Light Reflection and Refraction Notes - Spherical Mirrors, Lenses, Power of Lens and Numericals

• Optics is the study of the behavior and properties of light.
• An object reflects light that falls on it. Eyes receive the reflected light, and thus we see that object.
• We see through a transparent medium as light is transmitted through it.
• Light seems to travel in straight lines. A small source of light casts a sharp shadow of an opaque object. It is usually indicated as a ray of light.

REFLECTION OF LIGHT

Laws of Reflection of Light

1. Angle of incidence is equal to the angle of reflection.
2. The incident ray, the normal to the mirror at the point of incidence, and the reflected ray all lie in the same plane.

• These laws are applicable to all types of reflecting surfaces, including spherical surfaces.
• A highly polished surface, such as a mirror, reflects most of the light falling on it.
• The image formed by a plane mirror is virtual and erect. The size of the image is equal to that of the object. The image formed is as far behind the mirror as the object is in front of it. Further, the image is laterally inverted.

Image Formation by Curved Mirror

• Consider the curved surface of a large shining spoon as a curved mirror. The image at its concave side is real, inverted, and diminished. If the spoon is moved away, the image becomes smaller.
• The image at the convex side is virtual, erect, and diminished. If the spoon is moved away, the image becomes smaller.
• The most commonly used type of curved mirror is the spherical mirror.

SPHERICAL MIRRORS

These are mirrors with spherical reflecting surfaces. There are two types:

1. Concave mirror: Its reflecting surface is curved inwards (faces towards the center of the sphere).
2. Convex mirror: Its reflecting surface is curved outwards.

• Pole (P): It is the center of the reflecting surface of a spherical mirror. The reflecting surface is a part of a sphere.
• Centre of curvature (C): It is the center of the sphere of which the spherical mirror is a part. It is not part of the mirror and lies outside the reflecting surface. The centre of curvature lies in front of a concave mirror and behind a convex mirror.
• Radius of curvature (R): It is the radius of the sphere of which the mirror forms a part. The distance PC = radius of curvature.
• Principal axis: It is the imaginary straight line passing through the pole and centre of curvature of a spherical mirror. It is normal to the mirror at its pole.

Principal Focus (F) and Focal Length (f)

• Direct the reflecting surface of a concave mirror towards the Sun and direct the reflected light onto a paper.
• Move the paper to find a bright, sharp spot of light on it. Wait for a few minutes. The paper burns and catches fire.
• This is because the light from the Sun is converged (concentrated) at a point on the paper as a bright spot (image of the Sun). This point is the focus of the concave mirror. It produces heat and ignites the paper.
• The distance of this image from the position of the mirror is the focal length of the mirror. This can be represented as a ray diagram.

• When several rays parallel to the principal axis fall on a concave mirror, the reflected rays meet/intersect at a point on the principal axis of the mirror. This point is called the principal focus (F) of the concave mirror.
• In a convex mirror, the reflected rays appear to come from a point on the principal axis. This point is called the principal focus of the convex mirror.
• The distance between the pole and the principal focus of a spherical mirror is called the focal length (f).
• The reflecting surface of a spherical mirror is generally spherical and has a circular outline. The diameter of the reflecting surface is called its aperture (the distance MN). We consider only spherical mirrors whose aperture is much smaller than its radius of curvature.
• For spherical mirrors of small apertures, the radius of curvature is equal to twice the focal length (R = 2f), i.e., the principal focus lies midway between the pole and the centre of curvature.

Image Formation by Spherical Mirrors

• Find out the approximate focal length of a concave mirror.
• Mark 3 parallel lines P, F, and C on a table such that the distance between any two successive lines is equal to the focal length of the mirror.
• Place a stand with a concave mirror over the line P such that its pole lies over the line.
• Keep a bright object (e.g., a burning candle) at a position far beyond C. Place a paper screen and move it in front of the mirror to obtain a sharp, bright image of the candle flame.
• Repeat the activity by placing the candle (a) just beyond C, (b) at C, (c) between F & C, (d) at F, and (e) between P & F.
• The nature, position, and size of the image formed by a concave mirror depend on the position of the object in relation to points P, F, and C.

Representation of Images Formed by Spherical Mirrors Using Ray Diagrams

• In an extended object, each small portion acts like a point source. An infinite number of rays originate from each point. For clarity in the ray diagram, it is easier to consider only two rays to determine their directions after reflection.
• The intersection of at least two reflected rays gives the position of the image of the point object. Any two of the following rays can be considered to locate the image:

1. A ray parallel to the principal axis. After reflection, it passes through the principal focus in a concave mirror or appears to diverge from the principal focus in a convex mirror.

2. A ray through the principal focus of a concave mirror or directed towards the principal focus of a convex mirror. After reflection, it emerges parallel to the principal axis.

3. A ray through the centre of curvature of a concave mirror or directed in the direction of the centre of curvature of a convex mirror. It is reflected back along the same path because the incident rays fall on the mirror along the normal to the reflecting surface.

4. A ray incident obliquely to the principal axis, towards the pole (P), on the concave mirror or a convex mirror. It is reflected obliquely.

• In all these cases, the laws of reflection are followed, i.e., the angle of reflection equals the angle of incidence.

(a) Image Formation by a Concave Mirror

Ray Diagrams

• When the object is between F & P, the image is not obtained on the screen. Here, a virtual image can be seen in the mirror.

Uses of Concave Mirrors

• Used in torches, searchlights, and vehicle headlights to get powerful parallel beams of light.
• Used as shaving mirrors to see a larger image of the face.
• Used by dentists to see large images of the teeth of patients.
• Large concave mirrors are used to concentrate sunlight to produce heat in solar furnaces.

(b) Image Formation by a Convex Mirror

• Show a pencil in the upright position in front of a convex mirror. Its image in the mirror is erect and diminished.
• As the pencil is moved away from the mirror, the image becomes smaller and moves closer to the focus.
• Two positions of the object to study the image formed by a convex mirror are shown below.

(a) Formation of image when the object is at infinity

(b) Formation of image when the object is at a finite distance from the mirror

• In plane mirrors and concave mirrors of any size, we cannot see a full-length image of a distant object. But it is possible in a convex mirror with a wider field of view.
• A convex mirror is fitted in a wall of Agra Fort facing the Taj Mahal to observe the full image of the Taj Mahal.

Uses of Convex Mirrors

• Convex mirrors give an erect, diminished, virtual image. Also, they have a wider field of view as they are curved outwards. So, they are used as rear-view (wing) mirrors in vehicles. This enables the driver to see traffic behind them.

Sign Convention for Reflection by Spherical Mirrors

New Cartesian Sign Convention

• In this convention, the pole (P) of the mirror is taken as the origin. The principal axis of the mirror is taken as the x-axis (X’X) of the coordinate system.

The conventions are as follows:

1. The object is always placed to the left of the mirror, i.e., light from the object falls on the mirror from the left-hand side.
2. All distances parallel to the principal axis are measured from the pole of the mirror.
3. All distances measured to the right of the origin (along + x-axis) are taken as positive, while those measured to the left of the origin (along – x-axis) are taken as negative.
4. Distances measured perpendicular to and above the principal axis (along + y-axis) are taken as positive.
5. Distances measured perpendicular to and below the principal axis (along – y-axis) are taken as negative.

Sign conventions are applied to obtain the mirror formula and solve related numerical problems.

Mirror Formula and Magnification

• In a spherical mirror, the distance of the object from its pole is called the object distance (u).
• The distance of the image from the pole of the mirror is called the image distance (v).
• The distance of the principal focus from the pole is called the focal length (f).

• This formula is valid in all situations for all spherical mirrors for all positions of the object.

Magnification (m)

• It is the enlargement of the image formed by a spherical mirror, relative to the size of the object.
• It is the ratio of the height of the image (h′) to the height of the object (h).

• Magnification is also related to the object distance (u) and image distance (v). It can be expressed as:

• The height of the object is taken to be positive as the object is placed above the principal axis.
• The height of the image is taken as positive for virtual images and negative for real images.
• A negative sign in the value of the magnification indicates that the image is real. A positive sign indicates that the image is virtual.

Problem 1:

• A convex mirror used for rear-view on an automobile has a radius of curvature of 3.00 m. If a bus is located at 5.00 m from this mirror, find the position, nature, and size of the image.

Solution

• Radius of curvature, R = +3.00 m
• Object-distance, u = –5.00 m
• Image-distance, v = ?
• Height of the image, h′ = ?
• Focal length, f = R/2 = +3.00 m / 2 = +1.50 m (as the principal focus of a convex mirror is behind the mirror)

• The image is 1.15 m at the back of the mirror.

• The image is virtual, erect, and smaller by a factor of 0.23.

Problem 2:

• An object, 4.0 cm in size, is placed at 25.0 cm in front of a concave mirror of focal length 15.0 cm. At what distance from the mirror should a screen be placed in order to obtain a sharp image? Find the nature and the size of the image.

Solution

• Object-size, h = +4.0 cm
• Object-distance, u = –25.0 cm
• Focal length, f = –15.0 cm
• Image-distance, v = ?
• Image-size, h′ = ?

• v = –37.5 cm
• The screen should be placed at 37.5 cm in front of the mirror. The image is real.

• Height of the image, h′ = –6.0 cm
• The image is inverted and enlarged.

Refraction of Light

• Light seems to travel along straight-line paths in a transparent medium.
• Light does not travel in the same direction in all media. It appears that when travelling obliquely from one medium to another, the direction of propagation of light in the second medium changes. This phenomenon is called refraction of light.
• Examples:
• The bottom of a tank or a pond containing water appears to be raised.
• When a thick glass slab is placed over a printed matter, the letters appear raised.
• A pencil partly immersed in water in a glass tumbler appears to be displaced at the interface of air and water.
• A lemon kept in water in a glass tumbler appears to be bigger than its actual size, when viewed from the sides.

• Refraction is varied in different media such as kerosene, turpentine, transparent plastic slab, etc.

Experiments to Demonstrate the Refraction of Light

Experiment 1

• Place a coin at the bottom of a bucket filled with water.
• With our eye to a side above water, try to pick up the coin in one go. We do not succeed.
• Reason: Reflected light coming from the submerged coin in water (denser medium), on entering air (rarer medium), bends away from the normal due to refraction of light and image size becomes larger than its actual size. Thus, the coin appears to be closer than its actual distance.

Experiment 2

• Place a large shallow bowl on a table and put a coin in it.
• Move away slowly from the bowl. Stop when the coin just disappears from our sight.
• Ask a friend to pour water into the bowl without disturbing the coin. The coin becomes visible again. This is because the coin appears slightly raised above its actual position due to refraction of light.

Experiment 3

• Draw a thick straight line on a white paper. Place a glass slab over the line such that one of its edges makes an angle with the line.
• Look at the portion of the line under the slab from the sides. The line under the glass slab appears to be bent at the edges. It is due to the refraction of light.
• Place the glass slab normal to the line. The part of the line under the glass slab does not appear bent. It appears in a straight line. Because a ray of light perpendicular to the plane of a refracting medium does not change its angle due to refraction.
• Look at the line from the top of the glass slab. Part of the line appears to be raised. This is due to refraction of light.

Refraction through a Rectangular Glass Slab

• Fix a white paper on a drawing board and place a rectangular glass slab on its middle.
• Draw the outline (ABCD) of the slab.
• Fix two pins (E & F) vertically such that the line joining the pins is inclined to the edge AB.
• Look at the images of the pins E & F through the opposite edge. Fix two other pins (G & H) such that these pins and the images of E & F lie on a straight line.
• Remove the pins and the slab.
• Join the positions of the tips of the pins E & F and produce the line up to AB. Let EF meet AB at O. Similarly, join the positions of the tips of the pins G & H and produce it up to the edge CD. Let HG meet CD at O′.
• Join O and O′. Also produce EF up to P, as shown below:

• Here, the light ray has changed its direction at points O and O′. Both points O and O′ lie on surfaces separating two transparent media. Draw a perpendicular NN′ to AB at O and another perpendicular MM′ to CD at O′.
• The light ray at point O enters from a rarer medium (air) to a denser medium (glass). The light ray bends towards the normal. At O′, the light ray enters from glass to air (denser medium to rarer medium). The light here bends away from the normal.
• Compare the angle of incidence with the angle of refraction at both refracting surfaces AB & CD. A ray EO is obliquely incident on surface AB, called the incident ray.
• OO′ is the refracted ray and O′H is the emergent ray. The emergent ray is parallel to the direction of the incident ray. The extent of bending of the ray of light at the opposite parallel faces AB (air-glass interface) and CD (glass-air interface) of the rectangular glass slab is equal and opposite. This is why the ray emerges parallel to the incident ray. However, the light ray is shifted sideward slightly.
• When a light ray is incident normally to the interface of two media, it goes along the same straight line.
• Refraction is due to the change in the speed of light as it enters from one transparent medium to another.

Laws of Refraction of Light

1. The incident ray, the refracted ray, and the normal to the interface of two transparent media at the point of incidence all lie in the same plane.
2. The ratio of the sine of the angle of incidence (i) to the sine of the angle of refraction (r) is a constant, for the light of a given colour and pair of media. This is also known as Snell’s law of refraction. (It is true for angles 0 < i < 90°).

This constant value is called the refractive index of the second medium with respect to the first.

The Refractive Index (n)

• It is the ratio of the speeds of light in a pair of media.
• It is used to measure the change in direction of a light ray occurring in two media.
• Light travels fastest in vacuum (3×10⁸ m/s). In air, there is only a marginal decrease. It reduces considerably in glass or water.
• Consider a ray of light travelling from medium 1 into medium 2. The refractive index of medium 2 with respect to medium 1 (n₂₁) is the ratio of speed of light in medium 1 (v₁) to speed of light in medium 2 (v₂).

• Refractive index of medium 1 with respect to medium 2 is represented as n₁₂.

• The ratio of the speed of light in vacuum or air (medium 1) to that in medium 2 is called the absolute refractive index. It is represented as n₂.
• If c is the speed of light in air and v is the speed of light in the medium, the refractive index of the medium n_m is:

• The absolute refractive index of a medium is simply called its refractive index. For example:
• Refractive index of water, n_w = 1.33.

• Refractive index of crown glass, n_g = 1.52.

Absolute Refractive Index of Some Material Media

• The ability of a medium to refract light is expressed in terms of optical density. It is not the same as mass density. An optically denser medium may not have greater mass density. For example, kerosene is optically denser than water, but its mass density is less than water.

Refraction by Spherical Lenses

• The glasses used in spectacles and the magnifying glass used by watchmakers are examples of lenses.
• A transparent material bound by two surfaces, of which one or both surfaces are spherical, forms a lens.
• In a lens with only one spherical surface, the other surface would be plane.
• A lens with two spherical surfaces, bulging outwards, is called a double convex lens (simply a convex lens). It is thicker at the middle than at the edges. Such lenses converge light rays, so they are also called converging lenses.
• A double concave lens (simply a concave lens) is bounded by two spherical surfaces, curved inwards. It is thicker at the edges than at the middle. Such lenses diverge light rays, so they are also called diverging lenses.

• A convex lens or concave lens has two spherical surfaces. Each of them forms a part of a sphere. The centres of these spheres are called centres of curvature of the lens (C₁ & C₂).
• An imaginary straight line passing through the two centres of curvature of a lens is called its principal axis.
• The central point of a lens is its optical centre (O).
• A ray of light through the optical centre of a lens passes without any deviation.
• The effective diameter of the circular outline of a spherical lens is called its aperture.
• Thin lenses with small apertures: The lenses whose aperture is much less than its radius of curvature and the two centres of curvature are equidistant from the optical centre.

Incidence of Parallel Rays on a Lens

• Using a convex lens, focus the light from the Sun on a paper to obtain a sharp, bright, real image of the Sun.
• Hold the paper and lens in the same position for a while.
• The paper begins to burn and catches fire.
• The parallel rays of light from the Sun are converged by the lens at the sharp, bright spot on the paper. This generates heat and causes the burning of the paper.
• When several rays of light parallel to the principal axis fall on a convex lens, they undergo refraction from the lens and converge to a point on the principal axis. This point is called the principal focus of the convex lens.
• When several rays of light parallel to the principal axis fall on a concave lens, they undergo refraction from the lens and diverge from a point on the principal axis. This point is called the principal focus of the concave lens.
• If parallel rays are passed from the opposite surface of the lens, another principal focus is formed on the opposite side. A lens has two principal foci (F₁ & F₂).
• The distance of the principal focus from the optical centre of a lens is called its focal length (f). The distance between the position of the convex lens and the position of the image of the Sun gives the approximate focal length of the lens.

Lenses form images by refracting light.

Image Formation by Convex Lens

• Take a convex lens. Find its approximate focal length.
• Draw five parallel straight lines on a table such that the distance between the successive lines is equal to the focal length of the lens.
• Place the lens on the central line such that the optical centre of the lens lies just over the line.
• The two lines on either side of the lens correspond to F and 2F of the lens respectively. Mark them 2F₁, F₁, F₂ & 2F₂ respectively.
• Place a burning candle far beyond 2F₁ to the left. A clear sharp image is formed on a screen placed at F₂.
• Repeat this activity by placing the object just behind 2F₁, between F₁ & 2F₁, at F₁, between F₁ & O.

Nature, Position, and Relative Size of the Image Formed by a Convex Lens for Various Positions of the Object

Image Formation by Concave Lens

• Place a burning candle on one side of a concave lens.
• Look through the lens from the other side and observe the image. The image will not be obtained on a screen but can be observed through the lens.
• Move the candle far away from the lens. The image becomes highly diminished.
• A concave lens always gives a virtual, erect, and diminished image, irrespective of the position of the object.

Nature, Position, and Relative Size of the Image Formed by a Concave Lens for Various Positions of the Object

Image Formation in Lenses Using Ray Diagrams

• For drawing ray diagrams in lenses, any two of the following rays are considered:

1. A ray of light parallel to the principal axis: After refraction from a convex lens, passes through the principal focus on the other side of the lens. For a concave lens, the ray appears to diverge from the principal focus located on the same side of the lens.

2. A ray passing through a principal focus: After refraction from a convex lens, emerges parallel to the principal axis. A ray of light appearing to meet at the principal focus of a concave lens, after refraction, emerges parallel to the principal axis.

3. A ray passing through the optical centre: It emerges without any deviation.

Ray Diagrams for Image Formation in a Convex Lens

• The ray diagrams for position, size, and the nature of the image formation in a convex lens for a few positions of the object are shown below (Refer the table):

Ray Diagrams for Image Formation in a Concave Lens

• The ray diagrams for the image formation in a concave lens for various positions of the object (Refer the table):

Sign Convention for Spherical Lenses

• It is the same as the sign convention used in spherical mirrors.
• The rules are applied for signs of distances, but the measurements are taken from the optical centre of the lens.
• According to the convention, the focal length of a convex lens is positive and that of a concave lens is negative.

Lens Formula and Magnification

• The lens formula gives the relationship between object distance (u), image distance (v), and the focal length (f). The lens formula is expressed as:

• The lens formula is general and is valid in all situations for any spherical lens.
• The magnification (m) produced by a lens is the ratio of the height of the image (h') to the height of the object (h).

• Magnification is also related to the object distance (u) and the image distance (v).
• Magnification m = h'/h = v/u.

Problem 1: Concave Lens

• A concave lens has a focal length of 15 cm. At what distance should the object from the lens be placed so that it forms an image at 10 cm from the lens? Also, find the magnification produced by the lens.

Solution

• A concave lens always forms a virtual, erect image on the same side of the object.
• Image distance v = –10 cm
• Focal length f = –15 cm
• Object distance u = ?

• or, u = –30 cm
• Thus, the object distance is 30 cm.
• Magnification m = v/u

• The positive sign shows that the image is erect and virtual. The image is one-third of the size of the object.

Problem 2: Convex Lens

• A 2.0 cm tall object is placed perpendicular to the principal axis of a convex lens of focal length 10 cm. The distance of the object from the lens is 15 cm. Find the nature, position, and size of the image. Also find its magnification.

Solution

• Height of the object h = +2.0 cm
• Focal length f = +10 cm
• Object distance u = –15 cm
• Image distance v = ?
• Height of the image h' = ?

• or, v = +30 cm
• The positive sign of v shows that the image is formed at a distance of 30 cm on the other side of the optical centre. The image is real and inverted.

• or, h' = h (v/u)
• Height of the image, h' = (2.0) (+30/–15) = –4.0 cm
• Magnification m = v/u

• The negative signs of m and h' show that the image is inverted and real. It is formed below the principal axis. Thus, a real, inverted image, 4 cm tall, is formed at a distance of 30 cm on the other side of the lens. The image is two times enlarged.

Power of a Lens

• The ability of a lens to converge or diverge light rays depends on its focal length.
• A convex lens of short focal length bends the light rays through large angles and focuses closer to the optical centre.
• A concave lens of very short focal length causes higher divergence than one with a longer focal length.
• The degree of convergence or divergence of light rays by a lens is expressed as its power.
• The power of a lens (P) is defined as the reciprocal of its focal length (f).

• The SI unit of power of a lens is dioptre (D).
• 1 dioptre is the power of a lens whose focal length is 1 metre (1D = 1m up>–1).
• The power of a convex lens is positive and that of a concave lens is negative.
• Opticians prescribe corrective lenses indicating their powers. It is more convenient to use powers instead of focal lengths.
• • A lens of power +2.0 D has a focal length of +0.50 m. The lens is convex.
  • A lens of power –2.5 D has a focal length of –0.40 m. The lens is concave.
• In many optical instruments, several lenses are combined to increase magnification and sharpness of the image. The net power (P) of the lenses placed in contact is the algebraic sum of the individual powers P₁, P₂, P₃, … (P = P₁ + P₂ + P₃ + …).
• During eye-testing, an optician puts several corrective lenses of known power inside the testing spectacles’ frame. Thus, the required power of the lens is calculated by algebraic addition. E.g., two lenses of power +2.0 D and +0.25 D are equivalent to a single lens of power +2.25 D.
• The simple additive property of the powers of lenses can be used to design lens systems to minimise certain defects in images produced by a single lens. Such a lens system is commonly used in the design of lenses for cameras, microscopes, and telescopes.