Chapter 2: The Human Eye and the Colourful World

Solution

The least distance of distinct vision (also called the near point) for a normal, healthy human eye is 25 cm. This means the closest distance at which the eye can focus on an object without strain is 25 cm. Objects closer than this appear blurry.

Solution

The power of accommodation is the ability of the eye to change the focal length of its lens by adjusting the curvature through the action of ciliary muscles. Persistence of vision is the phenomenon where the image persists on the retina for 1/16th of a second. Dispersion is the splitting of white light.

Solution

The iris controls the amount of light entering the eye by adjusting the size of the pupil. In bright light, the iris contracts the pupil to reduce light entry. In dim light, it dilates the pupil to allow more light in. The cornea refracts light, the lens focuses it, and the retina receives the image.

Solution

In hypermetropia (long-sightedness), the near point shifts farther than 25 cm. Since this person's near point is 50 cm (greater than the normal 25 cm), they have hypermetropia. In myopia, distant objects are blurry. Presbyopia is age-related stiffening of the lens. Astigmatism is due to irregular corneal curvature.

Solution

Myopia is corrected using a concave lens. The lens must form a virtual image of a distant object (at infinity) at the person's far point (80 cm).

Using the lens formula with u = -∞ and v = -80 cm:

1/f = 1/v - 1/u = 1/(-80) - 1/(-∞) = -1/80

f = -80 cm

So a concave lens of focal length 80 cm is needed.

Solution

In the visible spectrum, red light has the maximum wavelength (approximately 620–750 nm), while violet has the minimum wavelength (approximately 380–450 nm). The order from longest to shortest wavelength is: Red > Orange > Yellow > Green > Blue > Indigo > Violet (ROYGBIV — reverse of VIBGYOR).

Solution

The sky appears blue due to Rayleigh scattering. Tiny molecules of nitrogen and oxygen in the atmosphere scatter shorter wavelengths (blue and violet) much more than longer wavelengths (red). The scattering intensity is proportional to 1/λ⁴ (inversely proportional to the fourth power of wavelength). Blue is scattered about 5.5 times more than red. Although violet is scattered even more, our eyes are more sensitive to blue, making the sky appear blue.

Solution

The Tyndall effect is the scattering of light by particles in a colloid or a fine suspension. When a beam of light passes through a colloidal solution (like fog, mist, or milk diluted in water), the colloidal particles scatter the light, making the path of the beam visible. This is different from dispersion (splitting into colours) or regular reflection.

Solution

A spectrum is the band of colours obtained when white light is dispersed (split) by a prism or any dispersive medium.

The seven colours in order of increasing wavelength are:

• Violet (shortest wavelength, ~380 nm)
• Indigo (~445 nm)
• Blue (~475 nm)
• Green (~510 nm)
• Yellow (~570 nm)
• Orange (~590 nm)
• Red (longest wavelength, ~650–750 nm)

This order is also the order of decreasing frequency and decreasing deviation by a prism. Remember: VIBGYOR (from most deviated to least deviated).

Solution

The ciliary muscles hold the eye lens in place and control its shape (curvature). Their function:

• When you look at a distant object: ciliary muscles relax → the lens becomes thin → focal length increases → parallel rays from the distant object are focused on the retina.
• When you look at a nearby object: ciliary muscles contract → the lens becomes thick and more curved → focal length decreases → diverging rays from the nearby object are focused on the retina.

This ability to adjust focal length is called the power of accommodation. Without ciliary muscles, the eye could only focus at one fixed distance.

Solution

Without an atmosphere, there would be no molecules to scatter sunlight. The sky would appear dark or black, and we would see stars even during the daytime. This is exactly what astronauts see from the Moon (which has virtually no atmosphere) — a black sky with the Sun and stars visible simultaneously.

Solution

Given: Near point = 75 cm. Normal near point = 25 cm.

The corrective convex lens must form a virtual image at 75 cm of an object placed at 25 cm.

u = -25 cm, v = -75 cm (virtual image on same side as object)

Lens formula:

1/f = 1/v - 1/u = 1/(-75) - 1/(-25) = -1/75 + 1/25

1/f = (-1 + 3)/75 = 2/75

f = 75/2 = 37.5 cm = 0.375 m

Power: P = 1/f = 1/0.375 = +2.67 D

The positive power confirms it is a convex lens, as expected for hypermetropia correction.

Solution

Given: Far point = 150 cm = 1.5 m

The concave lens must form a virtual image of a distant object (at infinity) at the far point.

u = -∞, v = -150 cm = -1.5 m

Lens formula:

1/f = 1/v - 1/u = 1/(-1.5) - 0 = -1/1.5

f = -1.5 m

Power: P = 1/f = 1/(-1.5) = -0.67 D

The negative power confirms it is a concave (diverging) lens.

Solution

Violet deviates the most because it has the shortest wavelength and therefore the highest refractive index in glass. The shorter the wavelength, the more the light is refracted (bent). Red light deviates the least because it has the longest wavelength and lowest refractive index.

Solution

At sunrise and sunset, the Sun is near the horizon. Sunlight has to travel through a much greater thickness of atmosphere compared to noon. During this long journey:

• The shorter wavelengths (blue, violet) are scattered away in all directions by atmospheric molecules. They get scattered so many times over the long path that very little blue/violet light reaches the observer.
• The longer wavelengths (red, orange) are scattered much less (scattering ∝ 1/λ⁴) and can travel through the thick atmosphere to reach our eyes.

As a result, the Sun and the sky near it appear reddish-orange at sunrise and sunset.

At noon, sunlight passes through a thin atmospheric layer, so all colours reach us and the Sun appears white.

Solution

Stars twinkle because of atmospheric refraction. Here's the step-by-step explanation:

• Stars are extremely far away and act as point sources of light.
• As starlight enters Earth's atmosphere, it passes through layers of air with varying temperatures and densities.
• Each layer refracts the light slightly differently. Since atmospheric conditions are constantly changing (wind, convection currents), the refraction fluctuates continuously.
• This causes the apparent position of the star to shift slightly, and the intensity of light reaching our eyes varies from moment to moment.
• We perceive this as twinkling.

Why planets don't twinkle: Planets are much closer and appear as small discs (extended sources), not points. Light from different parts of the disc averages out the fluctuations.

Solution

In a glass prism, the two refracting surfaces are inclined to each other at an angle (the angle of the prism). When white light enters the first surface, different colours are refracted by different amounts (dispersion begins). At the second surface, the colours are further separated because it is inclined. The net result: the colours emerge at different angles, producing a visible spectrum.

In a rectangular glass slab, the two surfaces are parallel. At the first surface, dispersion begins — the colours separate slightly. But at the second surface (which is parallel to the first), the refraction reverses the dispersion exactly. Each colour emerges parallel to its original direction. The colours recombine, and we see white light emerging with no visible spectrum.

In short: the slab undoes the dispersion; the prism does not because its surfaces are not parallel.

Solution

Presbyopia is an age-related defect of vision that usually develops after the age of 40. It happens because:

• The ciliary muscles weaken with age.
• The eye lens gradually loses its flexibility and becomes rigid.
• The power of accommodation decreases.

How it differs from myopia and hypermetropia:

• Myopia — distant objects are blurry; caused by elongated eyeball or excessive lens curvature; affects people of any age.
• Hypermetropia — nearby objects are blurry; caused by short eyeball or insufficient lens curvature; can occur at any age.
• Presbyopia — may affect both near and distant vision; caused specifically by aging of the lens and muscles. A person may develop a combination of myopia and hypermetropia.

Correction: Bifocal lenses — the upper part is a concave lens for distance vision, and the lower part is a convex lens for reading. Progressive lenses (with a gradual change in power) are also used.

Solution

Red colour is used for danger signals because red light is scattered the least by fog, smoke, and atmospheric particles.

Since scattering is inversely proportional to the fourth power of wavelength (∝ 1/λ⁴), and red has the longest wavelength among visible colours, it gets scattered the least. This means red light can travel the farthest distance through the atmosphere without being significantly diminished.

This makes red visible from a great distance even in foggy or hazy conditions — crucial for danger signals where visibility over long distances is essential for safety.

Solution

The Sun becomes visible about 2 minutes before it actually crosses the horizon (and stays visible about 2 minutes after it dips below). This is due to atmospheric refraction.

Explanation:

• The Earth's atmosphere has layers of air with gradually decreasing density (and refractive index) as we go higher.
• When the Sun is just below the horizon, its light enters the atmosphere at a very shallow angle.
• As the light travels from denser layers (near the surface) to rarer layers (higher up) — or more accurately, as it curves through the continuously varying density — the light path bends gradually toward the surface.
• This bending makes the Sun appear to be at a higher position than it actually is.
• So even when the Sun is geometrically below the horizon, its refracted image appears above the horizon.

This effectively lengthens our day by about 4 minutes (2 minutes at sunrise + 2 minutes at sunset).

Solution

A rainbow is formed when sunlight enters tiny water droplets suspended in the atmosphere. Inside each droplet, the light undergoes:

• Refraction — as it enters the droplet (from air to water)
• Dispersion — different colours refract by different amounts
• Total internal reflection — at the back surface of the droplet
• Refraction again — as it exits the droplet

The combined effect of dispersion and total internal reflection separates white light into its component colours, producing the rainbow. Neither reflection alone, refraction alone, nor scattering can produce a rainbow.

Solution

In myopia (short-sightedness), the eyeball is too long or the lens is too converging. This causes the image of a distant object to form in front of the retina instead of on it. The light converges too quickly. A concave lens is used to diverge the rays slightly before they enter the eye, pushing the image back onto the retina.

Solution

A person with normal vision has a near point at 25 cm. This is the minimum distance at which the eye can focus without strain.

Why 25 cm? At this distance, the ciliary muscles are contracted to their comfortable maximum, making the lens as thick as it can get without strain. The focal length is at its shortest, and the diverging rays from the nearby object are properly focused on the retina.

If the book is held much closer (say 10 cm):

• The rays from the object are highly divergent.
• The ciliary muscles would need to contract even more, but they are already at their limit.
• The lens cannot become thick enough to focus these rays on the retina.
• The image forms behind the retina, and the text appears blurry.
• Attempting to focus causes eye strain and headaches.

Solution

Violet light will be bent (deviated) more than red light.

Reason: The refractive index of glass for violet light (n = 1.53) is greater than that for red light (n = 1.51). A higher refractive index means the medium is optically denser for that colour, causing greater bending.

From Snell's law: sin i / sin r = n. For the same angle of incidence, a higher n gives a smaller angle of refraction — meaning the light bends more toward the normal. This is why violet is at the bottom of the spectrum (most deviated) and red is at the top (least deviated).

Solution

Given: Far point = 2 m (the person has myopia).

The corrective lens must form a virtual image of a distant object (at infinity) at the far point (2 m).

u = -∞, v = -2 m

Lens formula:

1/f = 1/v - 1/u = 1/(-2) - 1/(-∞) = -1/2 - 0 = -1/2

f = -2 m

Power: P = 1/f = 1/(-2) = -0.5 D

The negative sign indicates a diverging (concave) lens. This is correct — myopia is always corrected with concave lenses.