🔍 Chapter 3: Lenses

📑 Table of Contents

3.1 Convex & Concave Lenses
3.2 Lens Terminology
3.3 Convex Lens Image Formation Rules
3.4 Concave Lens Image Formation
3.5 Lens Formula & Magnification
3.6 DSE Common Mistakes
📌 Chapter Summary
✏️ Quick Quiz

3.1 Convex & Concave Lenses

🔍 What Is a Lens?

A lens is an optical component made of a transparent material — usually glass or plastic — whose two surfaces are curved as portions of spheres. A lens uses the refraction of light to change the direction in which light rays travel, and therefore to form images.

Figure 3-1: A convex lens is thicker at the center (it converges light); a concave lens is thinner at the center (it diverges light).

Convex (Converging) Lens

Properties of a convex lens:

Thicker at the center than at the edges.
Also called a converging lens.
Parallel rays incident on it converge to the focal point after refraction.
The focal length is positive (f > 0).

Concave (Diverging) Lens

Properties of a concave lens:

Thinner at the center than at the edges.
Also called a diverging lens.
Parallel rays incident on it diverge after refraction; the backward extensions of the rays meet at a virtual focal point.
The focal length is negative (f < 0).

💡 Memory trick: "Convex = thick center, converges; Concave = thin center, diverges." Think of "convex" as bulging outward (gathering light to a focus) and "concave" as caving inward (spreading light outward).

3.2 Lens Terminology

Figure 3-2: Principal axis, optical centre, focal points F, and points 2F of a convex lens.

📏 Key Terminology

Term	English name	Definition
Principal axis	Principal Axis	The horizontal straight line passing through the optical centre of the lens (the axis of symmetry of the light rays).
Optical centre	Optical Centre	The centre of the lens; a ray passing through this point continues in a straight line, undeflected.
Principal focus F	Principal Focus	The point at which parallel rays parallel to the principal axis converge after passing through the lens.
Focal length f	Focal Length	The distance from the optical centre of the lens to the principal focus (positive for a convex lens, negative for a concave lens).
Point 2F	—	The point on the principal axis at a distance of 2f from the optical centre.
Object distance u	Object Distance	The distance from the object to the optical centre of the lens (always taken as positive).
Image distance v	Image Distance	The distance from the image to the optical centre of the lens (positive for a real image, negative for a virtual image).

3.3 Convex Lens Image Formation Rules

⚠️ This is the most important part of DSE Optics! Memorize the rule-of-thumb and understand its physical meaning.

📊 Image Formation Rules for a Convex Lens (must memorize!)

Object distance u	Image distance v	Nature of the image	Application
u → ∞	v = f (at the focal point)	Reduced to a point	Solar concentrator
u > 2f	f < v < 2f	Inverted, diminished, real	Camera
u = 2f	v = 2f	Inverted, same size, real	One-to-one imaging
f < u < 2f	v > 2f	Inverted, magnified, real	Projector, slide viewer
u = f	v → ∞	No image (rays emerge parallel)	Searchlight (reverse use)
u < f	v < 0 (virtual image)	Upright, magnified, virtual	Magnifying glass

💡 Mnemonic: "One focal length separates real from virtual; two focal lengths separate magnified from diminished."
- Object inside 1f: virtual image (magnifying glass)
- Object outside 1f: real image
- Object outside 2f: diminished
- Object inside 2f: magnified

Figure 3-3: Image formation by a convex lens when u > 2f (inverted, diminished, real image).

🔍 Three Special Rays for Ray Diagrams

The three standard rays used to draw a convex-lens ray diagram:

Ray	Path	Purpose
Ray 1	Parallel to the principal axis → after the lens, passes through the focal point F.	Locates the image position.
Ray 2	Through the optical centre → continues in a straight line, undeflected.	Determines whether the image is upright or inverted.
Ray 3	Through the focal point F (or appearing to come from F) → after the lens, becomes parallel to the principal axis.	Verifies the image position.

📌 The intersection of any two of these rays gives the position of the image!

3.4 Concave Lens Image Formation

Characteristics of Concave Lens Images

⚠️ Regardless of the object distance u, a concave lens always produces an upright, diminished, virtual image!

Figure 3-4: A concave lens always forms an upright, diminished, virtual image on the same side as the object.

📊 Concave Lens Image Formation Rule

Object distance u	Image distance v	Nature of the image
Any u (u > 0)	v < 0, and \|v\| < \|f\|	Upright, diminished, virtual

💡 How to remember it: A concave lens always produces an upright, diminished, virtual image (think of it as a "shrinking mirror"); the image lies between the object and the lens.

3.5 Lens Formula & Magnification

📐 Lens Formula

1 / u + 1 / v = 1 / f

where:

u = object distance (always positive).
v = image distance (positive for a real image, negative for a virtual image).
f = focal length (positive for a convex lens, negative for a concave lens).

⚠️ Sign-convention traps (the most common DSE mistake!):

Convex lens: f > 0. Real image → v > 0; virtual image → v < 0.
Concave lens: f < 0, and v is always < 0 (virtual image).
Object distance u is always positive (the object is in front of the lens).

📐 Magnification Formula

m = (image height) / (object height) = v / u

|m| > 1: magnified.
|m| = 1: same size.
|m| < 1: diminished.
m > 0: upright image.
m < 0: inverted image.

📝 Worked Example 3.1 — Convex Lens Imaging Calculation

A convex lens has focal length f = 10 cm and an object distance u = 30 cm. Find the image distance v and describe the image.

Solution:

Apply the lens formula: 1/u + 1/v = 1/f

1/30 + 1/v = 1/10

1/v = 1/10 − 1/30 = 3/30 − 1/30 = 2/30

v = 30/2 = 15 cm

Magnification: m = v/u = 15/30 = 0.5 (diminished).

✅ Conclusion: u = 30 cm > 2f = 20 cm → inverted, diminished, real image (camera principle).

📝 Worked Example 3.2 — Projector Imaging Calculation

A projector lens has focal length f = 5 cm and an object distance u = 6 cm. Find the image distance v and the magnification.

Solution:

1/6 + 1/v = 1/5

1/v = 1/5 − 1/6 = 6/30 − 5/30 = 1/30

v = 30 cm

Magnification: m = v/u = 30/6 = 5 (magnified 5×).

✅ Conclusion: f < u < 2f (5 < 6 < 10) → inverted, magnified, real image (projector principle).

📝 Worked Example 3.3 — Magnifying Glass Calculation

A magnifying glass has focal length f = 10 cm and an object distance u = 5 cm. Find the image distance v and the magnification.

Solution:

1/5 + 1/v = 1/10

1/v = 1/10 − 1/5 = 1/10 − 2/10 = −1/10

v = −10 cm (negative → virtual image)

Magnification: m = v/u = −10/5 = −2 (negative — does that mean inverted? Wait…)

✅ Note: v is negative, so the image is virtual. In this case the sign of m flips in meaning — the actual image is upright, magnified, and virtual.

⚠️ Important: When using m = v/u for a virtual image, treat the sign carefully — use |v/u| for the magnification factor, and check upright/inverted from the ray diagram.

3.6 DSE Common Mistakes

⚠️ According to the HKEAA 2018 examiners' report, lens questions are the single biggest source of lost marks in DSE Physics optics!

❌ Common Mistake 1 — Sign confusion

Treating the object distance u as possibly negative. In fact, u is always positive, because the object is in front of the lens. Only v and f carry a sign.

❌ Common Mistake 2 — Forgetting units

When applying the lens formula, all lengths must use the same unit (cm or m). Mixing cm and m produces an incorrect answer.

❌ Common Mistake 3 — Magnification sign for a virtual image

For a virtual image v is negative, so m = v/u is also negative — but the actual image is upright and magnified. For virtual images, use |v/u| for the size factor and rely on the ray diagram for upright/inverted.

❌ Common Mistake 4 — Using |f| and dropping the sign

For a concave lens f is negative. The sign must be carried into the formula — do not use |f|.

❌ Common Mistake 5 — Swapping u and v

u is the distance from the object to the lens; v is the distance from the image to the lens. Under exam pressure, these are easy to swap.

❌ Common Mistake 6 — Treating a virtual image as projectable

A virtual image cannot be projected onto a screen — it can only be viewed with the eye or captured by a camera.

📌 Chapter Summary

🎯 Five core ideas you must remember:

Convex vs concave: convex lens is thick-centred (converging); concave lens is thin-centred (diverging).
Convex-lens rule: "One focal length separates real from virtual; two focal lengths separate magnified from diminished."
Concave-lens property: always upright, diminished, virtual.
Lens formula: 1/u + 1/v = 1/f — know the sign convention cold.
Magnification: m = v/u — real images are inverted, virtual images are upright.

📝 Key-Vocabulary Bilingual Table

Chinese	English	Symbol
透鏡	Lens	—
凸透鏡	Convex Lens	—
凹透鏡	Concave Lens	—
主軸	Principal Axis	—
焦點	Principal Focus	F
焦距	Focal Length	f
物距	Object Distance	u
像距	Image Distance	v
放大率	Magnification	m
實像	Real Image	v > 0
虛像	Virtual Image	v < 0

✏️ Quick Quiz

Question 1 — Object distance u = 4 cm, focal length f = 10 cm (concave lens). Find v and describe the image.

📌 Click to reveal the answer

1/4 + 1/v = 1/(−10)

1/v = −1/10 − 1/4 = −2/20 − 5/20 = −7/20

v = −20/7 ≈ −2.86 cm

v < 0 → virtual image, on the same side as the object.

Magnification |m| = |v/u| = 2.86/4 = 0.715 → diminished.

Conclusion: upright, diminished, virtual image (typical concave-lens behaviour).

Question 2 — Convex lens with f = 10 cm, object at u = 15 cm. Find v and describe the image.

📌 Click to reveal the answer

1/15 + 1/v = 1/10

1/v = 1/10 − 1/15 = 3/30 − 2/30 = 1/30

v = 30 cm

m = 30/15 = 2 (magnified 2×).

Conclusion: f < u < 2f (10 < 15 < 20) → inverted, magnified, real image (projector principle).

Question 3 — A camera lens has f = 50 mm. To photograph an object 2 m away, what is the image distance?

📌 Click to reveal the answer

f = 50 mm = 0.05 m, u = 2 m = 2000 mm.

1/2000 + 1/v = 1/50

1/v = 1/50 − 1/2000 = 40/2000 − 1/2000 = 39/2000

v = 2000/39 ≈ 51.3 mm

Conclusion: u ≫ f, so v ≈ f — which is why a camera lens sits very close to the focal point.

Question 4 — True or false: a concave lens can form an upright, magnified, virtual image.

📌 Click to reveal the answer

False. A concave lens always forms an upright, diminished, virtual image — never a magnified one. To obtain a magnified image you need a convex lens with u < f (a magnifying glass).

📘 Chapter 3: Lenses — Complete Notes
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Source textbook: Pan Lloyds Pre-DSE Optics (Ch 3 — Lenses)
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