Chapter 10 — Light Reflection And Refraction

Rectilinear propagation of light: Light travels in straight lines in a uniform medium. This is called rectilinear propagation. It explains why shadows form and how images are formed in pinhole cameras.

Simple demonstration: A small hole (pinhole) in an opaque screen produces well-defined spots of light on a screen placed behind it. Each hole lets through nearly straight rays that make a small bright patch; shifting the source or hole shifts the patch along a straight line. This shows light rays travel along straight lines.

Formation of shadows: When an opaque object blocks light from a source, the region behind the object that receives no light becomes a shadow. For a point source (or a source so small it can be treated as a point) the shadow has well-defined, sharp edges because rays from the single point source that reach the screen are either completely blocked or completely unblocked by the object.

Extended source: umbra and penumbra: For a source with finite size (extended source), some rays from different parts of the source are blocked while others pass the object. This creates two regions on the screen:

• Umbra — the fully dark part where all rays from the source are blocked.
• Penumbra — the partially lit/fuzzy region where only some rays from the source are blocked.

Thus larger sources produce wider penumbrae (fuzzier shadows); very small or distant sources produce sharper shadows.

Dependence of shadow size and sharpness: Using straight-line geometry (similar triangles) you can relate the object height and shadow height. If the source is effectively a point located at distance u from the object and the screen is at distance v from the same point (so the object is at distance u from the source and the screen at v), then rays from the source passing the top and bottom of the object determine the size of the shadow on the screen. The shadow height H and object height h are proportional to distances:

H = h × (v / u)

So:

• If the screen is moved farther from the source (v increases) the shadow size increases linearly.
• If the object is moved closer to the source (u decreases) the shadow size increases.

Practical notes and examples: Solar shadows are fuzzy at sunrise/sunset (large effective source — the Sun appears large or atmosphere scatters light) and sharp under a small bright lamp or laser pointer. Eclipses illustrate umbra and penumbra: total eclipse occurs for observers in the Moon's umbra; partial eclipse for those in the penumbra.

Definition: Reflection of light is the phenomenon in which light rays falling on a surface bounce back into the same medium. Reflection can be regular (specular) from smooth surfaces or irregular (diffuse) from rough surfaces.

Laws of reflection:

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

Types of reflection: Regular reflection (from plane/smooth surfaces) produces clear images; diffuse reflection (from rough surfaces) scatters light and does not produce a clear image.

Plane mirrors: A plane mirror forms a virtual, erect image which is laterally inverted, of the same size as the object, and at a distance behind the mirror equal to the object distance in front (image distance v = −u, using sign convention where object distance u is negative).

Spherical mirrors (concave and convex): Important points: pole (P), centre of curvature (C), radius of curvature (R), principal focus (F) and focal length (f). For spherical mirrors the focal length is f = R/2.

Image formation (qualitative):

• Concave mirror: Can produce real, inverted images (when object is beyond focus) or virtual, erect, magnified images (when object is between focus and mirror).
• Convex mirror: Always produces a virtual, erect, diminished image behind the mirror.

Mirror formula and magnification (quantitative): For spherical mirrors (concave/convex) the mirror equation relates object distance (u), image distance (v) and focal length (f): 1/v + 1/u = 1/f. Lateral magnification m is given by m = h_i/h_o = −v/u, where h_i is image height and h_o is object height. Sign conventions (Cartesian) must be followed: object distance u is taken negative for real objects in front of the mirror; v and f signs depend on mirror type.

Real vs virtual images: A real image is formed when reflected rays actually meet (can be projected on a screen); a virtual image is formed when reflected rays appear to diverge from a point behind the mirror (cannot be projected).

Practical points and uses: Reflection is exploited in everyday mirrors (plane), vehicle rear-view mirrors (convex), shaving/makeup mirrors (plane or concave for magnification), torch and headlight reflectors (concave), solar concentrators, telescopes and satellite dishes.

Definition: A spherical mirror is a mirror whose reflecting surface is a part of a spherical surface. There are two types: concave (inner surface reflective) and convex (outer surface reflective).

Important terms:

• Pole (P): the center of the mirror surface.
• Principal axis: the line through P and the centre of curvature.
• Centre of curvature (C): centre of the sphere of which the mirror is a part. Radius = R.
• Focal point (F): point where paraxial rays parallel to principal axis meet (concave) or appear to meet (convex). Focal length f = PF = R/2.
• Aperture: the effective diameter of the mirror.

Ray rules (principal rays) — useful for ray diagrams:

• Ray parallel to principal axis → after reflection passes through (concave) or appears to come from (convex) the focal point F.
• Ray through centre of curvature C → reflects back along the same path (normal incidence).
• Ray through focal point F → after reflection becomes parallel to principal axis (concave); for convex, a ray aimed toward F reflects parallel but appears to come from F behind the mirror.

Image formation by concave mirror (qualitative cases):

• Object at infinity → image at F: real, point-sized (highly diminished).
• Object beyond C (u > 2f) → image between C and F: real, inverted, diminished.
• Object at C (u = 2f) → image at C: real, inverted, same size.
• Object between C and F (f < u < 2f) → image beyond C: real, inverted, enlarged.
• Object at F (u = f) → image at infinity.
• Object between F and P (u < f) → image behind the mirror: virtual, erect, enlarged.

Image formation by convex mirror (general): For any object position, the image is virtual, erect, diminished and located behind the mirror between P and F' (the virtual focal point).

Sign convention (Cartesian, brief): Distances measured from the pole. Distances measured in the direction of incident light are positive; opposite direction negative. Heights above principal axis are positive. Using sign convention correctly gives correct algebraic values for u (object distance), v (image distance) and f (focal length).

Practical uses (brief): Concave mirrors concentrate light and form magnified images when close to the mirror (used in torches, shaving mirrors, solar concentrators, reflecting telescopes). Convex mirrors give a wide field of view and erect diminished images (used as rear-view mirrors, security mirrors, road safety mirrors).

Typical classroom activities: Draw ray diagrams for the standard object positions (beyond C, at C, between C and F, at F, between F and P) for a concave mirror and for various objects in front of a convex mirror. Verify image location using the mirror equation.

What the formulas relate
The mirror formula connects the object distance (u), image distance (v) and focal length (f) of a spherical mirror (concave or convex):

1/f = 1/v + 1/u

The magnification relates the heights of image (h') and object (h) to the distances:

m = h'/h = -v/u

(A negative m means the image is inverted; a positive m means the image is erect.)

Derivation (sketch for a concave mirror)

• Draw principal axis, pole P, centre of curvature C, focal point F and an object AB placed on the axis. Draw two rays from the top A: one parallel to the axis (reflects through F) and one through C (reflects back along itself). Their intersection gives image A'B'.
• Using similar triangles formed by rays and the axis, one obtains AB/A'B' = (u+v)/v and also AB/A'B' = -u/v (signs explained by convention). Simplifying these relations leads to 1/f = 1/v + 1/u.

Sign convention (Cartesian, CBSE style — concise)

• All distances are measured from the pole (P) along the principal axis.
• Distances measured in the direction of incident light are taken as positive; those opposite are negative.
• With the usual arrangement (object placed in front of the mirror): object distance u is taken negative. For a concave mirror the focal length f is negative (F lies in front of the mirror), while for a convex mirror f is positive (F lies behind the mirror). The image distance v and height h' may be positive or negative depending on where and how the image forms.

Physical meanings and common cases

• f = R/2 where R is the radius of curvature.
• Concave mirror:
• Object beyond C (u > 2f): real, inverted image between C and 2f, |m| < 1 (diminished).
• Object at C (u = 2f): real, inverted image at C, |m| = 1 (same size).
• Object between C and F: real, inverted image beyond C, |m| > 1 (enlarged).
• Object at F: image at infinity.
• Object between F and mirror: virtual, erect, enlarged image behind mirror (used in makeup mirrors).
• Convex mirror: always forms a virtual, erect, diminished image behind the mirror (used as rear-view and shop security mirrors).

How to use the formulas (practical steps)

1. Choose the Cartesian sign convention and assign signs to u and f accordingly.
2. Substitute numerical values into 1/f = 1/v + 1/u and solve for v.
3. Compute magnification m = -v/u and then image height h' = m × h.

Worked numeric example (concise)

Concave mirror of focal length 15 cm; object is placed 30 cm in front of mirror. Take u = −30 cm, f = −15 cm. Using 1/f = 1/v + 1/u:

1/(−15) = 1/v + 1/(−30) ⇒ −1/15 = 1/v − 1/30 ⇒ 1/v = −1/15 + 1/30 = −1/30 ⇒ v = −30 cm.

So image is 30 cm in front of the mirror (real, inverted). Magnification m = −v/u = −(−30)/(−30) = −1 ⇒ image is of same size but inverted.

Practical examples

• Concave mirrors in shaving/face mirrors and torch reflectors (produce enlarged or focused images).
• Dental mirrors and magnifying cosmetic mirrors (virtual, enlarged images).
• Convex mirrors as rear-view and security mirrors (virtual, erect, diminished images with wide field of view).
• Large concave mirrors used as solar concentrators (focus parallel rays to the focal point).

Definition: Refraction is the bending of a light ray when it passes from one transparent medium to another due to a change in its speed.

Why refraction occurs: Light travels at different speeds in different media. When a light ray meets the interface at an angle, one side of the wavefront changes speed before the other, causing the ray to change direction (bend).

Laws of refraction (Snell's laws):

• 1st law: The incident ray, the refracted ray and the normal to the interface at the point of incidence all lie in the same plane.
• 2nd law (Snell's law): The ratio of the sines of the angle of incidence i and angle of refraction r is constant for a given pair of media: n1 sin i = n2 sin r.

Refractive index:

• Absolute refractive index of a medium (with respect to vacuum): μ = c / v, where c is speed of light in vacuum and v is speed in the medium.
• Relative refractive index of medium 2 with respect to medium 1: μ_{21} = n2 / n1 such that n1 sin i = n2 sin r. Often for air→medium we use μ ≈ sin i / sin r.

Special cases & related phenomena:

• Bending toward normal: When light enters a denser medium (higher refractive index), it bends toward the normal (r < i).
• Bending away from normal: When light goes into a rarer medium (lower refractive index), it bends away from the normal (r > i).
• Apparent depth: Objects under a denser medium (e.g., water) appear nearer the surface. For near-vertical viewing, μ = real depth / apparent depth.
• Critical angle and total internal reflection (TIR): If light goes from denser (n1) to rarer (n2) medium and incidence angle exceeds the critical angle C, the ray is totally internally reflected. Critical angle: sin C = n2 / n1 (for n1 > n2).
• Lateral shift through a slab: A glass slab displaces a transmitted ray sideways. The amount depends on thickness and angles.

Experimental observations and uses: Refraction explains many everyday effects and technologies: why a straw looks bent in a glass of water, why pools look shallower, formation of mirages, focusing by lenses, and optical fibers using TIR for low-loss light transmission.

Typical classroom ray diagrams to draw:

• Ray entering a denser medium (bends toward normal).
• Ray leaving denser to rarer (bends away from normal).
• Ray through a parallel-sided glass slab showing incident, refracted, emergent rays and lateral shift.
• Ray at the critical angle showing emergence at 90° and beyond showing TIR.

Definition: The refractive index of a medium is a number that indicates how much the speed of light is reduced inside that medium compared to its speed in vacuum. It also determines how much light bends when it passes from one medium to another.

Absolute refractive index (n): n = c / v, where c is the speed of light in vacuum (≈ 3.00 × 10^8 m/s) and v is the speed of light in the medium. A larger n means light travels slower in that medium and the medium is said to be optically denser.

Relative refractive index: When light goes from medium 1 to medium 2, the relative refractive index of medium 2 with respect to medium 1 is given by n_{21} = v1 / v2. Using absolute indices, n_{21} = n2 / n1.

Snell's law (relation with sines): For a ray going from medium 1 into medium 2, Snell's law states n1 sin i = n2 sin r, where i is the angle of incidence and r the angle of refraction (angles measured from the normal). From this we get:

• sin i / sin r = n2 / n1 = v1 / v2

Wavelength and frequency: When light enters a medium, its frequency f remains the same but its speed and wavelength change. v = f λ, so λ_in_medium = v / f = (c / n) / f = λ0 / n, where λ0 is the wavelength in vacuum.

Optical density and bending: If light enters an optically denser medium (higher n), it slows down and bends toward the normal. If it enters a less dense medium (lower n), it speeds up and bends away from the normal.

Critical angle and total internal reflection (brief): When light tries to pass from a denser medium to a less dense medium, beyond a certain incident angle called the critical angle i_c, no refraction occurs and all light is reflected back. The critical angle satisfies sin i_c = n2 / n1 (for n1 > n2).

Examples of typical refractive indices and speeds: air (n ≈ 1.0003, v ≈ 2.998 × 10^8 m/s), water (n ≈ 1.33, v ≈ 2.26 × 10^8 m/s), glass (n ≈ 1.5, v ≈ 2.00 × 10^8 m/s), diamond (n ≈ 2.42, v ≈ 1.24 × 10^8 m/s).

Key ideas to remember: n = c / v; frequency is unchanged across media; wavelength decreases in denser media; higher n implies more bending toward the normal.

Basic idea
When a light ray travels from one medium to another (air → glass → air), its direction changes at each surface because of change in speed. This change of direction is called refraction. For a parallel-sided transparent slab (glass slab) the emergent ray is parallel to the incident ray but is laterally shifted (displaced) sideways.

What happens step by step

• Incident ray meets the first surface at angle of incidence i (measured from the normal). Using Snell's law it refracts into the glass at angle r (toward the normal if glass is denser).
• Inside the slab the ray travels in a straight line. At the second (exit) surface the ray goes from glass back to air and refracts away from the normal. The angle of emergence (in air) equals the original angle of incidence (i), so the emergent ray is parallel to the incident ray.
• Because the emergent ray is displaced sideways from the original path, the ray shows a lateral displacement (also called lateral shift) Δ which depends on the slab thickness, angles and refractive index.

Key points

• Angle of emergence = angle of incidence (for a parallel-sided slab).
• Emergent ray is parallel to the incident ray but shifted sideways (lateral displacement).
• Lateral displacement increases with slab thickness and with angle of incidence (up to a point), and decreases with larger refractive index (for fixed other parameters).

Derivation (concise)
Using geometry inside the slab: if slab thickness = t, incident angle = i and refracted angle inside glass = r, the lateral displacement Δ is given by Δ = t * sin(i - r) / cos r. This follows from projecting the internal path and comparing entry and exit points. Using Snell's law (sin i = μ sin r for air→glass with μ = refractive index of glass) you can express r in terms of i and μ if needed.

Apparent depth (special case for near-normal viewing)
When you look normally (i ≈ 0) into a glass slab of real thickness t, the slab appears shallower. Apparent thickness t' ≈ t / μ. The apparent shift (toward observer) = t - t/μ.

Why this matters
This phenomenon explains many everyday observations (objects under water or behind glass look shifted), and is used in optical devices and thickness measurements.

Definition: Apparent depth is the depth at which an object submerged in a transparent medium (like water or glass) appears to an observer looking from another medium (usually air). Due to refraction at the flat surface, the object appears shifted from its true position — typically nearer the surface.

Why it happens: Light rays from the object travel through the denser medium and refract (bend) when emerging into the rarer medium (e.g., air). Rays bend away from the normal, and the brain back-traces the emergent rays in straight lines. The intersection of these back-traced rays is a virtual image located at a shallower depth than the real object.

Derivation (paraxial / small-angle approximation):

• Consider an object at true (real) depth d below a horizontal interface between two media: medium 2 (refractive index n2, where the object is) and medium 1 (refractive index n1, where the observer is). A ray from the object at height h from the axis emerges and makes angles r (in medium 2) and i (in medium 1) with the normal.
• Geometry gives tan r = h/d and tan i = h/d'. For small angles (paraxial rays) tan ≈ sin ≈ angle, so sin r ≈ h/d and sin i ≈ h/d'.
• Snell's law: n2 sin r = n1 sin i. Using the approximations and canceling h, n2 (1/d) = n1 (1/d'). Rearranging gives the relation below.

General formula: d' = (n1 / n2) · d
where d = real depth (in medium with refractive index n2), d' = apparent depth (as seen from medium with refractive index n1).

Common case (observer in air): For air n1 ≈ 1, so
d' = d / n2. If the medium is water (n2 ≈ 4/3), then d' = d / (4/3) = 3d/4 — the object appears at 3/4 of its real depth.

Apparent shift: The shift toward the surface = Δd = d - d' = d (1 - n1/n2). For observer in air, Δd = d (1 - 1/n2).

Important notes:

• Apparent depth is a virtual image property — no real image is formed at that location.
• The formula above assumes paraxial rays (small angles) and a flat horizontal interface; for large incident angles exact trigonometric relations from Snell's law must be used and the simple linear relation becomes less accurate.
• If the observer is in the denser medium looking into a rarer medium, the apparent depth formula still applies with indices swapped: the image may appear deeper.

Practical consequence: Objects submerged in water (coins, fish, pool bottom) appear closer to the surface than they really are; this must be accounted for in activities like spearfishing or depth estimation.

Definition: Total internal reflection (TIR) is the phenomenon in which a light ray travelling in a denser medium (higher refractive index) and striking the boundary with a rarer medium (lower refractive index) is completely reflected back into the denser medium, with no refracted ray in the rarer medium.

Conditions for TIR:

• The light must travel from a denser medium (refractive index n1) to a rarer medium (refractive index n2), so n1 > n2.
• The angle of incidence i at the boundary must be greater than a specific angle called the critical angle (c). If i > c, TIR occurs.

Derivation of the critical angle: By Snell's law, n1 sin i = n2 sin r. The critical angle c is defined when the refracted ray grazes along the boundary (r = 90°), so sin r = 1. Putting r = 90° gives:

n1 sin c = n2 · 1  ⇒  sin c = n2 / n1

Thus c = sin^-1(n2 / n1). TIR occurs when i > c (and only if n1 > n2).

Behaviour of rays: For i < c, part of the light is refracted into the rarer medium and part is reflected. At i = c the refracted ray travels along the interface (grazing emergence). For i > c, no refracted ray exists and all the light is reflected back — angle of reflection equals angle of incidence.

Additional note: Although no energy is carried away into the rarer medium by a propagating refracted ray when i > c, an evanescent wave penetrates a short distance into the rarer medium and decays exponentially. This is important in some applications (evanescent coupling).

Typical numerical examples: For glass to air (n1 ≈ 1.50, n2 ≈ 1.00), c ≈ sin^-1(1/1.5) ≈ 41.8°. For diamond to air (n1 ≈ 2.42), c ≈ sin^-1(1/2.42) ≈ 24.4°.

Applications: TIR is the principle behind optical fibers, prism reflectors in binoculars and periscopes, light pipes, certain endoscopes and many devices that guide light with very low loss. It is also the reason diamonds sparkle: high refractive index gives a small critical angle so light is trapped and undergoes many internal reflections before leaving.

Classroom demonstration: Shine a laser beam from inside a semi-circular glass block toward the flat face; rotate the beam to see partial refraction for small incidence, grazing emergence at the critical angle, and TIR beyond it. This clearly shows the three regimes.

What is total internal reflection (TIR)?
TIR occurs when light travels from a denser medium (higher refractive index n1) to a rarer medium (lower refractive index n2) and the angle of incidence (measured from the normal) is greater than the critical angle θc. Under these conditions all the light is reflected back into the denser medium — there is no refracted ray.

Conditions for TIR
1) Light must go from denser to rarer medium (n1 > n2).
2) Angle of incidence θi > critical angle θc where θc = sin⁻¹(n2 / n1).

Why TIR is useful
Because TIR gives nearly 100% internal reflection with no loss by absorption (unlike metallic mirrors), it is exploited in many optical devices to guide, reflect and concentrate light with very high efficiency.

Main applications (summary)
• Optical fibres for communication, illumination & medical endoscopes — light is guided along a thin glass/plastic core by repeated TIR at the core–cladding boundary.
• Reflecting prisms in periscopes, binoculars and cameras — prisms use TIR instead of mirrors to produce bright, undistorted images and to change the direction of light compactly.
• Diamond sparkle and gem cutting — because diamond has a very high refractive index, its small critical angle causes many internal reflections, producing brilliance and fire.
• Retroreflectors and road safety devices — corner-cube prisms and some cat's-eye reflectors use internal reflections to return light back toward its source, improving visibility of signs and clothing at night.

Class 10 level explanation (how each application works)
• Optical fibres: A fibre has a core of refractive index n_core and a cladding of lower index n_clad. Light entering within the acceptance cone is trapped inside the core by repeated TIR at the core–cladding interface and travels long distances with little loss. This is the basis of high-speed data transmission (internet), cable TV and medical endoscopes.
• Prisms: Right-angled and Porro prisms are cut so that incoming rays strike an internal face at angles greater than the critical angle; these faces reflect light by TIR, replacing mirrors. This gives high reflectivity, correct image orientation (in binoculars) and durable optics.
• Diamonds: n ≈ 2.42, so θc ≈ 24.4°. When cut properly, rays entering a diamond encounter internal faces at angles greater than θc and are reflected internally several times before emerging — this increases brilliance.

Practical notes
• TIR is lossless except for small losses at surface imperfections; a thin cladding with slightly lower n protects the core and ensures TIR.
• The numerical aperture (NA) of a fibre determines how much light it can accept; larger NA → larger acceptance cone.

Definition and types
A lens is a transparent medium (usually glass or plastic) bounded by two surfaces (spherical or plane) that refracts light and forms images. Two basic types: convex (converging) lens (thicker at centre) and concave (diverging) lens (thicker at edges). Convex lenses have positive focal length (f > 0); concave lenses have negative focal length (f < 0).

Principal axis and focal points
The principal axis is the straight line through the centre. The principal focus (F) is the point where rays parallel to the axis meet (convex) or appear to diverge from (concave). The distance from optical centre to focus is the focal length f.

Three principal rays used in ray diagrams
- Ray 1: A ray parallel to the principal axis refracts through (or appears to come from) the focus.
- Ray 2: A ray through the optical centre passes undeviated (approximately).
- Ray 3: A ray through (or toward) the focus emerges parallel to the principal axis.

Image formation by a convex lens (thin lens approximation)
Convex lenses can form real or virtual images depending on object position (measured from lens centre):

• Object at infinity → Image at F (focal point). Image is real, highly diminished (point) and inverted.
• Object beyond 2F (object distance u > 2f) → Image between F and 2F (v between f and 2f). Image is real, inverted and diminished.
• Object at 2F (u = 2f) → Image at 2F (v = 2f). Image is real, inverted and same size.
• Object between F and 2F (f < u < 2f) → Image beyond 2F (v > 2f). Image is real, inverted and magnified.
• Object at F (u = f) → Image at infinity (rays emerge parallel). No image on screen; useful in collimators and telescopes.
• Object between lens and F (u < f) → Image is virtual, erect, magnified and on same side of lens as object (used in magnifying glass).

Image formation by a concave lens
A concave lens always produces a virtual, upright and diminished image located between the lens and its focal point (image is on the same side as the object). Rays diverge after refraction; when extended backward they meet at a virtual image.

Sign convention (classroom practical)
For many CBSE problems the simpler magnitude form is used: object distance u and image distance v are taken positive when measured from lens along the principal axis; focal length f is positive for convex and negative for concave. Be aware that advanced Cartesian sign convention assigns signs depending on direction of measurement; always check which convention your textbook uses.

Thin-lens approximation
We assume lens thickness is small compared with object and image distances so refraction can be treated as occurring at a single plane (optical centre).

Practical tips for drawing ray diagrams
1) Draw principal axis and lens centre. 2) Mark object position and focal points. 3) Draw at least two of the principal rays above to locate image intersection. 4) For virtual images, extend refracted rays backward with dotted lines to meet.

Overview: A lens is a transparent medium bounded by two spherical surfaces that refracts light to form images. Convex (converging) lenses can form real or virtual images; concave (diverging) lenses form only virtual images. Key quantities are focal length (f), image distance (v), object distance (u), magnification (m) and power (P).

Sign (Cartesian) convention (used in CBSE):

• Light travels left to right. The optical centre of the lens is the origin.
• Object distances (u) are measured from the lens: distances measured to the left of the origin are negative, to the right are positive.
• Image distances (v): real images (right of lens) are positive; virtual images (left) are negative.
• Focal length (f): + for convex lens, − for concave lens.

Lens formula (thin lens approximation):

Using the Cartesian sign convention, the lens formula is

1/f = 1/v - 1/u

Rearrangements useful in problems:

• v = uf / (u - f)
• 1/v = 1/f + 1/u

Derivation sketch (idea): Consider a thin convex lens and an object AB on the principal axis. Draw the standard rays (parallel ray refracted through focus, ray through centre undeviated). Using similar triangles formed by the rays and axis you obtain h_i/h_o = v/u. Using geometry and these ratios leads to 1/f = 1/v - 1/u.

Magnification (linear):

m = h_i / h_o = v / u

• m positive: image is erect (virtual for a single lens); m negative: image is inverted (real).
• |m| > 1: image magnified; |m| < 1: image diminished; |m| = 1: same size.

Power of a lens:

P = 1/f (f in metres)

• Unit: dioptre (D) where 1 D = 1 m^{-1}.
• Convex lens: P positive; concave lens: P negative.
• For thin lenses in contact: P_total = P_1 + P_2.

Image nature vs object position for a convex lens (useful summary):

• u > 2f: image real, inverted, diminished, v between f and 2f.
• u = 2f: image real, inverted, same size, v = 2f.
• f < u < 2f: image real, inverted, magnified, v > 2f.
• u = f: image at infinity (rays parallel).
• u < f: image virtual, erect, magnified (used in magnifying glass).

Tips for solving problems:

• Apply the sign convention consistently: often u is taken negative because object is left of lens.
• Use m = v/u to get image height once distances are known.
• To find focal length experimentally: measure several (u, v) pairs and use 1/v = 1/f + 1/u; plot 1/v vs 1/u; intercept gives 1/f.

What it means
A combination of thin lenses is two or more thin lenses placed along the same optical axis so that they act together to form a single optical system. The combined system has an effective focal length and produces images that depend on the focal lengths and separation of the lenses.

Two main arrangements

• Lenses in contact (separation d = 0): The effective focal length f_eff is given by 1/f_eff = 1/f1 + 1/f2. The combined power (in dioptres) is P = P1 + P2.
• Separated lenses (separation d > 0): Each lens forms an image which acts as the object for the next lens. The effective focal length of two thin lenses separated by distance d is

1/f_eff = 1/f1 + 1/f2 - d/(f1 f2)

where f1, f2 and d must be expressed in the same units. Equivalently, using powers (P = 1/f in metres):

P = P1 + P2 - d·P1·P2

Image formation procedure for separated lenses
1. Use the lens formula for the first lens to find image1: 1/v1 - 1/u1 = 1/f1 (or 1/v1 = 1/f1 + 1/u1 using magnitudes).
2. Treat image1 as the object for the second lens: object distance for lens2 is u2 = d - v1 (take care of sign convention).
3. Use lens formula for lens2 to find final image v2. Total lateral magnification = m = m1 × m2, where m1 = v1/u1 and m2 = v2/u2 (watch signs for inverted images).

Special points

• If lenses are in contact, powers simply add: P = P1 + P2.
• If P1 + P2 = 0 (equal and opposite powers in contact) the combination is optically neutral (f_eff = ∞) — it does not converge or diverge parallel rays.
• A converging and a diverging lens can produce a net converging or diverging system depending on their powers and separation.

Overview: This topic covers how to verify and illustrate the laws of reflection and refraction using simple experiments and how to draw ray diagrams to represent image formation by mirrors and refraction through slabs and prisms.

Experimental methods (what to do and why):

• Plane mirror (verify laws of reflection): Place a plane mirror on paper. Draw the mirror line and mark an incident ray striking the mirror. At the point of incidence draw the normal (perpendicular to mirror). Using a protractor measure the angle of incidence (i) and the reflected angle (r). Repeat for several incident angles. Observation: i = r and incident, reflected rays and normal lie in the same plane — this verifies the two laws of reflection.
• Pin (or needle) method for locating image in plane mirror: Fix two pins as object and view their virtual image in the mirror. From positions on either side, place pins to line up with the apparent image; the intersection of the lines of sight (extended) behind the mirror gives the virtual image position. Use this to show image distance = object distance and erect laterally inverted image.
• Refraction through a rectangular glass slab: Place a rectangular glass slab on paper. Draw an incident ray and mark entry and exit points. Draw normals at these points. Measure angle of incidence (i) and angle of refraction (r) at entry. Measure the emergent ray — it is parallel to incident but laterally shifted. Use repeated measurements to apply Snell's law and find refractive index n = sin i / sin r.
• Apparent depth method (find refractive index): Place a pin at the bottom of a beaker or trough of water or in a glass slab. Look from above and mark apparent position of pin. Measure actual depth and apparent depth. Refractive index n = real depth / apparent depth (when viewing normally).
• Semi-circular block (study critical angle & TIR): Use a semi-circular glass block and send a ray through the flat face so that it exits the curved face normally. Vary the incident angle at the curved surface to find the critical angle where refracted ray grazes along the surface; beyond this angle total internal reflection occurs.

Drawing ray diagrams (rules and common diagrams):

• Plane mirror: Draw incident ray and reflected ray with equal angles w.r.t. normal. Image is virtual, erect, same size and at equal distance behind mirror as object in front.
• Spherical mirrors (concave and convex) — principal rays: For concave mirror, use three principal rays: (1) Ray parallel to principal axis reflects through focus (F). (2) Ray through focus reflects parallel to axis. (3) Ray through center of curvature (C) reflects back on itself. Their intersection gives image location. For convex mirror, use same rays but the reflected rays appear to diverge from a virtual focus behind the mirror; image is virtual, erect and diminished.
• Refraction at a plane surface and through a slab: At each interface draw normal at point of incidence. Apply Snell's law to get direction change. For a slab, show the emergent ray parallel to incident ray and displaced sideways by the lateral shift. For a prism draw the incident, refracted and emergent rays and show deviation of ray toward base and dispersion if polychromatic light is considered.

Practical tips for experiments: use a sharp ray (laser or narrow beam from a slit), mark precise points and normals on paper, take repeated readings for different angles, plot graphs (sin i vs sin r) to reduce random errors and determine refractive index from slope.

Overview
Mirrors and lenses bend and reflect light to form images. Depending on shape and position, they produce virtual or real, erect or inverted, magnified or diminished images. These properties are exploited in many everyday devices — from rear‑view mirrors and shaving mirrors to cameras, microscopes and spectacles.

Mirrors

• Plane mirror: Produces a virtual, erect image of the same size as the object and laterally inverted. Used in bathrooms, dressing mirrors and periscopes (to change line of sight without changing the image size).
• Concave (converging) spherical mirror: Can form real, inverted images (when object is beyond the focal point) or magnified virtual images (when object is between the mirror and focal point). Uses: makeup and shaving mirrors (object within focal length gives magnified virtual image); dentist mirrors; torch/spotlight and car headlight reflectors (to produce a parallel beam by placing light at focus); solar concentrators and some telescope primary mirrors (to focus sunlight or distant light to a point).
• Convex (diverging) spherical mirror: Always forms a virtual, erect and diminished image behind the mirror. Uses: rear‑view and side‑view mirrors in vehicles (wider field of view), security mirrors in shops, road safety mirrors at blind corners.

Lenses

• Convex (converging) lens: Can produce a real inverted image (object beyond focal length) or a virtual magnified image (object within focal length). Uses: magnifying glass (object inside focal length gives enlarged virtual image); camera and projector objectives (form real images on film/sensor or screen); microscopes and telescopes (objective lenses form real images that are magnified by eyepieces); corrective lenses for hypermetropia (convex spectacles converge light to help focus on the retina).
• Concave (diverging) lens: Always forms a virtual, erect, diminished image. Uses: spectacles for myopia (concave lenses diverge light so distant objects are brought into focus on the retina); peepholes in doors (gives a wide-angle view); some beam‑expanding or correction systems in optics.

How these properties are used practically
Designers choose the mirror or lens type and the object‑to‑optical‑element distance to get the required image size, orientation and position. For example, a shaving mirror is a concave mirror used close to the face (object within focal length) to give an enlarged upright virtual image for detailed view; a car headlight uses a concave reflector with the bulb at the focus to produce a nearly parallel beam for long‑range illumination.

Important points for students
Memorize characteristic image types (virtual/real, erect/inverted, magnified/diminished) for each element and typical uses. Being able to draw ray diagrams for each case (convex lens, concave lens, concave mirror at object positions: beyond C, at C, between C and F, at F, inside F) helps predict image properties.