According to the laws of reflection, the angle of reflection is equal to the angle of incidence. The laws also state that the incident ray, reflected ray and the normal to the reflecting surface at the point of incidence all rest in the same plane. These laws can be applied for light reflecting on both spherical and plane surfaces. This section explores the laws of reflection of light for a curved surface or spherical mirrors. The normal for a curved surface or mirror is normal to the tangent of the surface at the point of incidence.

A spherical mirror’s geometric centre is known as its pole, and that of a spherical lens is its optical centre. A spherical mirror’s pole and centre of curvature are joined together by a line called the principal axis. In spherical lenses, the principal axis joins the optical centre with its principal focus.

a. Sign convention

A sign convention is adopted to measure distances to determine the relevant formulae for the reflection and refraction of light by different spherical surfaces. The Cartesian sign convention is used in the NCERT Solutions for Class 12 Ray Optics and Optical Instruments. While following this convention, all distances are to be taken from the pole of the mirror or the lens’ optical centre. If the distances are measured in the same direction as the ray of incidence, they are taken as positive and if they are measured in the opposite direction of the incident ray, they are taken as negative. Distances measured upwards with respect to the x-axis and the normal to the spherical surface’s principal axis are taken as positive, while the distances measured downwards are taken as negative. A commonly accepted convention makes things convenient as a single formula can be used for different types of cases.

b. Focal Length of Spherical Mirrors

This section of the NCERT Solutions for Ray Optics and Optical Instruments chapter covers what happens when a parallel beam of light is incident on a concave mirror and a convex mirror. The incident rays are taken to be paraxial, meaning that they touch the mirrors at points close to the mirror’s pole P, making an angle with the principal axis. For a concave mirror, the reflected rays converge at a point F on the principal axis. The reflected rays diverge from a point F on the principal axis of a convex mirror. The point F, from where the rays converge or diverge, is known as the principal focus of the mirror. If the parallel paraxial beam of light makes an angle with the principal axis, the reflected rays would converge or diverge from a point in a plane through F normal to the principal axis. This is known as the focal plane of the mirror.

c. The Mirror Equation

If the rays originating from a point meet at a different point after reflection or refraction, this point provides the image of the first point. If the rays converge at the point, the image is real. If the rays diverge from the point, the image is virtual. Therefore, an image is established by the point-to-point correspondence with the object through reflection or refraction. The mirror equation is given by: 1/v + 1/u = 1/f.

When a beam of light enters a medium of different density, some part of the light enters the medium, while the remaining is reflected back into the first medium and changes the direction of the light rays entering the second medium. This change in the direction of propagation of light is known as refraction of light. Snell’s laws of refraction state the following:

• The incident ray, the refracted ray, and the normal at the point of incidence lie in the same plane.
• The ratio of the sine of the angle of incidence (i) to the sine of angle of refraction (r) is constant, i.e. , where n21 is the refractive index of the second medium with respect to the first. If medium 2 is optically denser than medium 1, the refractive index will be greater than 1 and the light will bend towards the normal. However, the opposite is true if medium 1 is optically denser than medium 2.

When light travels and enters an optically rarer medium from a denser medium, it is partially reflected back into the same medium and partially refracted to the second one. This is called internal reflection. In such a case, the light bends away from the normal. The angle of refraction keeps increasing as the angle of incidence increases, until the angle of refraction reaches π/2. The refracted ray is then bent so far away from the normal that it touches the surface at the point between the two media. After a certain point of increasing the angle of incidence, refraction is not possible. This results in the total reflection of the incident ray called total internal reflection. For the given pair of media, the angle of incidence corresponding to an angle of refraction 90º is called the critical angle ic. Usually, when a surface reflects light, some fraction of the light gets transmitted, regardless of how smooth the reflecting surface is. However, in total internal reflection, there is no transmission of light.

1. Total Internal Reflection in Nature and Its Technological Applications
2. Mirage: On hot days, the air near the ground surface is hotter than the air above it. With a rise in air density, the refractive index of the air also rises. Cooler air is more dense and has a greater refractive index than hot air. Light from a tall object travels through a medium whose refractive index decreases closer to the ground, causing the ray of light to bend away from the normal and undergoing total internal reflection, if the critical angle is greater than the angle of incidence for the air near the ground. This makes it look like the light is coming from below the ground, tricking the human eye into believing that the light is being reflected from the ground by a pool of water near the tall object. These inverted images of distant tall objects cause an optical illusion, and the phenomenon is called a mirage.
3. Prism: Prisms are designed to bend light by 90º or by 180º. They work on the principle of total internal reflection to invert images without changing their size. In the first two cases, the critical angle ic for the material of the prism must be less than 45º.
4. Diamond: The brilliance of diamonds is created by expert diamond cutters who can cut it in a way that the light entering the diamond undergoes total internal reflection. The critical angle for diamond-air interface is ≅ 24.4°, which is very small, making it an ideal surface for total internal reflection. The more the total internal reflections occur, the more sparkles a diamond exhibits.

So far, we have learned about the refraction at a plane surface. Now we will discuss the refraction at a spherical surface between two transparent media.

a. Refraction at a Spherical Surface

Let’s look at what happens when refraction takes place on a spherical surface. When light travels from a rarer to a denser medium, it changes its direction and speed. The first medium has a refractive index n1 and the second medium has a refractive index n2. An object O is placed at a distance from the spherical surface and a ray of light from O is incident on the spherical mirror. The ray bends towards the normal if it’s moving from a rarer medium to a denser medium, forming an image.

The relation between the object and image distance in terms of the refractive index of the medium and the radius of the curved spherical surface’s curvature is given by:

b. Refraction by a Lens

The image formation by a double convex lens consists two convex lenses placed in front of each other. The image is formed in two steps: the first refracting surface forms an image I1 of the object O, which in turn acts as a virtual object for the second surface. The second surface forms the final image I. The focal point F of a lens is the point at which an image of the object at infinity is formed. The distance between the surface and its focal point is called focal length f. A lens has two foci, F and F′, on its either side.

The lens maker’s formula is given by: . This equation holds good for a concave lens as well, but R1 is negative and R2 is positive, making f negative. Thus, the equation is useful in designing lenses of required focal lengths.

c. Power of a Lens

The measure of how much a lens causes light to converge or diverge is called the power of a lens. A convex lens with short focal length bends the incident light more, converging it, while a concave lens with short focal length diverges the light. The power P of a lens is the tangent of the angle by which it converges or diverges a beam of light falling at a unit distant from the optical centre. The SI unit for power of a lens is dioptre (D): 1D = 1m–1. If a lens has focal length 1m, its power is 1 dioptre. A converging lens has a positive power value, while a diverging lens has a negative power value.

d. Combination of Thin Lenses in Contact

A thin lens is a transparent optical medium bound by two surfaces, at least one of which is spherical. Consider two thin lenses A and B with focal lengths f1 and f2, respectively, positioned in contact with each other. If the object is placed beyond the focus of the first lens A, the first lens produces a real image at I1, acting as a virtual object for the second lens B. Lens B produces the final image at I. The formation of image by the first lens is presumed only to determine the position of the final image. The direction of rays emerging from the first lens gets modified in accordance with the angle at which they strike the second lens. Since the lenses are thin, the optical centres of the lenses are assumed to be coincident. If several thin lenses of focal length f1, f2, f3,... are in contact, the effective focal length of their combination is given by: and so on. The power is given by P = P1 + P2 + P3 + …. and so on.

Consider a beam of light passing through a triangular prism ABC. According to Snell’s law, light bends towards the normal when it travels from a rarer medium to a denser medium and vice versa. Since glass is denser than air, a ray of light entering a prism bends towards the normal. Next, from the other side of the prism, the ray of light travels from the glass to the air. Since air is less dense than glass, the emergent ray bends away from the normal.

The angle between the emergent ray and the direction of the incident ray is called the angle of deviation δ. The total deviation δ is the sum of deviations at the two faces. This is what happens when a ray of white light enters a prism, separating into seven colours. The different light waves in white light undergo different degrees of deviation, separating white light into its colours.

It has been established that when a narrow beam of white light hits a glass prism, the emergent light is not white but of seven different colours. The colours of light appearing in sequence are violet, indigo, blue, green, yellow, orange and red. The red light bends the least, while the violet light bends the most. The phenomenon of splitting of light into its component colours is known as dispersion of light by a prism. The pattern of colour components of light is called the spectrum of light.

The properties and behaviour of light with different objects causes several spectacular phenomena. Some examples of natural phenomena caused by sunlight are the colour of the sky and clouds at different times of the day and rainbows. Some of them are explained below:

a. The Rainbow

A rainbow occurs when rays of sunlight pass through spherical droplets of water in the air after it has rained. This phenomenon occurs due to a combination of dispersion, refraction and reflection of sunlight by the water droplets. When sunlight enters a droplet, the light is separated into different wavelengths or colours of light. The wavelengths of light like red are bent the least while the shorter ones like violet are bent the most.

b. Scattering of Light

As sunlight reaches the Earth’s atmosphere, the particles present in the air scatter the light rays, changing their direction. Waves of shorter wavelengths are scattered more than the longer ones. Blue waves have shorter wavelengths than red, and this scattering causes the bluish hue of the sky. Violet waves have even shorter wavelengths than blue waves, so they get scattered even more than the blue ones. But since the human eye is more sensitive to blue than violet, the sky appears blue.

Several optical devices and instruments are designed using the properties and behaviour of light and different surfaces such as mirror, lenses and prisms. Periscope, kaleidoscope, binoculars, telescopes and microscopes are some devices that use the reflective and refractive properties of different surfaces.

a. The Eye

The first point of entry for light entering the eye is through the curved surface called the cornea. Light rays then pass through the pupil and are focused onto the retina by the eye lens. Muscles around the inside of the eyes change the size of the pupil according to the amount of light rays passing through. Nerve fibres in the retina—made up of rods and cones that detect the intensity and colour of light respectively—transmit electrical signals through the optic nerve to the brain. The ciliary muscles control the curvature and the focal length of the eye lens. Depending on the distance of the object, the ciliary muscles adjust the focal length and the curvature of the lens, so that the object is focused on to the retina. This property of the eye is called accommodation. The closer the object gets to the eye, the blurrier the image becomes as after a certain point the lens is unable to curve enough to focus the image on to the retina. The near point is the closest point at which the lens focuses light on the retina. The standard value for normal vision is taken as 25cm.

b. The Microscope

A microscope makes use of a converging lens that has a small focal length. The lens is placed near the object closer than or the same distance as its focal length and the eye is positioned close to the lens on the other side. The aim is to get an enlarged image of the object so that it can be viewed effortlessly, i.e. at 25cm or more. If the object is at a distance equal to the focal length, then the image is at infinity. If the object is closer than the focal length of the lens, the image appears at a point less than infinity. The image formed at infinity is the most comfortable for viewing by relaxed eyes. A simple microscope can only magnify objects up to a limited amount. A compound microscope is used for much larger magnifications in which there are two lenses, the effect of one lens compounding the other.

c. The Telescope

The telescope provides angular magnification of distant objects. It consists of an objective with a huge focal length and an aperture bigger than its eyepiece. Light rays from a distant object enter the objective and a real image is formed in the tube at its second focal point. The eyepiece enlarges this image, forming a final upside-down image. A pair of investing lenses in terrestrial telescopes produce the final erect image. A refracting telescope is useful in making terrestrial and astronomical observations.

1. What is refraction, according to Chapter 9 of NCERT Solutions for Class 12 Physics?

When a wave travels through two different media, the wave bends and changes direction due to the differences in density of the two substances. This phenomenon is known as refraction.

2. What are the important topics of Class 12 Physics Chapter 9 Ray Optics and Optical Instruments?

Some important topics of Ray Optics Class 12 Physics are:

• Reflection of light by spherical mirrors
• Refraction of light
• Total internal reflection
• Refraction of light by spherical surfaces and lenses
• Refraction through a prism
• Optical instruments

3. What is the Meaning of Ray Optics?

Ray optics is the propagation of light through different media, such as lenses, prisms and mirrors.

4. How can I Study for Ray Optics Class 12?

Download the NCERT Solutions for Class 12 Physics Ray Optics and Optical Instruments from the MSVgo website to study the Ray Optics chapter for the exam. The solutions are a one-stop solution to study for the exam as they have been curated by subject matter experts and are aligned with the latest CBSE curriculum. The solutions contain all the concepts and problems necessary to score high marks and boost confidence in the chapter.