Light is a form of energy that enables us to see the world around us. This chapter delves into two fundamental phenomena of light: reflection and refraction. Reflection is the bouncing back of light when it strikes a surface, allowing us to see objects and forming images in mirrors. Refraction is the bending of light as it passes from one transparent medium to another, explaining how lenses work and why objects appear distorted in water. Understanding these concepts is crucial for comprehending how optical instruments like cameras, telescopes, and our own eyes function.

Reflection of Light

When light falls on a surface, a part of it is sent back into the same medium. This phenomenon is called reflection of light.

Laws of Reflection

There are two laws of reflection:

1. The angle of incidence ($\angle i$) is equal to the angle of reflection ($\angle r$). The angle of incidence is the angle between the incident ray and the normal to the surface, and the angle of reflection is the angle between the reflected ray and the normal.
2. The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane.

Plane Mirror

A plane mirror is a flat, smooth reflecting surface. The image formed by a plane mirror is always:

• Virtual: Cannot be obtained on a screen.
• Erect: Upright with respect to the object.
• Laterally inverted: Left appears right and right appears left.
• Of the same size as the object.
• At the same distance behind the mirror as the object is in front.

Spherical Mirrors

Spherical mirrors are reflecting surfaces that are part of a sphere. They are of two types:

1. Concave Mirror: A spherical mirror whose reflecting surface is curved inwards (towards the centre of the sphere).
2. Convex Mirror: A spherical mirror whose reflecting surface is curved outwards.

Key Terms Related to Spherical Mirrors

• Pole (P): The centre of the spherical mirror's reflecting surface.
• Centre of Curvature (C): The centre of the sphere of which the mirror is a part.
• Radius of Curvature (R): The distance between the Pole and the Centre of Curvature (PC). It is the radius of the sphere of which the mirror is a part.
• Principal Axis: The straight line passing through the Pole and the Centre of Curvature.
• Principal Focus (F): For a concave mirror, it's a point on the principal axis where rays of light parallel to the principal axis converge after reflection. For a convex mirror, it's a point on the principal axis from which rays of light parallel to the principal axis appear to diverge after reflection.
• Focal Length (f): The distance between the Pole and the Principal Focus (PF). For spherical mirrors, $f = R/2$.
• Aperture: The diameter of the reflecting surface of the spherical mirror.

Ray Diagrams for Concave Mirror

To determine the position, nature, and size of the image formed by a concave mirror, we use ray diagrams. We typically use two of these four rules:

1. A ray parallel to the principal axis, after reflection, passes through the principal focus (F).
2. A ray passing through the principal focus (F), after reflection, becomes parallel to the principal axis.
3. A ray passing through the centre of curvature (C), after reflection, is reflected back along the same path.
4. A ray incident obliquely to the principal axis, directed towards the pole (P), is reflected obliquely, making equal angles with the principal axis.

Image Formation by Concave Mirror:

Position of Object	Position of Image	Size of Image	Nature of Image
At infinity	At F	Highly diminished	Real, Inverted
Beyond C	Between F and C	Diminished	Real, Inverted
At C	At C	Same size	Real, Inverted
Between C and F	Beyond C	Magnified	Real, Inverted
At F	At infinity	Highly magnified	Real, Inverted
Between F and P	Behind the mirror	Magnified	Virtual, Erect

Ray Diagrams for Convex Mirror

Rules for drawing ray diagrams for convex mirrors:

1. A ray parallel to the principal axis, after reflection, appears to diverge from the principal focus (F).
2. A ray directed towards the principal focus (F), after reflection, becomes parallel to the principal axis.
3. A ray directed towards the centre of curvature (C), after reflection, is reflected back along the same path.
4. A ray incident obliquely to the principal axis, directed towards the pole (P), is reflected obliquely, making equal angles with the principal axis.

Image Formation by Convex Mirror:

Position of Object	Position of Image	Size of Image	Nature of Image
At infinity	At F (behind mirror)	Highly diminished	Virtual, Erect
Between infinity and P	Between P and F (behind mirror)	Diminished	Virtual, Erect

Uses of Spherical Mirrors

• Concave Mirrors: Used in torches, searchlights, and vehicle headlights to get powerful parallel beams of light. Used as shaving mirrors and by dentists to see larger images of teeth. Large concave mirrors are used in solar furnaces to concentrate sunlight.
• Convex Mirrors: Used as rear-view mirrors in vehicles because they give a wider field of view and always form erect, diminished images.

New Cartesian Sign Convention for Spherical Mirrors

• The pole (P) of the mirror is taken as the origin (0,0).
• The principal axis is taken as the x-axis.
• All distances measured to the right of the origin are taken as positive.
• All distances measured to the left of the origin are taken as negative.
• Distances measured perpendicular to and above the principal axis are taken as positive.
• Distances measured perpendicular to and below the principal axis are taken as negative.
• The object is always placed to the left of the mirror.

Sign Convention Summary:

• Object distance (u): Always negative.
• Image distance (v): Negative for real images (formed in front of mirror), positive for virtual images (formed behind mirror).
• Focal length (f): Negative for concave mirrors, positive for convex mirrors.
• Object height (h): Always positive (object usually placed above principal axis).
• Image height (h'): Positive for erect images (virtual), negative for inverted images (real).

Mirror Formula

The mirror formula gives the relationship between object distance (u), image distance (v), and focal length (f) of a spherical mirror: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$

Magnification

Magnification (m) produced by a spherical mirror indicates how much the image is magnified relative to the object. It is defined as the ratio of the height of the image ($h'$) to the height of the object ($h$). $m = \frac{h'}{h}$ It is also related to object and image distances: $m = -\frac{v}{u}$

• If $m$ is negative, the image is real and inverted.
• If $m$ is positive, the image is virtual and erect.
• If $|m| > 1$, the image is magnified.
• If $|m| < 1$, the image is diminished.
• If $|m| = 1$, the image is of the same size as the object.

Example 1: Mirror Problem

An object 4 cm high is placed at a distance of 25 cm in front of a concave mirror of focal length 15 cm. At what distance from the mirror should a screen be placed to obtain a sharp image? Find the nature and size of the image.

Given: Object height, $h = +4$ cm Object distance, $u = -25$ cm (always negative) Focal length of concave mirror, $f = -15$ cm (negative for concave)

To find: Image distance, $v$ Image height, $h'$ Nature and size of the image

Solution: Using the mirror formula: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ $\frac{1}{v} + \frac{1}{-25} = \frac{1}{-15}$ $\frac{1}{v} - \frac{1}{25} = -\frac{1}{15}$ $\frac{1}{v} = \frac{1}{25} - \frac{1}{15}$ To subtract the fractions, find a common denominator, which is 75. $\frac{1}{v} = \frac{3}{75} - \frac{5}{75}$ $\frac{1}{v} = -\frac{2}{75}$ $v = -\frac{75}{2} = -37.5 cm$

The image distance $v$ is -37.5 cm. The negative sign indicates that the image is formed in front of the mirror, which means it is a real image. A screen should be placed at a distance of 37.5 cm in front of the mirror.

Now, let's find the magnification using $m = -\frac{v}{u}$: $m = - \frac{-37.5}{-25} = -1.5$

Since $m = \frac{h'}{h}$, we can find $h'$: $-1.5 = \frac{h'}{4}$ $h' = -1.5 \times 4 = -6 cm$

The image height $h'$ is -6 cm. The negative sign indicates that the image is inverted. Since $|m| = 1.5 > 1$, the image is magnified.

Conclusion: The image is formed at 37.5 cm in front of the mirror. It is real, inverted, and magnified (6 cm tall).

Refraction of Light

Refraction is the phenomenon of bending of light as it passes obliquely from one transparent medium to another. This bending occurs because the speed of light changes as it enters a different medium.

Laws of Refraction

1. The incident ray, the refracted ray, and the normal to the interface of the two transparent media at the point of incidence, all lie in the same plane.
2. Snell's Law: The ratio of the sine of the angle of incidence ($\sin i$) to the sine of the angle of refraction ($\sin r$) is a constant for a given pair of media and a given colour of light. $\frac{\sin i}{\sin r} = \text{constant} = n_{21}$ Here, $n_{21}$ is the refractive index of the second medium with respect to the first medium.

Refractive Index

The refractive index of a medium is a measure of how much it slows down light.

• Absolute refractive index (n): The ratio of the speed of light in vacuum ($c$) to the speed of light in the medium ($v$). $n = \frac{c}{v}$ The refractive index of vacuum is 1. For air, it is approximately 1.0003.
• Relative refractive index ($n_{21}$): The refractive index of medium 2 with respect to medium 1. $n_{21} = \frac{\text{Speed of light in medium 1}}{\text{Speed of light in medium 2}} = \frac{v_1}{v_2}$ Also, $n_{21} = \frac{n_2}{n_1}$. Similarly, $n_{12} = \frac{v_2}{v_1} = \frac{n_1}{n_2}$. Note that $n_{21} = 1/n_{12}$.

A medium with a higher refractive index is said to be optically denser, and a medium with a lower refractive index is optically rarer.

• When light goes from a rarer to a denser medium, it bends towards the normal.
• When light goes from a denser to a rarer medium, it bends away from the normal.

Refraction Through a Rectangular Glass Slab

When a light ray passes through a rectangular glass slab, it undergoes two refractions: first, while entering the glass from air, and second, while emerging from the glass into air. The emergent ray is parallel to the incident ray but is slightly displaced laterally. This displacement is called lateral displacement.

Spherical Lenses

A spherical lens is a transparent material bounded by two spherical surfaces, or one spherical and one plane surface. Lenses are of two main types:

1. Convex Lens (Converging Lens): Thicker in the middle and thinner at the edges. It converges parallel rays of light.
2. Concave Lens (Diverging Lens): Thinner in the middle and thicker at the edges. It diverges parallel rays of light.

Key Terms Related to Spherical Lenses

• Optical Centre (O): The central point of a lens. A ray of light passing through the optical centre goes undeviated.
• Principal Axis: The imaginary straight line passing through the optical centre and perpendicular to both faces of the lens.
• Principal Focus (F): Lenses have two principal foci, F1 and F2, on either side of the lens.
• For a convex lens, F2 is the point on the principal axis where rays parallel to the principal axis converge after refraction. F1 is the point on the principal axis such that rays passing through it become parallel to the principal axis after refraction.
• For a concave lens, F2 is the point on the principal axis from which rays parallel to the principal axis appear to diverge after refraction. F1 is the point on the principal axis towards which rays appear to converge, become parallel to the principal axis after refraction.
• Focal Length (f): The distance between the optical centre and either of the principal foci (OF1 or OF2). Lenses also have two centres of curvature, 2F1 and 2F2, at twice the focal length.
• Aperture: The effective diameter of the circular outline of the spherical lens.

Ray Diagrams for Convex Lens

Rules for drawing ray diagrams for convex lenses:

1. A ray of light parallel to the principal axis, after refraction, passes through the principal focus (F2) on the other side.
2. A ray of light passing through the principal focus (F1) will emerge parallel to the principal axis after refraction.
3. A ray of light passing through the optical centre (O) goes undeviated.

Image Formation by Convex Lens:

Position of Object	Position of Image	Size of Image	Nature of Image
At infinity	At F2	Highly diminished	Real, Inverted
Beyond 2F1	Between F2 and 2F2	Diminished	Real, Inverted
At 2F1	At 2F2	Same size	Real, Inverted
Between F1 and 2F1	Beyond 2F2	Magnified	Real, Inverted
At F1	At infinity	Highly magnified	Real, Inverted
Between F1 and O	On the same side as object	Magnified	Virtual, Erect

Ray Diagrams for Concave Lens

Rules for drawing ray diagrams for concave lenses:

1. A ray of light parallel to the principal axis, after refraction, appears to diverge from the principal focus (F2) on the same side of the lens.
2. A ray of light directed towards the principal focus (F1) will emerge parallel to the principal axis after refraction.
3. A ray of light passing through the optical centre (O) goes undeviated.

Image Formation by Concave Lens:

Position of Object	Position of Image	Size of Image	Nature of Image
At infinity	At F1 (on same side)	Highly diminished	Virtual, Erect
Between infinity and O	Between F1 and O (on same side)	Diminished	Virtual, Erect

Uses of Lenses

• Convex Lenses: Used in spectacles to correct hypermetropia, in magnifying glasses, cameras, telescopes, and microscopes.
• Concave Lenses: Used in spectacles to correct myopia, in flashlights, peepholes in doors, and some telescopes.

New Cartesian Sign Convention for Spherical Lenses

The sign convention for lenses is similar to that for mirrors, with the optical centre (O) taken as the origin.

• The object is always placed to the left of the lens.
• All distances measured to the right of the optical centre are taken as positive.
• All distances measured to the left of the optical centre are taken as negative.
• Distances measured perpendicular to and above the principal axis are taken as positive.
• Distances measured perpendicular to and below the principal axis are taken as negative.

Sign Convention Summary:

• Object distance (u): Always negative.
• Image distance (v): Positive for real images (formed on the right side of the lens), negative for virtual images (formed on the left side of the lens).
• Focal length (f): Positive for convex lenses, negative for concave lenses.
• Object height (h): Always positive.
• Image height (h'): Positive for erect images (virtual), negative for inverted images (real).

Lens Formula

The lens formula gives the relationship between object distance (u), image distance (v), and focal length (f) of a spherical lens: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

Magnification

Magnification (m) produced by a spherical lens is defined as the ratio of the height of the image ($h'$) to the height of the object ($h$). $m = \frac{h'}{h}$ It is also related to object and image distances: $m = \frac{v}{u}$

• If $m$ is negative, the image is real and inverted.
• If $m$ is positive, the image is virtual and erect.
• If $|m| > 1$, the image is magnified.
• If $|m| < 1$, the image is diminished.
• If $|m| = 1$, the image is of the same size as the object.

Power of a Lens

The power of a lens (P) is a measure of its ability to converge or diverge light rays. It is defined as the reciprocal of its focal length (f) in metres. $P = \frac{1}{f}$ The SI unit of power of a lens is dioptre (D). One dioptre is the power of a lens whose focal length is 1 metre ($1 \text{ D} = 1 \text{ m}^{-1}$).

• For a convex lens, focal length (f) is positive, so its power (P) is positive.
• For a concave lens, focal length (f) is negative, so its power (P) is negative. If several lenses are combined, the total power ($P_{total}$) is the algebraic sum of the individual powers: $P_{total} = P_1 + P_2 + P_3 + ...$

Example 2: Lens Problem

A concave lens has a focal length of 20 cm. At what distance should an object be placed from the lens so that it forms an image at 15 cm from the lens? Also, calculate the magnification produced by the lens.

Given: Focal length of concave lens, $f = -20$ cm (negative for concave) Image distance, $v = -15$ cm (concave lenses always form virtual images on the same side as the object, so $v$ is negative)

To find: Object distance, $u$ Magnification, $m$

Solution: Using the lens formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ $\frac{1}{-15} - \frac{1}{u} = \frac{1}{-20}$ $-\frac{1}{15} - \frac{1}{u} = -\frac{1}{20}$ $-\frac{1}{u} = -\frac{1}{20} + \frac{1}{15}$ To add the fractions, find a common denominator, which is 60. $-\frac{1}{u} = -\frac{3}{60} + \frac{4}{60}$ $-\frac{1}{u} = \frac{1}{60}$ $u = -60 cm$

The object distance $u$ is -60 cm. This means the object should be placed 60 cm in front of the lens.

Now, let's find the magnification using $m = \frac{v}{u}$: $m = \frac{-15}{-60} = \frac{1}{4} = +0.25$

Conclusion: The object should be placed 60 cm in front of the concave lens. The magnification is +0.25. The positive sign indicates that the image is virtual and erect. Since $|m| < 1$, the image is diminished.

Common Mistakes

1. Sign Convention Errors: This is the most frequent mistake. Students often forget to assign correct signs to $u$, $v$, $f$, $h$, and $h'$ based on the New Cartesian Sign Convention. Remember $u$ is always negative, and focal length depends on the mirror/lens type.
2. Mixing Mirror and Lens Formulas: The mirror formula is $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$, while the lens formula is $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$. Swapping the '+' and '-' signs is a common error.
3. Incorrect Magnification Formula: For mirrors, $m = -\frac{v}{u}$, but for lenses, $m = \frac{v}{u}$. The negative sign difference is crucial.
4. Units for Power: For power of a lens, focal length MUST be in metres to get the power in dioptres. Converting cm to m is often overlooked.
5. Ray Diagram Inaccuracies: While drawing, students might not make rays pass exactly through F or $C/2F$, leading to incorrect image positions/sizes. Always use a ruler and ensure rays are straight.
6. Confusing Real/Virtual with Erect/Inverted: Real images are always inverted, and virtual images are always erect (for single mirrors/lenses). This relationship helps cross-check answers.
7. Not Stating Nature of Image: Board exam questions often ask for the "nature, position, and size" of the image. Just calculating $v$ and $h'$ is not enough; you must interpret them (e.g., real, inverted, magnified).
8. Refractive Index Direction: For $n_{21}$, it's the refractive index of medium 2 with respect to medium 1, meaning light travels from 1 to 2.

Quick Revision

• Reflection: Bouncing back of light.
• Laws: $\angle i = \angle r$, incident ray, reflected ray, normal are coplanar.
• Plane Mirror: Virtual, erect, same size, laterally inverted image.
• Spherical Mirrors: Concave (converging), Convex (diverging).
• Mirror Formula: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$
• Magnification (Mirror): $m = \frac{h'}{h} = -\frac{v}{u}$
• Sign Convention: Distances left of $P/O$ are negative, right are positive. Above principal axis positive, below negative. Concave $f$ is negative, Convex $f$ is positive.
• Refraction: Bending of light as it passes from one medium to another.
• Laws: Incident ray, refracted ray, normal are coplanar. $\frac{\sin i}{\sin r} = \text{constant}$ (Snell's Law).
• Refractive Index (n): $n = \frac{c}{v}$. $n_{21} = \frac{v_1}{v_2} = \frac{n_2}{n_1}$.
• Optical Density: Higher $n$ means optically denser. Light bends towards normal (rarer to denser) and away from normal (denser to rarer).
• Spherical Lenses: Convex (converging), Concave (diverging).
• Lens Formula: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$
• Magnification (Lens): $m = \frac{h'}{h} = \frac{v}{u}$
• Sign Convention: Convex $f$ is positive, Concave $f$ is negative.
• Power of a Lens (P): $P = \frac{1}{f}$ (f in metres). Unit is Dioptre (D). Convex lens has positive power, Concave lens has negative power.
• Key Relationships:
• $f = R/2$ for mirrors.
• Real images are inverted; Virtual images are erect.
• Negative $m$ means real, inverted. Positive $m$ means virtual, erect.
• $|m| > 1$ (magnified), $|m| < 1$ (diminished), $|m| = 1$ (same size).