Light – Reflection & Refraction (Chapter 9) — Detailed Notes

Straight-Line Propagation & Dual Nature of Light

Visibility of Objects
- Objects are seen because they reflect light into our eyes; a dark room becomes visible only after illumination.
- Transparent media transmit light, allowing us to “see through” them.
Evidence for Straight-Line Travel (Ray Model)
- Sharp shadows from small sources imply rectilinear propagation.
- A ray is the idealised straight path of light.
Wave & Quantum Views (Connections to Higher Studies)
- Diffraction (bending around very small obstacles) shows the limitations of the ray model; light then behaves as a wave.
- Early 20th-century experiments revealed particle-like behaviour; modern quantum theory reconciles wave & particle aspects.

Reflection of Light

Highly polished surfaces (mirrors) reflect most incident light.

Laws of Reflection

$\angle i = \angle r$ (equality of angles of incidence & reflection).
Incident ray, reflected ray and normal lie in the same plane.
Valid for all reflecting surfaces—plane or spherical.

Plane-Mirror Images

Virtual, erect, same size, laterally inverted, positioned as far behind mirror as object is in front.

Spherical Mirrors

Concave (converging): reflecting surface curves inwards (toward sphere centre).
Convex (diverging): reflecting surface curves outwards.

Essential Terms (common to both)

Pole (P) – geometric centre of mirror surface.
Centre of Curvature (C) – centre of the sphere of which mirror surface is a part (in front of concave, behind convex).
Radius of Curvature (R) – distance $PC$ .
Principal Axis – line through $P$ and $C$ ; normal to mirror at $P$ .
Principal Focus (F)
- Concave: point where rays parallel to principal axis actually meet after reflection.
- Convex: point from which reflected rays appear to diverge.
Focal Length (f) – $PF$ ; for small-aperture mirrors $R = 2f$ .
Aperture – effective diameter of reflecting surface.

Image Formation by Concave Mirror (Object on Principal Axis)

Object Position	Image Position	Size	Nature
At $\infty$	At $F$ (point)	Highly diminished	Real, inverted
Beyond $C$	Between $F$ & $C$	Diminished	Real, inverted
At $C$	At $C$	Same	Real, inverted
Between $C$ & $F$	Beyond $C$	Enlarged	Real, inverted
At $F$	At $\infty$	Infinitely large	Real, inverted
Between $P$ & $F$	Behind mirror	Enlarged	Virtual, erect

Image Formation by Convex Mirror

Object Position	Image Position	Size	Nature
At $\infty$	At $F$ (behind mirror)	Highly diminished (point)	Virtual, erect
Finite distance	Between $P$ & $F$ (behind)	Diminished	Virtual, erect

Ray Diagram Rules (Any 2 are sufficient)

Ray parallel to principal axis → passes through (concave) or appears from (convex) $F$ .
Ray through $F$ (or toward $F$ for convex) → reflects parallel to axis.
Ray through $C$ (or toward $C$ for convex) → retraces path (normal incidence).
Ray through pole $P$ → obeys law of reflection (symmetrical about axis).

Sign Convention (New Cartesian) for Mirrors

Origin at pole $P$ ; principal axis is $+x$ direction toward incident light.
- Distances right of $P$ : $+$ ; left: $-$ .
- Distances above axis: $+$ ; below: $-$ .

Mirror Formula & Magnification

Mirror formula: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ .
Magnification: $m = \frac{h'}{h} = -\frac{v}{u}$ .
- m<0 ⇒ real, inverted; m>0 ⇒ virtual, erect.

Practical Uses

Concave: torches, searchlights, car headlights (parallel beam); shaving/makeup mirrors (magnified upright view); dentists’ mirrors; solar furnaces (concentrate heat).
Convex: rear-view mirrors of vehicles (wide field, erect diminished image); wall-mounted tourist mirrors (full view of large monuments).

Refraction of Light

Phenomenon: bending of light at interface between two media due to change in speed.
Everyday evidence: raised pool bottom, bent pencil in water, magnified lemon, letters raised under glass slab.

Laws of Refraction (Snell’s Laws)

Incident ray, refracted ray & normal lie in one plane.
$\dfrac{\sin i}{\sin r} = \text{constant}$ (for given colour & media pair).
- Constant is refractive index $n_{21}$ .

Refractive Index

Relative: $n{21} = \dfrac{v1}{v2}$ where $v1$ , $v_2$ are light speeds in media 1 & 2.
Absolute: $n = \dfrac{c}{v}$ where $c = 3\times10^{8}\,\text{m s}^{-1}$ .
Higher $n$ ⇒ optically denser (not necessarily mass-denser).
Typical values (air $\approx1$ , water $1.33$ , glass $1.5$ , diamond $2.42$ ).

Rectangular Glass Slab

Two refractions (air→glass, glass→air).
Emergent ray parallel to incident; lateral shift occurs.

Lenses

Classification & Geometry

Convex (double-convex / converging): thicker at centre; brings parallel rays to focus.
Concave (double-concave / diverging): thinner at centre; spreads rays.
Key points
- Two centres of curvature $C1, C2$ .
- Optical centre (O): ray through O suffers no deviation.
- Principal axis: line through $C1, C2$ & $O$ .
- Two equal principal foci $F1, F2$ ; focal length $f = OF1 = OF2$ .

Ray Diagram Rules for Lenses

Ray parallel to principal axis → passes through (convex) or appears from (concave) principal focus on opposite side.
Ray through (or toward) principal focus → emerges parallel to axis.
Ray through optical centre O → undeviated.

Image Formation by Convex Lens

Object Position	Image Position	Size	Nature
$\infty$	At $F_2$	Point, highly diminished	Real, inverted
Beyond $2F_1$	Between $F<em>2$ & $2F</em>2$	Diminished	Real, inverted
At $2F_1$	At $2F_2$	Same	Real, inverted
Between $F<em>1$ & $2F</em>1$	Beyond $2F_2$	Enlarged	Real, inverted
At $F_1$	$\infty$	Infinitely large	Real, inverted
Between $F_1$ & O	Same side as object	Enlarged	Virtual, erect

Image Formation by Concave Lens (any position)

Image always virtual, erect, diminished, located between $F_1$ & O on same side as object.

Sign Convention for Lenses

Origin at optical centre O.
- Measurements toward incident light (left of O) are negative; opposite direction positive.
Convex focal length f>0; concave f<0.

Lens Formula & Magnification

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$
Magnification $m = \frac{h'}{h} = \frac{v}{u}$
- m>0 ⇒ virtual, erect; m<0 ⇒ real, inverted.

Power of a Lens

$P = \frac{1}{f}$ (f in metres).
SI unit: dioptre (D).
- Convex: P>0; Concave: P<0.
Combined lenses in contact: $P = P1 + P2 + P_3 +\dots$ .

Practical & Ethical Implications

Magnifying glasses, spectacle lenses, microscopes, telescopes: rely on controlled refraction & combination of lens powers.
Safety caution: Never look directly at the Sun or its reflection/focus through mirrors/lenses—risk of eye damage or fire (demonstrated by burning paper in Activities 9.2 & 9.11).

Worked Examples & Numerical Data (Selected)

Mirror: For concave mirror with $R = 20\,\text{cm}$ → $f = 10\,\text{cm}$ .
Lens: Concave lens $f = -15\,\text{cm}$ , image at $v = -10\,\text{cm}$ → $u = -30\,\text{cm}$ , $m = +0.33$ (virtual, one-third size).
Power: Lens with $f = 0.50\,\text{m}$ has $P = +2.0\,\text{D}$ .

Real-World Connections & Devices

Automobile headlights (concave mirror) produce parallel beams.
Vehicle rear-view mirrors (convex) provide wide, diminished images.
Solar furnaces (large concave mirrors) concentrate solar energy.
Dentist’s mirror (concave) gives enlarged upright view when tool is within $P$ and $F$ .
Optical instruments: Cameras, microscopes, telescopes employ multiple lenses; designers use powers for quick calculation & chromatic/ spherical aberration minimisation.

Ethical, Philosophical & Safety Notes

Scientific exploration illustrates models evolve (ray → wave → quantum).
Experiments with intense light require safety precautions (eye protection, fire hazards).
Responsible innovation: understanding optics underpins technologies from medical devices to renewable energy concentrators.

Key Formulae (Quick Reference)

Mirror: $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ ; $m = -\frac{v}{u}$ .
Lens: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ ; $m = \frac{v}{u}$ .
Radius–focus: $R = 2f$ (spherical mirror).
Snell: $\dfrac{\sin i}{\sin r} = n{21} = \dfrac{v1}{v_2}$ .
Absolute index: $n = \dfrac{c}{v}$ .
Power: $P=\dfrac{1}{f}\,(\text{metres})$ ; $1\,\text{D} = 1\,\text{m}^{-1}$ .

Theory Questions (Concepts & Definitions) Solutions

Define the following terms related to spherical mirrors and lenses:
- Pole (P): The geometric centre of the mirror surface.
- Centre of Curvature (C): The centre of the sphere of which the mirror surface is a part (in front of concave mirror, behind convex mirror). For lenses, C1C1 and C2C2 are centres of curvature of the two spherical surfaces.
- Radius of Curvature (R): The distance PCPC for a mirror. For a lens, it's the radius of the sphere from which the lens surface is cut.
- Principal Axis: The straight line passing through the Pole (P) and the Centre of Curvature (C) of a mirror. For a lens, it's the line connecting the two centres of curvature (C1,C2C1,C2) and the Optical Centre (O).
- Principal Focus (F):
  - Concave Mirror: A point on the principal axis where rays parallel to the principal axis actually meet after reflection.
  - Convex Mirror: A point on the principal axis from which reflected rays appear to diverge after reflection (located behind the mirror).
  - Convex Lens: A point on the principal axis where rays parallel to the principal axis converge after refraction (on the opposite side).
  - Concave Lens: A point on the principal axis from which refracted rays appear to diverge after refraction (on the same side as the object).
- Focal Length (f): The distance PFPF for mirrors. For lenses, it is the distance OF1=OF2OF1=OF2, where O is the optical centre.
- Optical Centre (O): The geometric centre of a lens. A ray of light passing through the optical centre of a lens goes undeviated.
State the Laws of Reflection.
- The angle of incidence (hetaihetai) is equal to the angle of reflection (hetarhetar) (hetai=hetarhetai=hetar).
- The incident ray, the reflected ray, and the normal to the surface at the point of incidence all lie in the same plane. This is valid for all reflecting surfaces—plane or spherical.
State the Laws of Refraction, including Snell's Law.
- The incident ray, the refracted ray, and the normal to the interface of two transparent media at the point of incidence all lie in one plane.
- For a given pair of media and a given colour of light, the ratio of the sine of the angle of incidence to the sine of the angle of refraction is constant. This is Snell's Law: sin⁡isin⁡r=constantsinrsini=constant. This constant is known as the refractive index (n21n21).
What is the relationship between the Radius of Curvature (R) and Focal Length (f) for a spherical mirror?
- For small-aperture spherical mirrors, the radius of curvature (RR) is twice the focal length (ff): R=2fR=2f.
Distinguish between real and virtual images.
- Real Image: Formed when light rays actually converge and meet at a point after reflection or refraction. Real images can be obtained on a screen and are always inverted.
- Virtual Image: Formed when light rays appear to diverge from a point after reflection or refraction. Virtual images cannot be obtained on a screen and are always erect.
Differentiate between concave and convex mirrors based on their reflecting surfaces and convergence/divergence of light.
- Concave Mirror (converging mirror): Its reflecting surface curves inwards (towards the centre of the sphere). It converges parallel rays of light after reflection.
- Convex Mirror (diverging mirror): Its reflecting surface curves outwards. It diverges parallel rays of light after reflection.
Differentiate between convex and concave lenses based on their thickness at the centre/edges and their effect on parallel rays of light.
- Convex Lens (double-convex / converging lens): It is thicker at the centre and thinner at the edges. It converges parallel rays of light to a single point (focus) after refraction.
- Concave Lens (double-concave / diverging lens): It is thinner at the centre and thicker at the edges. It diverges parallel rays of light after refraction.
Define Refractive Index (absolute and relative).
- Relative Refractive Index (n21n21): It is the ratio of the speed of light in medium 1 (v1v1) to the speed of light in medium 2 (v2v2): n21=v1v2n21=v2v1.
- Absolute Refractive Index (nn): It is the ratio of the speed of light in vacuum (cc) to the speed of light in a given medium (vv): n=cvn=vc. The speed of light in vacuum is approximately 3×108 m s−13×108m s−1.
- A higher refractive index (nn) means the medium is optically denser, implying that the speed of light is lower in that medium.
What is the Power of a Lens? State its SI unit and formula (P=1fP=f1). How does the power of a convex lens differ from that of a concave lens?
- Power of a Lens: It is a measure of the degree of convergence or divergence of light rays that a lens can produce. It is defined as the reciprocal of its focal length.
- Formula: P=1fP=f1 (where ff must be in metres).
- SI Unit: Dioptre (D). (1 Dioptre = 1 m−11m−1).
- Convex Lens: Has a positive focal length, so its power (PP) is positive (P>0P>0).
- Concave Lens: Has a negative focal length, so its power (PP) is negative (P<0P<0).
Explain the New Cartesian Sign Convention for spherical mirrors and lenses.
- Origin: All distances are measured from the pole (P) for mirrors and from the optical centre (O) for lenses.
- Incident Light Direction: The principal axis is taken as the +x+x direction, and the incident light is always considered to travel from left to right (or from a given direction towards the optical component).
- Distances along Principal Axis: Distances measured to the right of the origin (P or O) along the principal axis are taken as positive ($+$), while those measured to the left are taken as negative ($-\text{}).DistancesPerpendiculartoPrincipalAxis:Distancesmeasuredupwards(abovetheprincipalaxis)andperpendiculartoitaretakenaspositive().DistancesPerpendiculartoPrincipalAxis:Distancesmeasuredupwards(abovetheprincipalaxis)andperpendiculartoitaretakenaspositive($+$). Distances measured downwards (below the principal axis) and perpendicular to it are taken as negative ($-\text{}).

Real-Life Questions & Applications Solutions

Why do objects appear raised when viewed through water (e.g., a pool bottom, a coin in a glass)?
- This phenomenon is due to the refraction of light. When light rays from an object submerged in water (an optically denser medium) travel into the air (a rarer medium), they bend away from the normal at the water-air interface. Our eyes then perceive these bent rays as originating from a point shallower than the actual position, making the object appear raised.
Why are convex mirrors preferred as rear-view mirrors in vehicles?
- Convex mirrors are preferred because they always form a virtual, erect, and diminished image, irrespective of the distance of the object. Crucially, their outward curved surface provides a significantly wider field of view compared to plane mirrors, allowing drivers to see a much larger area of the traffic behind them, thus enhancing safety.
Why are concave mirrors used in torches, searchlights, and car headlights?
- In these devices, the light source (bulb) is placed at the principal focus (FF) of the concave mirror. According to the ray diagram rules, any ray passing through the focus of a concave mirror becomes parallel to the principal axis after reflection. This property allows the concave mirror to produce a strong, parallel beam of light for illumination over long distances.
How is a concave mirror used by a dentist?
- A dentist uses a concave mirror to get a magnified and erect (upright) virtual image of the tooth being examined. This magnification occurs when the tooth (object) is placed between the pole (PP) and the principal focus (FF) of the concave mirror. This allows the dentist to see a larger, detailed view of the tooth.
Explain why a lemon kept in water in a glass tumbler appears larger than its actual size when viewed from the sides.
- This is due to refraction of light through the water and the curved glass surface, which collectively act somewhat like a convex lens. Light rays from the lemon passing from the water (denser medium) to the glass and then to the air (rarer medium) bend. The curvature of the tumbler filled with water causes the light rays to diverge in such a way that when they reach our eyes, the lemon appears to be enlarged, similar to how an object placed within the focal length of a convex lens is magnified.
Why is it dangerous to look directly at the Sun or its reflection/focus through mirrors or lenses?
- Concave mirrors and convex lenses are designed to converge (concentrate) light rays. When sunlight (which is very intense) is focused by such an optical component, it concentrates a large amount of solar energy into a very small, intense point (at the focus). Looking directly into this concentrated light can cause severe and irreversible damage to the retina of the eye, leading to blindness. It can also cause fires by igniting combustible materials placed at the focal point due to the intense heat generated.
Briefly explain the principle behind solar furnaces.
- Solar furnaces utilize large concave mirrors (or a system of many plane mirrors arranged to act as a large concave mirror). These mirrors are used to concentrate a vast amount of incident solar energy (sunlight) onto a very small area at their principal focus. The intense concentration of sunlight generates extremely high temperatures, which can be used for various industrial processes like melting metals, producing steam for electricity, or other high-temperature applications.
If you look at your reflection in a shiny spoon, what kind of image do you see on the inner and outer surfaces?
- Inner surface of the spoon (concave mirror): If you look very close (when your face is within its focal length), you will see an enlarged, erect, virtual image. If you move farther away, the image will become real and inverted, and its size will vary depending on your distance from the focal point and centre of curvature.
- Outer surface of the spoon (convex mirror): This surface acts as a convex mirror. A convex mirror always forms a diminished, erect, virtual image, regardless of the object's distance. So, you will see a smaller, upright image of yourself.
Why does a pencil appear bent or broken when partially immersed in water?
- This illusion is caused by the refraction of light. Light rays coming from the part of the pencil immersed in water (denser medium) bend away from the normal as they cross the water-air interface to enter our eyes (rarer medium). This bending causes the submerged part of the pencil to appear at a different position than its actual location