CDS Physics · Light and Optics

Refraction and Lenses

Why light bends between two media, the refractive index, total internal reflection, where a convex lens forms its image, and the lens formula and power.

All Light and Optics notes

Why this matters

Seventeen CDS questions from 2016 (II) to 2026 (II), the largest page in the chapter. Six are about refraction itself, one about total internal reflection, and ten about lenses: where the image forms, the lens formula and power in dioptres.

Concept 1 of 4: Refraction and the refractive index

Light travels at different speeds in different materials: fastest in vacuum, slower in water, slower still in glass. When a ray crosses from one material into another at a slant, the change of speed turns it, the way a car swerves when one front wheel runs onto sand first. The refractive index tells you how many times slower light is in a material than in vacuum.

Definition

Refraction rules:

Cause: the change in the speed of light at the boundary.
Refractive index $n = c/v$ , the speed in vacuum (or air) over the speed in the medium. It is greater than 1 for every material medium; water about 1.33, glass about 1.5, diamond about 2.42.
Rarer to denser (air into glass): light slows down and bends towards the normal. Denser to rarer: speeds up and bends away from the normal.
Along the normal (angle of incidence 0°): the ray goes straight through, angle of refraction 0°, though its speed still changes.
Snell's law: $n_1 \sin i = n_2 \sin r$ .

Refractive index and Snell's law

n = \frac{c}{v} \qquad n_1 \sin i = n_2 \sin r

nrefractive index of the medium
cspeed of light in vacuum, about $3 \times 10^{8}$ m/s
vspeed of light in the medium
i, rangles of incidence and refraction, from the normal

Worked example

The refractive index of water is 4/3 and of a glass is 1.6. In which does light travel faster, and how many times faster?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2020 · CDS (I) 2020 — General Knowledge · Q11Easy

Example 1 · Light and Optics · Refraction and Lenses

If the speed of light in air is

3 \times 10^{8}\ \text{m/s}

, then the speed of light in a medium of refractive index

\frac{3}{2}

Refractive index is c over v, not v over c

n = c/v

is always greater than 1. Writing

v/c

gives a number below 1, which no material has. To find the speed in a medium, divide

c

n

; never multiply.

Slower means towards the normal

Going into a denser medium, light slows down and bends towards the normal. The two halves always go together: an option pairing 'slows down' with 'away from the normal' is wrong.

Drill 5 more on refraction and the refractive index

Concept 2 of 4: Total internal reflection

When light goes from glass out into air, it bends away from the normal. Tilt the ray more and more, and at one angle, the critical angle, the outgoing ray skims along the surface. Tilt it further and no light gets out at all: every bit of it is reflected back inside. This is total internal reflection, and because nothing escapes, the reflected beam is as bright as the incoming one.

Definition

Conditions and uses:

Light must go from a denser to a rarer medium (glass to air, water to air).
The angle of incidence must be greater than the critical angle $C$ , where $\sin C = 1/n$ .
All the light is reflected, so there is no loss of intensity: better than any mirror.
Critical angle: glass about 42°, water about 49°, diamond about 24°.
Uses and examples: optical fibres (endoscopes, telecommunication), the sparkle of a diamond, totally reflecting prisms in binoculars, and the mirage on a hot road.

Critical angle

\sin C = \frac{1}{n}

Ccritical angle, inside the denser medium
nrefractive index of the denser medium relative to the rarer one

Worked example

Find the critical angle for light going from glass of refractive index 1.5 into air.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2016 · CDS (II) 2016 — General Knowledge · Q37Easy

Example 2 · Light and Optics · Refraction and Lenses

In total internal reflection, the light travels from

Higher refractive index, smaller critical angle

From

\sin C = 1/n

, a larger

n

gives a smaller

C

. Diamond (24°) traps light more easily than glass (42°), which is why it sparkles.

Concept 3 of 4: Images formed by a convex lens

A convex lens is thicker in the middle and bends parallel light together at a focus F on the far side. Where its image appears depends on how far the object is from the lens, measured in focal lengths. Far away, the image is small and near F. Bring the object closer and the image moves out and grows. At F the light comes out parallel and no image forms. Closer than F, the lens acts as a magnifying glass.

Definition

Convex lens, object at each position (F = focus, 2F = twice the focal length):

At infinity: image at F, real, inverted, highly diminished.
Beyond 2F: image between F and 2F, real, inverted, diminished.
At 2F: image at 2F, real, inverted, same size. Object and image are then $4f$ apart, the least separation for a real image.
Between F and 2F: image beyond 2F, real, inverted, enlarged.
At F: image at infinity.
Between F and the lens: image on the same side, virtual, erect, enlarged (a magnifying glass).
A concave lens always gives a virtual, erect, diminished image.
The aperture is the diameter of the lens's circular outline. It does not depend on the radius of curvature.

Object position	Image (convex lens)
Beyond 2F	Between F and 2F; real, inverted, smaller
At 2F	At 2F; real, inverted, same size
Between F and 2F	Beyond 2F; real, inverted, enlargedQ CDS 2020 (II): an object 40 cm from a lens of focal length 25 cm gives an inverted, magnified image.
At F	At infinityQ CDS 2022 (II): an object at the focus of a convex lens has its image at infinity.
Between F and the lens	Same side; virtual, erect, enlarged

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2021 · CDS (II) 2021 — General Knowledge · Q1Moderate

Example 3 · Light and Optics · Refraction and Lenses

Where should an object be placed in front of a convex lens to get a real and enlarged image of the object ?

Enlarged and real needs F to 2F

A convex lens gives a real, enlarged image only with the object between F and 2F. At 2F the image is the same size; beyond 2F it is smaller; inside F it is enlarged but virtual.

Object at the focus: no image at a finite distance

Light from the focus leaves the lens parallel, so the image is at infinity. It is not at the focus on the other side, which is where light from infinity goes.

Drill 4 more on images formed by a convex lens

Concept 4 of 4: Lens formula, magnification and power

The image table tells you roughly where an image is; the lens formula gives the exact distance. The magnification says how much bigger the image is, and its sign says whether it is upright or upside down. The power of a lens says how strongly it bends light: a lens with a short focal length bends light sharply and has a large power.

Definition

Lens rules (New Cartesian convention: distances measured from the optical centre, along the light are positive):

Lens formula: $\dfrac{1}{v} - \dfrac{1}{u} = \dfrac{1}{f}$ . A convex lens has positive $f$ , a concave lens negative $f$ .
Magnification $m = v/u$ . Positive $m$ : virtual and erect. Negative $m$ : real and inverted. $|m| > 1$ : enlarged.
Power $P = 1/f$ , with $f$ in metres; unit the dioptre (D). Convex lens: positive power; concave: negative.
The bigger the size of the power (ignoring the sign), the more sharply the lens bends light.
Lenses in contact: powers add, $P = P_1 + P_2$ .

Lens formula, magnification and power

\frac{1}{v} - \frac{1}{u} = \frac{1}{f} \qquad m = \frac{v}{u} \qquad P = \frac{1}{f\,(\text{in m})}

u, vobject and image distances from the lens
ffocal length (positive for convex)
mmagnification
Ppower, in dioptres

Worked example

An object is 30 cm in front of a convex lens of focal length 20 cm. Find the image distance, the magnification and the power of the lens.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2023 · CDS (I) 2023 — General Knowledge · Q10Easy

Example 4 · Light and Optics · Refraction and Lenses

The power of a lens of focal length 10 cm is

Focal length in metres before taking the reciprocal

P = 1/f

needs

f

in metres. A 25 cm lens has

P = 1/0.25 = 4

D, not 1/25 = 0.04 D. Most wrong options are off by a factor of 100.

Bending power goes by size, not sign

A +2.5 D lens bends light more than a −2.0 D lens, and a −3 D lens more than a +2 D lens. Compare the sizes of the powers; the sign only says converging or diverging.

Positive magnification means virtual

For a lens

m = v/u

. A positive

m

means an erect, virtual image, and

|m| > 1

means it is enlarged. A real image from a single lens always has a negative

m

Drill 4 more on lens formula, magnification and power

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Refraction and the refractive index
Refractive index and Snell's law
$n = \frac{c}{v} \qquad n_1 \sin i = n_2 \sin r$
Total internal reflection
Critical angle
$\sin C = \frac{1}{n}$
Lens formula, magnification and power
Lens formula, magnification and power
$\frac{1}{v} - \frac{1}{u} = \frac{1}{f} \qquad m = \frac{v}{u} \qquad P = \frac{1}{f\,(\text{in m})}$

Reference tables (1)

Images formed by a convex lens5 rows

Object position	Image (convex lens)
Beyond 2F	Between F and 2F; real, inverted, smaller
At 2F	At 2F; real, inverted, same size
Between F and 2F	Beyond 2F; real, inverted, enlargedQ CDS 2020 (II): an object 40 cm from a lens of focal length 25 cm gives an inverted, magnified image.
At F	At infinityQ CDS 2022 (II): an object at the focus of a convex lens has its image at infinity.
Between F and the lens	Same side; virtual, erect, enlarged

Watch out for (8)

Refractive index is c over v, not v over c
→ Refraction and the refractive index
Slower means towards the normal
→ Refraction and the refractive index
Higher refractive index, smaller critical angle
→ Total internal reflection
Enlarged and real needs F to 2F
→ Images formed by a convex lens
Object at the focus: no image at a finite distance
→ Images formed by a convex lens
Focal length in metres before taking the reciprocal
→ Lens formula, magnification and power
Bending power goes by size, not sign
→ Lens formula, magnification and power
Positive magnification means virtual
→ Lens formula, magnification and power

Test yourself on Light and Optics

15 past CDS questions from this chapter, timed at 15 minutes and marked the way the exam marks it. You see your score and every answer the moment you finish. Free to start.

Take the Light and Optics chapter test Drill all 17 questions