The Human Eye and the Colourful World — NCERT Solutions, AI Tutor & Practice

Power of accommodation is the ability of the eye to adjust the focal length of its lens so that objects at different distances form sharp images on the retina.

How it works:

• The ciliary muscles control the curvature (and hence the focal length) of the crystalline eye lens.
• When viewing distant objects, the ciliary muscles relax, the lens becomes thin, its focal length increases.
• When viewing nearby objects, the ciliary muscles contract, the lens becomes thick, its focal length decreases.

Thus the eye lens (with the help of ciliary muscles) is responsible for accommodation.

Far point: The farthest point up to which the eye can see objects clearly. For a normal eye it is at infinity.

Near point: The closest point at which the eye can see an object clearly and comfortably without strain. For a normal adult eye it is about 25 cm.

Least distance of distinct vision: The minimum distance at which an object can be seen most clearly without strain. This distance equals the near point, i.e. about 25 cm for a normal eye.

Defect: Myopia (short-sightedness) is corrected using a concave (diverging) lens.

The lens must form the image of a very distant object (at infinity) at the person's far point, so that the eye can see it.

• Object distance: $u = -\infty$
• Image distance: $v = -1.5\ \text{m}$ (far point, on same side as object)

Using the lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{-1.5} - \frac{1}{-\infty} = -\frac{1}{1.5}$ $f = -1.5\ \text{m}$

Power: $P = \frac{1}{f\,(\text{in m})} = \frac{1}{-1.5} = -0.67\ \text{D}$

A concave lens of power $-0.67\ \text{D}$ is required.

Defect: Hypermetropia (long-sightedness) is corrected using a convex (converging) lens.

The lens must form the image of an object placed at 25 cm at the person's near point (1 m), so the eye can see it.

• Object distance: $u = -25\ \text{cm} = -0.25\ \text{m}$
• Image distance: $v = -100\ \text{cm} = -1\ \text{m}$ (near point, virtual, same side)

Using the lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{-1} - \frac{1}{-0.25} = -1 + 4 = 3$ $f = \frac{1}{3}\ \text{m} = +0.33\ \text{m}$

Power: $P = \frac{1}{f} = +3\ \text{D}$

A convex lens of power $+3\ \text{D}$ is required.

Twinkling of stars is caused by atmospheric refraction of starlight.

Understanding:

• Stars are extremely far away, so they act as point sources of light.
• As starlight enters the Earth's atmosphere, it is refracted continuously through layers of air of changing density (due to temperature variations and air movement).
• The apparent position of the star keeps shifting slightly, and the amount of light entering the eye keeps changing — sometimes brighter, sometimes fainter. This is seen as twinkling.

Why planets don't twinkle:

• Planets are much closer, so they appear as extended sources — a collection of many point sources.
• The variations in light from these individual points average out, cancelling the flickering effect.

Hence planets do not appear to twinkle.

This is explained by the scattering of light by air molecules (Tyndall/Rayleigh scattering).

Key principle: Scattering is stronger for shorter wavelengths (blue/violet) than for longer wavelengths (red).

At sunrise/sunset:

• Sunlight travels through a greater thickness of atmosphere to reach us (it enters at a low angle near the horizon).
• Most of the shorter-wavelength blue light is scattered away during this long path.
• Mainly the longer-wavelength red light reaches our eyes, so the Sun appears reddish.

At noon:

• The Sun is overhead, so light passes through the least thickness of atmosphere.
• Comparatively little light is scattered, so the Sun appears nearly white.

Dispersion of light: The splitting of white light into its component colours (VIBGYOR) when it passes through a medium such as a glass prism.

How a prism produces a spectrum:

• White light is a mixture of seven colours, each with a different wavelength.
• Different colours travel at different speeds in glass, so they bend by different amounts (they have different refractive indices in glass).
• Violet (shortest wavelength) bends the most; red (longest wavelength) bends the least.
• Because of this unequal bending, the colours separate and emerge in a band called the spectrum.

The sequence of colours obtained is Violet, Indigo, Blue, Green, Yellow, Orange, Red (VIBGYOR).

A rainbow is a natural spectrum formed by the dispersion of sunlight by tiny water droplets present in the atmosphere after rain.

Formation (step by step):

1. Sunlight enters a water droplet and gets refracted (bent). During this refraction, dispersion begins as different colours bend differently.
2. The light is then internally reflected off the back inner surface of the droplet.
3. The light is refracted again as it leaves the droplet, further separating the colours.

Thus each droplet acts like a tiny prism. The combined effect of millions of droplets produces the coloured arc — a rainbow — with red on the outer edge and violet on the inner edge.

Condition: A rainbow is always seen in the direction opposite to the Sun, with the observer's back to the Sun.

Atmospheric refraction: The bending of light as it passes through layers of the Earth's atmosphere that have different densities (and therefore different refractive indices). The air near the surface is denser than the air higher up.

Advance sunrise and delayed sunset:

• When the Sun is slightly below the horizon, its light travels from the rarer (upper) atmosphere into the denser (lower) atmosphere and is refracted, bending towards the observer.
• This makes the Sun appear raised above its actual position.
• Because of this, we can see the Sun about 2 minutes before it actually rises (advance sunrise) and for about 2 minutes after it has actually set (delayed sunset).

Thus the day appears about 4 minutes longer than it actually is due to atmospheric refraction.

The blue colour of the sky is due to the scattering of sunlight by the tiny molecules of air (Rayleigh scattering).

Explanation:

• Scattering is much stronger for shorter wavelengths. Blue light (short wavelength) is scattered far more than red light (long wavelength).
• As sunlight passes through the atmosphere, blue light is scattered in all directions by air molecules.
• This scattered blue light reaches our eyes from all parts of the sky, so the sky appears blue.

(Violet is scattered even more, but our eyes are more sensitive to blue, and sunlight contains more blue, so we perceive blue.)

Without atmosphere: If the Earth had no atmosphere, there would be no scattering, so the sky would appear dark/black even in daytime, as seen by astronauts in space.

Defect: Myopia — corrected using a concave (diverging) lens.

The lens must form the image of a distant object (at infinity) at the far point of the eye (80 cm).

• Object distance: $u = -\infty$
• Image distance: $v = -80\ \text{cm} = -0.8\ \text{m}$

Lens formula: $\frac{1}{f} = \frac{1}{v} - \frac{1}{u} = \frac{1}{-0.8} - 0 = -1.25$ $f = -0.8\ \text{m}$

Power: $P = \frac{1}{f} = \frac{1}{-0.8} = -1.25\ \text{D}$

Required: a concave lens of power $-1.25\ \text{D}$ (focal length $-80\ \text{cm}$).

(a) Myopia (short-sightedness):

• The person can see nearby objects clearly but distant objects appear blurred.
• Causes: (i) excessive curvature of the eye lens (too much converging power), or (ii) the eyeball being too long (elongated).
• The image of a distant object forms in front of the retina.
• Correction: a concave (diverging) lens.

(b) Hypermetropia (long-sightedness):

• The person can see distant objects clearly but nearby objects appear blurred.
• Causes: (i) the eye lens having too little converging power (focal length too long), or (ii) the eyeball being too short.
• The image of a nearby object forms behind the retina.
• Correction: a convex (converging) lens.