Leaving Certificate Physics

Geometrical Optics 🔭

Everything you need to know about mirrors, lenses, reflection, and refraction — explained step by step.

🌈 Light & Optics 📐 Reflection & Refraction

This is one of the biggest chapters in Leaving Cert Physics. It splits into four main parts: plane mirrors, curved mirrors, refraction, and lenses. Don't worry — we'll go through each one step by step.

The Big Picture

      🗺️ What this chapter covers:
      Plane mirrors — flat mirrors, laws of reflection, virtual images
Curved mirrors — concave mirrors (cave shape) and convex mirrors (bulging shape)
Refraction — light bending when it changes medium, Snell's Law, total internal reflection, optical fibres
Lenses — converging and diverging lenses, the eye, correcting vision defects

    

All Formulas at a Glance

📐 Mirror / Lens Formula

1/f = 1/u + 1/v

f = focal length | u = object distance | v = image distance
Used for both mirrors and lenses. Remember: positive v = real image, negative v = virtual image.

📐 Magnification

M = image height / object height = v / u

If M > 1: image is bigger. If M < 1: image is smaller. If M = 1: same size.

📐 Snell's Law (Refractive Index)

n = sin i / sin r

n = refractive index | i = angle of incidence | r = angle of refraction
Angles are always measured from the normal.

📐 Refractive Index — Speed of Light

n = c_air / c_medium

Speed of light in air = 3 × 10⁸ m/s. Light slows down in denser media.

📐 Refractive Index — Critical Angle

n = 1 / sin C

C = critical angle. The bigger the refractive index, the smaller the critical angle.

📐 Refractive Index — Real/Apparent Depth

n = Real Depth / Apparent Depth

Why a swimming pool looks shallower than it is!

📐 Power of a Lens

P = 1/f P_total = P₁ + P₂

P = power in m⁻¹ (also called dioptres). f must be in metres.
Convex (converging) lens: positive power. Concave (diverging) lens: negative power.

💡 "Real is Positive" Convention (RIP)

For concave mirrors and convex lenses: f is positive.
For convex mirrors and concave lenses: f is negative.
If v comes out positive → real image. If v is negative → virtual image.

Chapter 1

🪞 Plane Mirrors & Reflection

A flat mirror — the simplest type of reflection

A plane mirror is just a flat, silvered surface. When light hits it, it bounces back in a very predictable way. This bouncing is called reflection.

📖 Definition — Reflection

Reflection is the bouncing of light off a surface.

Types of Reflection

Diffuse (Irregular) Reflection

When light hits a rough surface (like a wall or a book), it scatters in all directions. This is why we can see objects — light bounces off them in all directions, some reaching our eyes.

Specular (Regular) Reflection

When light hits a smooth, flat, silvered surface like a mirror, it reflects in one predictable direction. This follows the Laws of Reflection.

Laws of Reflection

📖 The Two Laws of Reflection — Learn These Off!

Law 1: The incident ray, the normal at the point of incidence, and the reflected ray all lie on the same plane.

Law 2: The angle of incidence equals the angle of reflection. (i = r)

💡 What's the "Normal"?

The normal is an imaginary line drawn perpendicular (at 90°) to the mirror surface at the point where the light hits. All angles are measured from the normal — NOT from the mirror surface!

Image in a Plane Mirror

📖 Properties of Image in a Plane Mirror

✅ The image is virtual (it appears to be behind the mirror — light rays don't actually go there)
✅ The image is upright (not inverted)
✅ The image is the same size as the object
✅ The image is as far behind the mirror as the object is in front
✅ The image is laterally inverted (left and right are swapped — like the word AMBULANCE)

Why Does the Image Appear Behind the Mirror?

Light rays reflect off the mirror and diverge (spread out) as they travel toward your eye. Your brain follows these diverging rays backwards and "puts" the object where the rays appear to come from — which happens to be behind the mirror. Your brain is being tricked!

Real Image vs Virtual Image

Real Image

Formed by the actual intersection of light rays
Can be projected onto a screen
Always inverted (upside down)
Always in front of the mirror or on the opposite side of the lens to the object

Virtual Image

Formed by the apparent intersection of light rays
Cannot be projected onto a screen
Always upright
Always behind the mirror or on the same side as the object (for lenses)

Exam-Ready Answers

When the driver looks in the rear-view mirror, the laterally-inverted image of the word appears the correct way around — making it readable. It's deliberately written backwards so it reads correctly in a mirror.

A periscope uses two plane mirrors (or prisms) set at 45° angles. Light enters from above, hits the top mirror, reflects downward, hits the bottom mirror, and then reflects into the observer's eye. The image is upright and the same size as the object.

A luminous object emits its own light (e.g. the Sun, a candle, a light bulb). An illuminated object only becomes visible because light from another source reflects off it (e.g. the Moon, a book, you!).

Chapter 2

🔵 Concave Mirrors

Curved inward — like the inside of a bowl or a cave

📖 Concave Mirror

A concave mirror curves away from the observer — it is shaped like a cave (concave = cave). It can form either real or virtual images depending on where the object is placed.

Key Terms to Know

Term	Symbol	Meaning
Principal Axis	—	The imaginary horizontal line through the centre of the mirror
Centre of Curvature	C	The centre of the sphere that the mirror is part of
Focus (Focal Point)	f	The point where parallel rays converge after reflection. Halfway between C and the mirror.
Focal Length	f	Distance from the mirror to the focus
Object Distance	u	Distance from object to mirror
Image Distance	v	Distance from image to mirror

⚠️ Remember: The focus is halfway between the centre of curvature and the mirror. So if C = 20 cm, then f = 10 cm.

Ray Diagram Rules for Concave Mirror

      Three rules (use any two):
      A ray parallel to the principal axis → reflects back through the focus (f)
A ray hitting the mirror at the intersection of the principal axis and mirror → reflects at an equal angle on the other side
A ray passing through the focus → reflects back parallel to the principal axis

    

Object OUTSIDE the Focus

📖 Image Properties: Object OUTSIDE f

The image is: Real, Inverted, and either magnified or diminished depending on distance. It forms in front of the mirror where the rays actually meet.

Object INSIDE the Focus

📖 Image Properties: Object INSIDE f

The image is: Virtual, Upright, and Magnified. The rays never actually meet — they only appear to diverge from a point behind the mirror. This is how a cosmetic/shaving mirror or dentist's mirror works.

Quick Summary: Concave Mirror Images

Object Position	Image Type	Orientation	Size	Location
Beyond C	Real	Inverted	Diminished	Between f and C
At C	Real	Inverted	Same size	At C
Between C and f	Real	Inverted	Magnified	Beyond C
At f	No image	—	—	Reflected rays are parallel
Inside f	Virtual	Upright	Magnified	Behind mirror

Uses of Concave Mirrors

      🔦 Torches / car headlights / searchlights / floodlights — bulb placed at focus → parallel beam of light

      💄 Cosmetic mirrors / dentist's mirrors — object inside focus → magnified, upright image

      ☀️ Solar furnaces / solar cookers — focuses sunlight to a point to generate heat

2023 Exam Question — Spoon

📐 2023 Deferred HL

A person holds a spoon at arm's length. On one side the image is inverted, on the other it is upright. Explain why.

Answer: The concave side (inside of spoon) forms an inverted image because the object is outside the focal length. The convex side (outside of spoon) always forms an upright image regardless of where the object is placed.

Chapter 2

🔴 Convex Mirrors

Curved outward — always gives a small, wide-angle view

📖 Convex Mirror

A convex mirror bulges outward towards the observer. The image is always virtual, upright, and diminished — regardless of where the object is placed. The focus is behind the mirror.

Ray Diagram Rules for Convex Mirror

Two rules:

A ray parallel to the principal axis reflects as if it came from the focus on the other side (behind the mirror)
A ray hitting the centre of the mirror reflects at an equal angle on the opposite side

The rays never actually meet — they appear to diverge from a point behind the mirror, hence a virtual image.

Convex Mirror Image — Always the Same!

📖 Image in Convex Mirror (No Matter Where Object Is)

✅ Virtual (behind the mirror)
✅ Upright
✅ Diminished (smaller than the object)

Uses of Convex Mirrors

      🚗 Car door/wing mirrors — gives a wide field of view of traffic behind

      🏪 Shop security mirrors — wide angle view to deter shoplifters

      🚦 Concealed entrances — allows drivers to see oncoming traffic at blind corners

      Why convex mirrors are used for these: They give a wide field of view even though the image is smaller.

Why Is a Convex Mirror Used as a Car Wing Mirror?

📖 Exam Answer

A convex mirror is used because it gives a wider field of view than a plane mirror, allowing the driver to see more of the road behind.

Concave vs Convex Mirror Summary

Feature	Concave Mirror	Convex Mirror
Shape	Curves away from you (cave)	Bulges towards you
f value	Positive (+)	Negative (−)
Image type	Real OR virtual (depends on position)	Always virtual
Image orientation	Real → inverted; virtual → upright	Always upright
Image size	Can be any size	Always diminished
Main use	Mirrors, searchlights, cosmetics	Car mirrors, security mirrors

Chapter 2 — Calculations

📐 Mirror Maths: f, u, v & Magnification

Step-by-step calculations for curved mirrors

📐 The Mirror Formula

1/f = 1/u + 1/v

Rearranged versions:
1/v = 1/f − 1/u
v = 1 ÷ (1/f − 1/u)

⚠️ Concave mirror: f is positive (+)
⚠️ Convex mirror: f is negative (−)
✅ v is positive → real image v is negative → virtual image

📐 Magnification

M = image height / object height = v / u

Worked Examples

📐 Ordinary Level [2004] — Concave Mirror

A concave mirror has focal length 20 cm. An object is placed 30 cm in front. Where is the image?

What we know

f = 20 cm (positive, concave mirror), u = 30 cm, v = ?

Apply the formula

1/f = 1/u + 1/v → 1/v = 1/f − 1/u
1/v = 1/20 − 1/30
1/v = 3/60 − 2/60 = 1/60
v = 60 cm

✅ v is positive → real image formed 60 cm in front of the mirror.

📐 Higher Level [2014] — Convex Decorative Sphere

A person's face is 30 cm from a shiny spherical decoration of diameter 20 cm. Find the image position.

Find the focal length

Diameter = 20 cm → radius = 10 cm → centre of curvature C = 10 cm
Focus f = C/2 = 5 cm. But this is a CONVEX mirror → f = −5 cm

Apply the formula

1/v = 1/f − 1/u = 1/(−5) − 1/30
= −6/30 − 1/30 = −7/30
v = −30/7 ≈ −4.3 cm

✅ v is negative → virtual image 4.3 cm behind the mirror.

📐 Ordinary Level [2021] — Magnification

Object height 2 cm, image height 3 cm, object 17 cm in front of concave mirror. Find M and v.

Magnification

M = image height / object height = 3/2 = 1.5

Image distance (using M = v/u)

v = M × u = 1.5 × 17 = 25.5 cm

📐 Higher Level [2022] — Object 20 cm, f = 12 cm

Object 20 cm in front of concave mirror, f = 12 cm. Find image position and height if object is 4 cm tall.

Image distance

1/v = 1/12 − 1/20 = 5/60 − 3/60 = 2/60 → v = 30 cm

Magnification

M = v/u = 30/20 = 1.5

Image height

Image height = M × object height = 1.5 × 4 = 6 cm

📐 Higher Level [2017] — Rembrandt's Mirror

Concave mirror, f = 60 cm, image is half the size of object. Find image distance and object distance.

Use M = v/u

M = 1/2, so 1/2 = v/u → u = 2v

Substitute into mirror formula

1/f = 1/u + 1/v
1/60 = 1/(2v) + 1/v = 1/(2v) + 2/(2v) = 3/(2v)
2v = 180 → v = 90 cm

Object distance

u = 2v = 2 × 90 = 180 cm

💡 Why was the upright image no use to Rembrandt? Because to get an upright image the object must be inside the focus — which means the image is virtual and cannot be projected onto a copper sheet.

⚠️ Common Mistake

When dividing fractions like 1/f − 1/u, find a common denominator! Don't just subtract the numbers directly. For example, 1/20 − 1/30 = 3/60 − 2/60 = 1/60, NOT 1/10.

Mandatory Experiment

🔬 Measuring Focal Length of a Concave Mirror

A required experiment — you must know procedure, sources of error, and results

📖 Apparatus

Concave mirror, screen, lamp-box with crosswire, metre stick, retort stand

Procedure

1

Find approximate focal length first: Point the concave mirror at a distant object (like a window). Move a screen in front of it until a sharp image appears. That distance = approximate focal length.

2

Set up: Place the lamp-box (with crosswire) well outside the focal length. Set up the screen in front of the mirror.

3

Find image: Move the screen until a clear, sharp, inverted image of the crosswire appears on it.

4

Measure u: Measure the distance from the crosswire (object) to the mirror with the metre stick.

5

Measure v: Measure the distance from the screen (image) to the mirror.

6

Calculate f: Use 1/f = 1/u + 1/v. Get f for this measurement.

7

Repeat: Move the object to a different position, repeat for at least 5 different values of u. Average all the f values.

Sources of Error / Precautions

      🎯 Difficulty judging sharpest focus — repeat measurements and average
📏 Parallax error when measuring u and v — ensure line of sight is perpendicular to the metre stick
📐 Measure from the back of the mirror (not the front surface)
⚠️ Never place object inside focal length — no real image can be formed, so always find approximate f first

    

Why Find Approximate f First?

📖 Exam Answer

To avoid accidentally placing the object inside the focal length, which would result in only a virtual image — which cannot be formed on a screen.

Sample Data (Exam Question)

📐 Example Calculation

u (cm)	v (cm)	f (cm)
20.0	61.0	15.1
30.0	29.5	14.9
40.0	24.0	15.0
50.0	20.5	14.5

Average f ≈ 15.0 cm

For each row: 1/f = 1/u + 1/v → solve for f.

Chapter 3

💡 Refraction & Snell's Law

Light bends when it changes speed — this is why a pool looks shallower than it is

📖 Definition — Refraction

Refraction is the bending of a wave (light) as it passes from a medium of one refractive index to a medium of a different refractive index. The speed of light changes, causing the direction to change.

Key Rule: Which Way Does Light Bend?

🔹 Light going from less dense → more dense medium (e.g. air → glass): bends towards the normal (angle decreases)

🔹 Light going from more dense → less dense medium (e.g. glass → air): bends away from the normal (angle increases)

Laws of Refraction

📖 The Two Laws of Refraction

Law 1: The incident ray, the normal, and the refracted ray all lie on the same plane.

Law 2 (Snell's Law): The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant (the refractive index).

📐 Snell's Law — Refractive Index

n = sin i / sin r

n = refractive index of the medium (no units — it's a ratio)
i = angle of incidence (in the less dense medium, usually air)
r = angle of refraction

💡 Both angles are measured from the normal!

What Does Refractive Index Mean?

📖 Definition — Refractive Index

The refractive index of a medium is the ratio of sin i to sin r when light travels from a vacuum (or air) into that medium. It tells you how much the medium slows light down. Glass ≈ 1.5, water ≈ 1.33, diamond ≈ 2.42.

💡 Inverse Refractive Index

If the refractive index going from air to glass is ₐηᵍ = 1.5, then going from glass to air it's the inverse: ᵍηₐ = 1/1.5 ≈ 0.67

Why Do Different Colours Refract Differently?

Each colour (wavelength) of light has a slightly different refractive index in glass. Blue light refracts more than red light. This is why white light passing through a prism (or raindrop) separates into a rainbow — each colour bends by a different amount.

⚡ Worked Calculations

📐 Ordinary Level [2018] — Find Refractive Index

Angle of incidence = 40°, angle of refraction = 25°. Find the refractive index.

Apply Snell's Law

n = sin i / sin r = sin 40° / sin 25°
= 0.643 / 0.423
n = 1.52

📐 Ordinary Level — Find Angle of Refraction

Refractive index of glass = 1.5. Angle of incidence = 35°. Find the angle of refraction.

Rearrange Snell's Law

sin r = sin i / n = sin 35° / 1.5 = 0.5736 / 1.5 = 0.382
r = sin⁻¹(0.382) = 22.5°

📐 Higher Level — Find Refractive Index When Light EXITS a Block

[2012 HL] Light exits a glass block. Angle inside glass = θ (from surface, NOT normal), angle in air = 30°. Refractive index = 1.5. Find θ.

⚠️ Key insight

The angle is given FROM THE SURFACE. Angle from normal = 90° − θ (inside glass) and 90° − 30° = 60° (in air).
Light goes from glass → air, so use: n_glass × sin(angle in glass) = n_air × sin(angle in air)
1.5 × sin(90°−θ) = 1 × sin 60°
sin(90°−θ) = 0.866/1.5 = 0.577
90°−θ = 35.3°, so θ = 54.7°

Chapter 3

🔺 Critical Angle & Total Internal Reflection

What happens when light CAN'T escape from a denser medium

📖 Definition — Critical Angle

The critical angle is the angle of incidence (in the denser medium) which causes the angle of refraction to be exactly 90° — meaning the refracted ray travels along the surface.

📖 Definition — Total Internal Reflection (TIR)

Total internal reflection occurs when:
1. Light is travelling from a denser medium to a less dense medium (e.g. glass → air)
2. The angle of incidence is greater than the critical angle

In this case, ALL the light is reflected back into the denser medium — none escapes.

The Three Scenarios

Angle of Incidence (in glass)	What Happens?
Less than critical angle	Light refracts out AND weak reflected ray
Exactly equal to critical angle	Refracted ray travels along the surface (90°)
Greater than critical angle	Total Internal Reflection! All light reflects back in

Formula: Critical Angle and Refractive Index

📐 Critical Angle Formula

n = 1 / sin C

n = refractive index | C = critical angle

To find n given C: n = 1 / sin C
To find C given n: sin C = 1/n → C = sin⁻¹(1/n)

⚡ Worked Calculations

📐 Type 1 — Given C, find n [2015 OL]

Critical angle = 42°. Find the refractive index.

Apply formula

n = 1/sin C = 1/sin 42° = 1/0.669 = n = 1.49

📐 Type 2 — Given n, find C [2010 HL]

Refractive index = 1.46. Find the critical angle.

Rearrange: sin C = 1/n

sin C = 1/1.46 = 0.685
C = sin⁻¹(0.685) = 43.2°

📐 Higher Level [2015 HL] — Diamond

Refractive index of haematite = 3.2. Find critical angle.

sin C = 1/n

sin C = 1/3.2 = 0.3125
C = sin⁻¹(0.3125) = 18.2°

💡 The higher the refractive index, the smaller the critical angle — so TIR happens more easily.

Using a Prism for TIR

      A 45° glass prism has a critical angle of about 42°. When light hits the inside surface of the prism at 45°, that's greater than 42°, so TIR occurs.

      🔄 Rotate light 90°: Enter one face, TIR off the slanted face, exit the other face.

      ↩️ Rotate light 180°: Use two 45° prisms back-to-back — used in binoculars and periscopes.

Applications of Total Internal Reflection

      🚗 Reflective cat's-eye road signs

      🔭 Binoculars and periscopes (prisms)

      🏥 Endoscopes (in medicine)

      🌐 Optical fibres (broadband internet)

Chapter 3

🌐 Optical Fibres

How light travels around corners using total internal reflection

📖 Definition — Optical Fibre

An optical fibre consists of a glass (or plastic) pipe coated with a second material of lower refractive index. Light travels along the fibre using total internal reflection at the boundary between the two materials.

How Light Travels Through an Optical Fibre

1

Light enters one end of the fibre.

2

It strikes the boundary between the inner core and the outer cladding at an angle greater than the critical angle.

3

Total internal reflection occurs — all the light reflects back inward.

4

The reflected light travels to the opposite wall and undergoes TIR again.

5

This process repeats all the way along the fibre until light emerges at the far end.

⚠️ Why must the outer cladding have a LOWER refractive index?
Total internal reflection only occurs when light tries to go from a denser → less dense medium and the angle exceeds the critical angle. If the cladding had a higher refractive index, light would refract outward instead of reflecting inward — the light would escape.

The Outer Cladding Also...

📖 Role of the Outer Cladding

1. Has a lower refractive index — essential for TIR to occur
2. Acts as a protective layer against scratches and damage to the surface of the core

Applications

      🌐 Broadband / telecommunications (transmitting data as light pulses)

      🏥 Endoscopes in medicine (looking inside the body without surgery)

Advantages Over Copper Wire

      ✅ Cheaper raw material (glass/sand is more abundant than copper)

      ✅ Occupy less space (fibres are very thin — about the width of a hair)

      ✅ Less interference from electrical/magnetic fields

      ✅ Do not corrode

      ✅ Flexible — can reach inaccessible places

      ✅ Can carry much more information (higher bandwidth)

Material Used

📖 Exam Answer

Optical fibres are made from glass (or plastic / sand / silicon).

⚡ Worked Calculations

📐 HL [2009] — Speed in Optical Fibre

Optical fibre has refractive index = 1.55. Find speed of light in the fibre.

n = c_air / c_medium → c_medium = c_air / n

c = 3 × 10⁸ / 1.55 = 1.94 × 10⁸ m/s

📐 HL [2017] — Hibernia Express Cable

Signal travels 5500 km from London to New York in 29 milliseconds. Find the refractive index of the glass.

Step 1 — Convert units

Distance = 5500 km = 5500 × 10³ m = 5.5 × 10⁶ m
Time = 29 ms = 29 × 10⁻³ s

Step 2 — Speed of light in glass

v = d/t = (5.5 × 10⁶) / (29 × 10⁻³) = 1.9 × 10⁸ m/s

Step 3 — Refractive index

n = c_air / c_glass = (3 × 10⁸) / (1.9 × 10⁸) = 1.58

📐 OL [2017] — Critical Angle for Optical Fibre

Optical fibre has refractive index = 1.5. Find the critical angle.

sin C = 1/n = 1/1.5 = 0.667

C = sin⁻¹(0.667) = 41.8°

Chapter 3

🧮 All Refractive Index Formulas

Four different formulas, four different situations — learn to pick the right one

The refractive index appears in four different formulas. You need to recognise which one to use from the context of the question.

📐 Formula 1 — Snell's Law

n = sin i / sin r

Use when: Given angles of incidence and refraction, or asked to find an angle.
i = angle of incidence (in air/vacuum), r = angle of refraction

📐 Formula 2 — Speed of Light

n = c_air / c_medium

Use when: Given or asked for the speed of light in a medium.
c_air = 3 × 10⁸ m/s always.

📐 Formula 3 — Critical Angle

n = 1 / sin C

Use when: Given the critical angle, asked for refractive index (or vice versa).

📐 Formula 4 — Real/Apparent Depth

n = Real Depth / Apparent Depth

Use when: Given how deep something actually is and how deep it appears (e.g. a swimmer in a pool).
Apparent depth is always LESS than real depth (pools look shallower).

⚠️ IMPORTANT: Formulas 2, 3, and 4 are NOT in the Formulae & Tables book — you must know them! Formula 1 (Snell's Law) is given to you.

⚡ Real/Apparent Depth Calculations

📐 Example — Diver in Pool

A diver is 12 m below the surface. Refractive index of water = 1.33. What is the apparent depth?

n = Real Depth / Apparent Depth → rearrange

Apparent Depth = Real Depth / n = 12 / 1.33 = 9.02 m

💡 The diver appears to be only 9 m deep, not 12 m deep!

Why Does a Pool Look Shallower?

Light rays from the diver refract away from the normal as they exit the water into air. Your brain traces these diverging rays backwards and "places" the diver at a shallower depth than they actually are.

Quick Decision Guide

🔍 Which formula do I use?

Angles given → n = sin i / sin r
Speed of light given → n = c_air / c_medium
Critical angle given → n = 1 / sin C
Pool/swimmer depth → n = Real Depth / Apparent Depth

Mandatory Experiment

🔬 Verifying Snell's Law & Measuring Refractive Index

How to show that sin i / sin r is a constant

📖 Apparatus

Glass block, ray-box, protractor, sheet of paper

Procedure

1

Place the glass block on the paper and trace around its outline.

2

Shine a ray of light from the ray-box into the glass block at an angle.

3

Mark two dots on the incident ray and two dots on the exit ray (exiting the other side).

4

Remove the block. Draw the normal at the entry point. Connect the dots to trace the full path of the ray.

5

Measure angle of incidence i and angle of refraction r using the protractor.

6

Repeat for different values of i (use a wide range — small to large angles).

7

Record i, r, sin i, and sin r in a table.

8

Plot a graph of sin i (y-axis) against sin r (x-axis).

9

The slope of the straight line = refractive index of the glass.

Conclusion

📖 How the Graph Verifies Snell's Law

A straight line through the origin shows that sin i ∝ sin r — this confirms Snell's Law (the ratio is constant). The slope of the graph gives the refractive index.

Sources of Error / Precautions

      ⚠️ Don't use small angles (below 20°) — larger percentage errors when angles are small
📌 Mark two dots far apart on the incident and exit rays — more accurate direction
📐 Measure angles carefully with the protractor, relative to the normal
🔄 Repeat and average for accuracy

    

Why Not Use Angles Below 20°?

📖 Exam Answer

Small angles of incidence result in large percentage errors when measuring the angle — a small mistake in measuring a small angle makes a big difference to the result.

Graphical Method vs Calculation Method

📖 Exam Answer — Why Graph is Better

Using a graph is more accurate because outliers (errors) can be easily identified — a point that falls well off the straight line can be ignored. In a table of calculations, a faulty reading would still be included in the average.

Chapter 4

🔍 Convex (Converging) Lenses

Thicker in the middle — bends light rays inward

📖 Convex (Converging) Lens

A convex lens is thicker in the middle than at the edges. It causes parallel rays of light to converge (come together) at a point called the focus. Also called a "converging lens".

Why Does It Converge Light?

Light refracts (bends) twice as it passes through the lens — once entering and once leaving. At each surface, it bends toward the principal axis, so all parallel rays converge at the focus on the other side.

Ray Diagram Rules (Convex Lens)

      Three rules (use any two):
      A ray parallel to the principal axis → refracts through the focus on the other side
A ray hitting the centre of the lens → goes straight through (no bending)
A ray passing through the focus on the object side → refracts out parallel to the principal axis

    

Images Formed by Convex Lens

Object Position	Image Type	Orientation	Size	Uses
Beyond 2f	Real	Inverted	Diminished	Camera
At 2f	Real	Inverted	Same size	Photocopier
Between f and 2f	Real	Inverted	Magnified	Projector
At f	No image	—	—	Searchlight beam
Inside f	Virtual	Upright	Magnified	Magnifying glass

💡 Key Rules for Convex Lens

✅ Real image → on the opposite side to the object, inverted
✅ Virtual image → on the same side as the object, upright
✅ Object inside f → virtual image (magnifying glass effect)

Sign Convention for Lenses (Same as Mirrors)

Convex lens

f = positive (+)

Concave lens

f = negative (−)

Real image

v = positive (+)

Virtual image

v = negative (−)

Uses of Convex Lenses

      🔎 Magnifying glass (object inside f)    📸 Cameras

      👓 Spectacles/contact lenses (long-sightedness)    🎥 Projectors

      🔭 Telescopes and binoculars

Chapter 4

🔎 Concave (Diverging) Lenses

Thinner in the middle — spreads light rays outward

📖 Concave (Diverging) Lens

A concave lens is thinner in the middle than at the edges. It causes rays of light to diverge (spread apart). Also called a "diverging lens".

Ray Diagram Rules (Concave Lens)

Two rules:

A ray parallel to the principal axis refracts as if it came from the focus on the same side as the object
A ray hitting the centre of the lens passes straight through

The rays diverge after the lens and never actually meet. They only appear to come from a point on the same side as the object → always a virtual image.

Concave Lens Image — Always the Same!

📖 Image in Concave Lens (Always)

✅ Virtual (on same side as object)
✅ Upright
✅ Diminished (smaller than the object)

⚠️ A diverging (concave) lens cannot be used as a magnifying glass. Why? Because the image is always diminished — it makes things look smaller, not bigger!

Uses of Concave Lenses

      👓 Spectacles/contact lenses for short-sightedness (myopia) — the concave lens spreads the rays, allowing the eye to focus them correctly on the retina
    

Convex vs Concave Lens — Summary

Feature	Convex (Converging)	Concave (Diverging)
Shape	Thicker in middle	Thinner in middle
Effect on rays	Converges (brings together)	Diverges (spreads apart)
f value	Positive (+)	Negative (−)
Image type	Real or virtual	Always virtual
Image size	Can be any size	Always diminished
Used to correct	Long-sightedness	Short-sightedness

Chapter 4 — Calculations

📐 Lens Maths: f, u, v & Magnification

Same formula as mirrors — just different sign rules

📐 The Lens Formula (Same as Mirror Formula!)

1/f = 1/u + 1/v

Convex lens: f is positive
Concave lens: f is negative
v positive → real image (opposite side to object)
v negative → virtual image (same side as object)

⚡ Worked Calculations

📐 Ordinary Level [2020]

Converging lens, f = 15 cm. Object 20 cm in front. Find v and M.

Find v

1/v = 1/f − 1/u = 1/15 − 1/20
= 4/60 − 3/60 = 1/60
v = 60 cm (positive → real image)

Find M

M = v/u = 60/20 = 3 (image is 3× larger than object)

📐 Higher Level [2006] — Two Object Positions for Same Magnification

Converging lens, f = 8 cm. Find two positions where image is 4× the size of object.

Part 1 — Real image (M = 4, so v = 4u):

Substitute v = 4u into lens formula

1/8 = 1/u + 1/4u = 4/4u + 1/4u = 5/4u
4u = 40 → u = 10 cm

Part 2 — Virtual image (v is negative, so v = −4u):

Substitute v = −4u into lens formula

1/8 = 1/u + 1/(−4u) = 4/4u − 1/4u = 3/4u
4u = 24 → u = 6 cm

💡 u = 6 cm < f = 8 cm → object is inside the focus, so virtual image. ✓

📐 OL [2023] — Find image position

Converging lens, f = 15 cm. Object 25 cm in front. Find image position.

1/v = 1/f − 1/u

1/v = 1/15 − 1/25 = 5/75 − 3/75 = 2/75
v = 37.5 cm

Chapter 4

👁️ The Eye

Your eye is an optical instrument — and it can break down

Parts of the Eye and Their Functions

Part	Function
Cornea	Together with the lens, forms part of the focusing system. Fixed lens with the most refractive power.
Lens	Together with the cornea, focuses light onto the retina. Changes shape to focus near and far objects.
Iris	Acts like a shutter — controls the amount of light entering the eye (like the aperture of a camera)
Pupil	The hole in the iris that allows light to enter
Retina	Light-sensitive screen at the back of the eye where the image is formed
Optic Nerve	Carries information from the retina to the brain in electrical form
Ciliary Muscles	Muscles that change the shape (and therefore focal length) of the lens

Power of Accommodation

📖 Power of Accommodation

The power of accommodation of the eye is its ability to focus a real image of an object onto the retina, whether the object is near or far away. It does this by the ciliary muscles changing the shape of the lens — making it fatter (more curved = more powerful) for near objects, and flatter (less curved = less powerful) for distant objects.

Common Eye Defects

Short-Sightedness (Myopia)

Problem: Can see nearby objects clearly but distant objects are blurry. The eye is too powerful — light focuses in front of the retina.

Correction: Concave (diverging) lens — spreads the rays out slightly before they enter the eye, so they focus correctly on the retina.

Power: Negative (−)

Long-Sightedness (Hypermetropia)

Problem: Can see distant objects clearly but nearby objects are blurry. The eye is not powerful enough — light focuses behind the retina.

Correction: Convex (converging) lens — converges the rays slightly before entering the eye.

Power: Positive (+)

💡 Memory Trick

Short = Concave Long = conVex

Short-sighted → Concave lens | Long-sighted → Convex lens

Underwater Vision

The cornea has almost the same refractive index as water (≈1.33). When underwater, light doesn't refract as it enters the cornea — so the cornea doesn't act as a lens. The eye loses most of its focusing ability and everything looks blurry.

Goggles create a layer of air in front of the eyes. Light now travels from water → air → cornea, so it refracts again at the cornea (air and cornea have different refractive indices). The cornea acts as a lens again and vision is restored.

Chapter 4

⚡ Power of Lenses

Power lets us combine lenses — essential for correcting vision

📖 Definition — Power of a Lens

The power of a lens is the reciprocal (1 ÷) of its focal length. A short focal length = high power (bends light a lot). Unit: m⁻¹ (also called dioptres, D).

📐 Power Formulas

P = 1/f f = 1/P

⚠️ f must be in METRES — not cm!
Convex lens: power is positive (+)
Concave lens: power is negative (−)

📐 Two Lenses in Contact

P_total = P₁ + P₂

Just add the powers! Remember the signs.

📐 Correcting Vision Defects

P_{corrected eye} = P_{defective eye} + P_{corrective lens}

Rearranged: P_{corrective lens} = P_{corrected eye} − P_{defective eye}

⚡ Worked Calculations

📐 Basic [HL 2012] — Find focal length from power

Lens has power of −2 m⁻¹. Find focal length.

f = 1/P

f = 1/(−2) = −0.5 m
The negative sign confirms it's a concave (diverging) lens.

📐 HL [2004] — Two converging lenses

Two converging lenses, each with f = 10 cm. Find total power.

Convert f to metres, find P for each

f = 10 cm = 0.1 m
P₁ = 1/0.1 = 10 m⁻¹ P₂ = 10 m⁻¹

P_total = P₁ + P₂

P_total = 10 + 10 = 20 m⁻¹

📐 HL [2022] — Converging + Diverging

Converging lens f = 5 cm, diverging lens f = 15 cm. Find total focal length.

Use 1/f_total = 1/f₁ + 1/f₂

f₁ = 5 cm, f₂ = −15 cm (diverging = negative)
1/f_total = 1/5 + 1/(−15) = 3/15 − 1/15 = 2/15
f_total = 7.5 cm

📐 HL [2002] — Short-Sightedness Correction

Normal eye power = 60 m⁻¹. Short-sighted eye power = 65 m⁻¹. Find corrective lens power.

P_corrected = P_defective + P_lens

60 = 65 + P_lens
P_lens = −5 m⁻¹ (negative → concave lens → corrects short-sightedness ✓)

📐 OL [2022] — Eye + Glasses

Eye power = 62 m⁻¹. Glasses power = −2 m⁻¹. Find total power and focal length of glasses.

Total power

P_total = 62 + (−2) = 60 m⁻¹

Focal length of glasses lens

f = 1/P = 1/(−2) = −0.5 m

📐 OL [2019] — Long-Sightedness Correction

Power needed = 38 m⁻¹. Eye's maximum power = 32 m⁻¹. Find corrective lens power and focal length.

P_corrective = P_needed − P_eye

38 = 32 + P_lens → P_lens = 6 m⁻¹ (positive → convex lens → corrects long-sightedness ✓)

Focal length

f = 1/P = 1/6 = 0.167 m

⚠️ Sign of P tells you the lens type!

P_corrective is negative → concave lens → corrects short-sightedness
P_corrective is positive → convex lens → corrects long-sightedness

Mandatory Experiment

🔬 Measuring Focal Length of a Convex Lens

Very similar to the concave mirror experiment!

📖 Apparatus

Converging (convex) lens, screen, lamp-box with crosswire, metre stick, retort stand

Procedure

1

Find approximate focal length: Focus the image of a distant object (e.g. a window) onto a screen. The distance from lens to screen ≈ focal length.

2

Set up: Place lamp-box outside the focal length. Position screen on the other side of the lens.

3

Find image: Move the screen until a sharp, inverted image of the crosswire appears.

4

Measure u: Distance from lamp (object) to lens.

5

Measure v: Distance from lens to screen (image).

6

Calculate f using 1/f = 1/u + 1/v.

7

Repeat for different u values. Average all f values.

Key Differences from Mirror Experiment

      📏 Measure from the centre of the lens (not back of lens)

      🖥️ Screen is on the opposite side of the lens to the object

      ⚠️ If object is too close to lens (inside focal length), no real image forms on screen

Sources of Error / Precautions

      Difficulty judging sharpest focus — repeat and average
Parallax error when measuring — keep line of sight perpendicular to metre stick
Measure from the centre of the lens
Find approximate focal length first to avoid placing object inside f

    

Why Can't You Use Object Distance < 10 cm?

📖 Exam Answer

If the object is inside the focal length, only a virtual image forms — and a virtual image cannot be projected onto a screen. So no image would appear.

Sample Data

📐 2009 Exam Example

u (cm)	v (cm)	f (cm)
20.0	65.2	15.3
30.0	33.3	15.8
40.0	25.1	15.4

Average f ≈ 15.5 cm

Revision

📖 All Definitions — Exam Ready

Every definition you could be asked in the Leaving Cert, word-perfect

Reflection

The bouncing of light off a surface.

Laws of Reflection (2 laws)

1. The incident ray, the normal, and the reflected ray all lie on the same plane.
2. The angle of incidence equals the angle of reflection.

Real Image

A real image is formed by the actual intersection of light rays. It can be projected onto a screen. It is always inverted.

Virtual Image

A virtual image is formed by the apparent intersection of light rays. It cannot be projected onto a screen. It is always upright.

Refraction

Refraction is the bending of a wave (light) as it passes from a medium of one refractive index to a medium of a different refractive index.

Laws of Refraction (Snell's Law)

1. The incident ray, the normal, and the refracted ray all lie on the same plane.
2. The ratio of the sine of the angle of incidence to the sine of the angle of refraction is a constant (= the refractive index).

Refractive Index

The refractive index of a medium is the ratio of sin i to sin r when light travels from a vacuum (or air) into that medium. n = sin i / sin r.

Critical Angle

The critical angle is the angle of incidence (in the denser medium) which causes the angle of refraction to be exactly 90° — meaning the refracted ray travels along the surface.

Total Internal Reflection

Total internal reflection occurs when light in a denser medium strikes the boundary with a less dense medium at an angle of incidence greater than the critical angle — all the light is reflected back into the denser medium.

Optical Fibre

An optical fibre consists of a glass pipe coated with a second material of lower refractive index. Light travels along it by total internal reflection at the boundary between the two materials.

Power of Accommodation

The power of accommodation of the eye is its ability to focus a real image of an object onto the retina, whether the object is near to or far away. Achieved by the ciliary muscles changing the shape of the lens.

Power of a Lens

The power of a lens is the reciprocal of its focal length. P = 1/f. Unit: m⁻¹. Converging lens: positive. Diverging lens: negative.

Parallax Error

A parallax error is the apparent shift in an object's position when viewed from different angles. Avoided by ensuring your line of sight is perpendicular to the scale.

Short-Sightedness (Myopia)

A short-sighted person can see nearby objects clearly but distant objects are blurred. Corrected with a concave (diverging) lens.

Long-Sightedness (Hypermetropia)

A long-sighted person can see distant objects clearly but nearby objects are blurred. Corrected with a convex (converging) lens.

🔢 All Formulas Summary

Formula	When to Use
1/f = 1/u + 1/v	Mirror or lens: find image distance, object distance, or focal length
M = v/u	Magnification from distances
n = sin i / sin r	Refractive index from angles
n = c_air / c_medium	Refractive index from speed of light
n = 1 / sin C	Refractive index from critical angle
n = Real Depth / Apparent Depth	Pool/swimming depth problems
P = 1/f	Power of a lens (f in metres)
P_total = P₁ + P₂	Two lenses in contact

Revision Quiz

🎯 Quick Quiz

Test your knowledge of all four sections!

Try all 14 questions. Click to reveal if you're right. Good luck! 🍀

1. State the second law of reflection.

2. What type of image is formed by a plane mirror?

3. An object is placed INSIDE the focal point of a concave mirror. What kind of image forms?

4. A concave mirror has focal length 10 cm. Object is 30 cm in front. Where is the image?

5. What is refraction?

6. Critical angle = 42°. What is the refractive index?

7. Why must the outer cladding of an optical fibre have a LOWER refractive index than the core?

8. What type of lens is used to correct short-sightedness?

9. Converging lens, f = 15 cm. Object at 20 cm. What is the image distance v?

10. What is the power of a lens with focal length 0.25 m?

11. A normal eye has power 60 m⁻¹. A person's eye has power 65 m⁻¹. What corrective lens is needed?

12. Why is a convex mirror used as a car wing mirror instead of a plane mirror?

13. What is the function of the retina in the eye?

14. In the experiment to verify Snell's Law, why is a graph of sin i vs sin r plotted instead of just i vs r?

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