateral argument.

For JEE Main, dispersive power and achromatic vs direct-vision combinations appear regularly. Memorise: achromatic = zero dispersion, direct vision = zero deviation.

NEET tests the qualitative property that violet bends most, red bends least. Combined with TIR critical angle questions for different colours.

Common Mistakes to Avoid

1. Using $\delta = (\mu - 1)A$ for any prism. This formula is only for thin prisms with small $A$ . Don’t apply it for $A = 60^\circ$ — use the full prism formula.

2. Confusing achromatic with direct-vision combinations. Achromatic kills dispersion (still has deviation). Direct-vision kills deviation (still has dispersion).

3. Forgetting that violet refracts more than red. $\mu$ is larger for violet, so $\delta$ is larger for violet. The order in the spectrum from least to most deviated is ROYGBIV.

4. Treating the angle of incidence on the second face as $A$ minus the refraction angle on the first face. It is not $A - r_1$ ; it equals $A - r_1$ only because of the geometry of the triangle inside the prism. Verify by drawing.

5. Mixing up $\delta_m$ and $\delta$ . $\delta_m$ is a specific value (the minimum). General $\delta$ is a function of $i$ .

Practice Questions

Q1. A prism with $A = 60^\circ$ and $\mu = \sqrt{2}$ . Find $\delta_m$ .

$\sin((60+\delta_m)/2) = \sqrt{2}\sin 30^\circ = 0.707$ . $(60+\delta_m)/2 = 45^\circ$ , so $\delta_m = 30^\circ$ .

Q2. A thin prism of $A = 6^\circ$ and $\mu = 1.5$ in contact with another thin prism for achromatic combination. If second has $\mu_v - \mu_r = 0.018$ , $\mu = 1.6$ , find $A_2$ given first prism has $\mu_v - \mu_r = 0.013$ .

Equate dispersions: $0.013(6) = 0.018\,A_2 \Rightarrow A_2 \approx 4.33^\circ$ (opposite orientation).

Q3. Why does a prism produce a spectrum but a glass slab does not?

A slab refracts each colour twice but in opposite directions, cancelling the dispersion. A prism refracts both times in the same rotational sense, so dispersion adds.

Q4. Refractive index of glass for red is $1.514$ and for violet is $1.523$ . Find dispersive power if mean $\mu = 1.519$ .

$\omega = (1.523 - 1.514)/(1.519 - 1) = 0.009/0.519 \approx 0.017$ .

Q5. For a prism at minimum deviation, what is the relation between $r_1$ and $r_2$ ?

$r_1 = r_2 = A/2$ . The ray inside the prism is parallel to the base.

Q6. Will a prism placed in water disperse light more or less than in air?

Less. The relative refractive index $\mu_{\text{glass/water}}$ is smaller than $\mu_{\text{glass/air}}$ , so deviations and dispersions both shrink.

Q7. Two prisms have the same refracting angle but different refractive indices. Which produces a wider spectrum?

The one with larger $\mu_v - \mu_r$ (i.e. larger dispersive power), regardless of mean $\mu$ .

Q8. What does a rainbow demonstrate?

Dispersion through water droplets, with one internal reflection. Primary rainbow has red on top, violet at bottom.

FAQs

Q: Why does violet bend more than red?

Refractive index increases as wavelength decreases. Violet has shorter wavelength than red, so it sees a larger $\mu$ and bends more.

Q: What is “deviation without dispersion”?

The achromatic combination of two prisms — net dispersion is zero but a deviation remains. Used in achromatic doublet lenses.

Q: Can a prism produce a single colour?

Only if the input light is already monochromatic (laser, sodium lamp). White light always disperses.

Q: Does dispersion happen at any refracting surface?

Yes, but the effect is much smaller in slabs because the second refraction undoes the first. Prisms are designed so the two refractions add up.

Q: How is $\delta_m$ used to measure $\mu$ in the lab?

Use a spectrometer to measure $A$ and $\delta_m$ , then plug into the prism formula. This is one of the standard CBSE Class 12 lab experiments.

Q: Why are most camera lenses achromatic doublets?

Single lenses suffer from chromatic aberration — different colours focus at different distances. An achromatic doublet (two glasses with different dispersive powers) cancels this.

Q: Is dispersion useful in spectroscopy?

Yes. By measuring how much each wavelength is deviated, we identify chemical elements via their characteristic spectra.

Q: Does a hollow glass prism filled with water produce a spectrum?

A small one. The relative refractive index of water with respect to air is around $1.33$ , less than glass, so the deviation and dispersion are smaller but visible.