Geometric Optics: Mirrors, Reflection and Refraction

• Luminous object: an object that emits its own light, such as the Sun or a glowing torch bulb.
• Non-luminous object: an object that does not produce the light by which we see it.
• Ray: a line showing light's direction of travel in the ray model.

An illuminated chair reflects light into your eyes. Transparent glass also lets light pass through it. Seeing through glass therefore involves transmission, not only reflection.

Why do shadows and pinhole images form?

Light travels along straight paths in a uniform medium within the ray approximation. An opaque object blocks those direct paths and creates a shadow behind it. A wall or screen makes the shadow visible. The shadow region exists even without a screen.

A pinhole camera admits narrow bundles through a small opening. Light from the object's top reaches the screen's bottom. Light from the bottom reaches the screen's top. The resulting image is real and inverted.

A smaller hole reduces overlapping bundles but admits less light. An extremely tiny hole also introduces diffraction, so smaller does not always mean sharper. Diffraction is the spreading of waves around edges or through openings.

The angle of incidence equals the angle of reflection. Both angles are measured from the normal, not from the mirror surface. The normal is perpendicular to the surface at the point where light strikes.

The incident ray, reflected ray and normal lie in one plane. A ray arriving at 30° to the normal leaves at 30°. The same incident ray makes 60° with the mirror surface.

A smooth mirror gives regular reflection. A rough surface scatters reflected rays because different points have different surface normals. Each ray still follows the reflection law.

Yes. A real image forms where light rays actually meet. A virtual image forms where backward extensions of rays appear to meet. You can see or photograph a virtual image. You cannot receive that image directly on a screen placed at its apparent location.

A plane mirror forms an upright, virtual image of the same size as the object. The image lies as far behind the mirror as the object lies in front. Moving closer to the mirror brings the image closer too.

What does lateral inversion mean?

Mirror writing appears reversed when compared with writing facing you. This effect is commonly called lateral inversion. However, a plane mirror reverses the direction perpendicular to its surface: front and back. It does not literally exchange every left-hand direction with a right-hand direction.

Reversed lettering on an ambulance can appear readable in a driver's mirror. Compare the lettering's orientation, rather than imagining that the mirror physically moves letters sideways.

A concave mirror has a reflecting surface that curves inward. A convex mirror has a reflecting surface that bulges outward. Their image rules differ because they redirect rays differently.

• Pole, P: the centre of the mirror's reflecting surface.
• Centre of curvature, C: the centre of the sphere containing that surface.
• Principal axis: the straight line through P and C.
• Principal focus, F: where rays parallel to the principal axis meet, or appear to come from, after reflection.
• Focal length, f: the distance from P to F, with a sign when using a sign convention.

For a spherical mirror, f ≈ R/2 for rays close to the principal axis. Here R is the signed radius of curvature. The approximation assumes small ray angles and a small usable mirror aperture.

What images does a concave mirror form?

For a real object beyond the focus, the image is real and inverted. Its size depends on the object's position.

A nearby face can therefore look enlarged in a concave shaving mirror. Move the face beyond the focus, and the image becomes inverted. “Concave always means real” is a useful misconception to reject.

Why are convex mirrors used on vehicles?

A convex mirror forms a virtual, upright, smaller image of a real object. The image lies behind the mirror, between P and F. Smaller images allow a wider field of view. This helps drivers see more surrounding traffic, although objects appear farther away.

Use the mirror formula 1/v + 1/u = 1/f for paraxial rays. Here u is object distance, v is image distance, and f is focal length. Use the same distance unit throughout.

In the Cartesian convention, measure distances from the pole. Choose the incoming light's direction as positive. A real object in front has u < 0. A concave mirror has f < 0. A real image in front has v < 0. A virtual image behind has v > 0.

Worked example

A concave mirror has f = −10 cm. Place an object 30 cm in front, so u = −30 cm.

1. Rearrange: 1/v = 1/f − 1/u. 2. Substitute: 1/v = −1/10 + 1/30 = −1/15 cm⁻¹. 3. Therefore, v = −15 cm.

The image forms 15 cm in front of the mirror. Magnification is m = −v/u, giving m = −0.5. The minus sign means inverted. The magnitude 0.5 means half the object's height.

Refraction occurs when light crosses a boundary between media with different optical properties. Its speed changes. For an oblique ray, the direction usually changes too.

The absolute refractive index is n = c/v. Here c is light's speed in vacuum, and v is its speed in the material. Refractive index has no unit. A larger n means a lower light speed at the stated wavelength.

Optical density concerns refractive index, not mass per unit volume. A material can be optically denser without having a greater mass density. Air's index is approximately 1.00, but vacuum defines the absolute reference.

Which way does light bend?

An oblique ray entering a higher-index medium bends towards the normal. Entering a lower-index medium, it bends away from the normal when refraction occurs. At normal incidence, the transmitted ray does not bend. Its speed and wavelength can still change.

Snell's law is n₁ sin θ₁ = n₂ sin θ₂. The indices describe the two media. Both angles are measured from the normal. Frequency remains unchanged across a stationary boundary; speed and wavelength change together.

This explains the bent-looking spoon. Light from the submerged part bends as it leaves water. Your eyes trace the emerging rays backward along straight lines. The submerged part therefore appears displaced.

At 0° incidence, the transmitted ray continues along the normal.

The bottom appears shallower because rays leaving water bend away from the normal. Their backward extensions locate a virtual image above the actual bottom.

For viewing close to the normal through a flat surface, apparent depth ≈ real depth/n. Here n is the water's refractive index relative to the air above. The approximation does not describe every oblique viewing angle.

For a 30 cm depth and n = 1.33, apparent depth ≈ 30/1.33 = 22.6 cm. The water has not lifted the bottom. Refraction has changed its apparent position.

Total internal reflection occurs when light cannot emerge as a propagating refracted ray into the second medium. Two conditions must hold in the ideal ray model:

• Light travels from a higher refractive index to a lower refractive index.
• The incidence angle exceeds the critical angle.

At the critical angle, the refracted ray runs along the boundary at 90° to the normal. The formula is sin θc = n₂/n₁, with n₁ > n₂. The critical angle is measured inside the higher-index medium.

These are rounded, representative values. Refractive index depends on wavelength and material conditions. The water angle uses a more precise index near 1.333. Calculating from the rounded 1.33 instead gives about 48.8°.

Diamond's small critical angle makes total internal reflection possible across a broad range of internal incidence angles. Internal reflections contribute to sparkle. Cut, illumination and colour dispersion also matter. A critical angle alone does not explain all diamond brightness.

An optical fibre has a core surrounded by lower-index cladding. Suitable rays undergo total internal reflection at the core–cladding boundary. This guides light along the fibre. Real fibres still lose some light through absorption, scattering and other effects.

A simple mirror periscope uses two plane mirrors, each inclined at 45° to the tube's axis. Each mirror turns the ray through a right angle. Two reflections let the observer see over an obstacle.

Filling an ideal reflection-only periscope with a transparent medium of n = 1.5 does not change the reflection law. Its mirror angles remain 45° for the same right-angle turns. Real entrance and exit surfaces can introduce additional refraction.

A lens refracts light at its surfaces. A converging lens brings parallel rays towards a focus. A diverging lens spreads parallel rays as though they came from a focus.

Lens power measures focusing strength: P = 1/f, with f expressed in metres. The unit is the dioptre, written D, equivalent to m⁻¹. A converging lens has positive power. A diverging lens has negative power.

For f = +0.50 m, P = 1/0.50 = +2 D. Do not substitute 50 cm directly and call the result dioptres. Convert centimetres to metres first.

Dispersion is the separation of light into its component colours. A material's refractive index varies with wavelength. In an ordinary glass prism, violet light bends more than red light.

The visible spectrum is continuous, not seven isolated bands. The conventional sequence is violet, indigo, blue, green, yellow, orange and red. VIBGYOR helps you remember these seven colour names.

A rainbow involves refraction, internal reflection and dispersion within water droplets. Sunlight enters a droplet, changes direction, reflects inside, and leaves through another refracting boundary. Different colours emerge in different directions. For the usual primary rainbow, the Sun is behind the observer.

The cornea and lens focus light into a real, inverted image on the retina. Light enters through the cornea and pupil before passing through the lens. The iris controls the pupil's opening. The retina contains light-sensitive cells that begin the conversion of light into nerve signals.

Rods help vision in dim light. Cones support colour vision and fine detail in brighter light. The blind spot has no rods or cones where the optic nerve leaves the retina. The optic nerve carries visual signals towards the brain.

How can the eye focus at different distances?

Accommodation changes the lens's shape and focal length. Ciliary muscles help control this adjustment. The retina does not slide backwards and forwards to focus.

The conventional near point for a young adult with normal vision is about 25 cm. A near point is the closest distance for clear, comfortable viewing. Actual near points differ between people and change with age.

For myopia, a diverging lens reduces the incoming rays' convergence. The eye can then bring those rays to a focus on the retina. For hypermetropia, a converging lens adds the missing focusing power.

Presbyopia is not simply another name for hypermetropia. Age-related changes reduce the lens's ability to adjust. Some people use bifocal or progressive lenses for near and distant needs. Their distance correction depends on their other refractive errors.

A cataract is clouding of the eye's lens, not a misplaced focus alone. Cataract surgery removes the cloudy lens and usually replaces it with an artificial lens. Spectacle rules for myopia do not explain cataracts.

Apparent motion lets a sequence of still images produce a sense of movement. A film shown at 24 frames per second presents rapidly changing images. The visual system interprets suitable changes as motion.

Do not explain cinema through a universal 1/16-second retinal storage rule. Persistence of vision alone does not account for apparent motion. Frame rate, image changes and viewing conditions affect the experience.

How does Braille represent text?

Braille is a tactile reading and writing system developed by Louis Braille, who was blind. A traditional Braille cell has six dot positions. These positions produce 63 non-empty patterns, plus one blank pattern. Readers feel raised-dot combinations with their fingertips.

Air molecules scatter short-wavelength visible light more strongly than long-wavelength light. Scattered blue light reaches your eyes from many directions. This makes a clear daytime sky look blue.

Near sunset, sunlight travels through more atmosphere before reaching you. More blue light is scattered out of the direct beam. The remaining direct light often looks red or orange. Clouds and airborne particles can modify the colours.

Why do planets usually twinkle less?

Moving air creates changing refractive conditions along the path of starlight. A distant star behaves approximately like a point source. Its apparent position and brightness fluctuate, producing twinkling.

A planet usually presents a larger apparent disc. Variations across different parts of that disc partly average out. Planets therefore usually twinkle less than stars. “Planets can never twinkle” is too absolute.

For mirror questions, identify the mirror and object position before choosing an image rule. For refraction questions, compare refractive indices and measure angles from the normal. For eye questions, distinguish focusing defects from lens clouding.

National Defence Academy and Naval Academy Examination, 2026 I

• Question 64: moving object and concave-mirror image. An object beyond C moves towards a concave mirror with acceleration towards it. While the object remains beyond C, its image moves from F towards C with acceleration in that direction. Image motion need not match object motion.
• Question 68: a periscope filled with liquid. For the ideal reflection-only construction, n = 1.5 does not change the 45° mirror orientation. The law of reflection remains unchanged.
• Question 69: a convex mirror. For a real object, the image remains virtual, upright and smaller. It lies between the pole and focus behind the mirror.

National Defence Academy and Naval Academy Examination, 2025

• 2025 I: convex-mirror image position. The image lies between P and F behind the mirror for a real object.
• 2025 I: an oblique ray entering water. Light entering water from air bends towards the normal. Normal incidence is the no-bending exception.
• 2025 II: raised-looking tank bottom. Refraction makes the bottom appear shallower. Reflection does not produce this apparent-depth effect.

Five checks are placed beside their relevant lesson sections. Answered and correct counts belong only to this page session. Reset lets you try the checks again. These checks are separate from the final eight practice MCQs.

Answer eight MCQs on reflection, mirrors, refraction, total internal reflection, lenses, dispersion and vision. Each answer includes a connected explanation.