Refraction and Lenses

What you will learn

Explain refraction, trace thin-lens images, and calculate image position, magnification and power. Establish directions and signs before using formulas. Examples and checks are original.

1 Light crossing a boundary

Within a homogeneous transparent medium, light travels in straight lines in the ray model. Homogeneous means that the relevant optical properties are uniform throughout. At a boundary between different refractive indices, transmitted light changes speed and usually direction. This is refraction.

Draw a normal perpendicular to the boundary at the point where the ray arrives. Measure incidence angle i and refraction angle r from this normal, never from the surface. For normal incidence, i = 0° and r = 0°: light goes straight through although its speed changes. Therefore, a speed change does not always produce a visible bend.

The absolute refractive index is n = c/s, where c is light's speed in vacuum and s is its speed in the medium. Index has no unit. A higher index means lower light speed. “Optically denser” describes this optical comparison, not mass per unit volume. At a stationary boundary, frequency stays unchanged; wavelength changes with speed.

2 Predicting the bend

The incident ray, refracted ray and normal lie in one plane. For a given colour travelling from medium 1 into medium 2, Snell's law is:

n₁ sin i = n₂ sin r

Thus sin i/sin r = n₂/n₁ for nonzero angles. It equals n₂ alone only when n₁ = 1, approximately the air-to-material case. Compare both media:

• Entering a higher-index medium: the transmitted ray bends towards the normal, so r < i for oblique incidence
• Entering a lower-index medium: it bends away, so r > i when a refracted ray exists

From higher to lower index, sufficiently oblique rays can undergo total internal reflection instead of transmission. Do not assume every incident ray must emerge.

3 A parallel glass slab

For a slab with parallel faces and air on both sides, draw a normal at each crossing. An oblique ray bends towards the normal on entering glass and away on leaving. The emergent ray is parallel to the original incident direction, but shifted sideways. This is lateral displacement. Parallel emergence does not mean refraction was absent. At normal incidence there is no sideways shift. A prism has nonparallel refracting faces, so this slab result cannot simply be transferred to it.

4 The lens model

Here we use thin glass lenses in air, with lens material optically denser than air. Rays stay close to the principal axis and make small angles with it: these are paraxial rays. Thick lenses and large-angle rays require more detail.

A convex lens is thicker at its centre and converges an axis-parallel beam. A concave lens is thinner at its centre and makes that beam diverge. Mark optical centre O, the principal axis through O, and equal-distance focal points F_L on the left and F_R on the right. Light travels left to right. Focal-length magnitude |f| is the distance OF; 2F marks twice that distance. Convex-lens parallel rays meet at F_R; concave-lens rays have backward extensions meeting at F_L.

5 Construct an image with two rays

Place an upright object arrow on the left, with its base on the axis. From its tip draw two of these rays:

Convex lens: A ray parallel to the axis emerges through F_R. A ray through O continues undeviated in this thin-lens model. A ray whose incident line passes through F_L emerges parallel to the axis.

Concave lens: An axis-parallel ray emerges diverging as if from F_L. A ray through O continues undeviated. A ray directed towards F_R emerges parallel to the axis.

Use solid lines with arrows for actual light travel. If outgoing rays meet, their intersection locates the image tip: the image is real and can be caught on a screen there. If they spread apart, extend their outgoing lines backwards using dashed lines. Their apparent intersection locates a virtual image, which cannot be caught directly on a screen at that position. Dashed extensions are construction lines, not light travelling backwards.

For a convex lens with the object beyond F_L, these rays meet on the right below the axis. With the object between O and F_L, their backward extensions meet on the left above the axis. For a concave lens and a real object, the backward extensions meet between O and F_L above the axis.

6 All the standard image cases

These cases assume a real object on the left: incident rays actually spread from it. Size comparisons refer to height.

Convex lens

• Object very far away: image near F_R, real, inverted and highly diminished; an axial point at infinity focuses at F_R
• Beyond 2F_L: image between F_R and 2F_R, real, inverted and diminished
• At 2F_L: image at 2F_R, real, inverted and the same size
• Between F_L and 2F_L: image beyond 2F_R, real, inverted and enlarged
• At F_L: rays from each object point emerge parallel; no finite screen position receives a focused image, conventionally called an image at infinity
• Between F_L and O: image on the object's side, virtual, upright and enlarged

Concave lens

• Any finite real-object distance: image between O and F_L, virtual, upright and diminished
• Axial object point at infinity: virtual image at F_L; a very distant extended object gives an upright, highly diminished image near F_L

A convex lens can therefore produce either real or virtual images. A magnifying glass uses the inside-focus case. The usual statement that a concave lens always makes a virtual image requires the real-object condition.

7 Signs and formulas

Measure all distances from O. With incident light left to right, rightward distances are positive and leftward distances negative. Hence a real object has u < 0; a real image on the right has v > 0; a virtual image on the left has v < 0. Convex f > 0 and concave f < 0. Heights above the axis are positive; below it, negative.

Thin-lens formula: 1/f = 1/v − 1/u

Magnification: m = hᵢ/hₒ = v/u

Here hᵢ and hₒ are image and object heights. Positive m means upright; negative m means inverted. Compare |m| with 1 to judge size. Use matching length units in the lens formula. Lens magnification is v/u; the mirror formula has an extra minus sign.

Power: P = 1/f, with f in metres and P in dioptres (D). Convex power is positive, concave power negative. A shorter focal-length magnitude means stronger convergence or divergence, not necessarily a larger image for every object position.

8 Three worked numerical examples

Example 1 Refraction. Light enters a material of n = 1.60 from air with n ≈ 1 at i = 30°. Take c = 3.00 × 10⁸ m/s.

Speed s = c/n = 1.875 × 10⁸ m/s. Snell's law gives sin r = (1 × sin 30°)/1.60 = 0.3125, so r ≈ 18.2°. The smaller angle confirms bending towards the normal. The speed symbol s here is different from image distance v.

Example 2 Convex lens. An upright 4 cm object is 36 cm left of a convex lens with focal length 12 cm.

u = −36 cm; f = +12 cm. 1/v = 1/f + 1/u = 1/12 − 1/36 = 1/18. Thus v = +18 cm and m = 18/(−36) = −0.50. hᵢ = m hₒ = −0.50 × 4 = −2 cm.

The image is 18 cm right of the lens, real, inverted and half-sized. This agrees with the beyond-2F case. For power, f = +0.12 m, so P ≈ +8.33 D.

Example 3 Concave lens. An upright 8 cm object is 48 cm left of a concave lens with focal-length magnitude 16 cm.

u = −48 cm; f = −16 cm. 1/v = −1/16 − 1/48 = −1/12. Thus v = −12 cm and m = (−12)/(−48) = +1/4. hᵢ = (1/4) × 8 = +2 cm.

The image is 12 cm left of the lens, virtual, upright and diminished, between O and F_L. With f = −0.16 m, P = −6.25 D.

9 Common mistakes and recap

Measure angles from the normal. Compare optical indices, not material weight. Check object position before describing a convex-lens image. Assign signs before calculation and convert centimetres to metres before finding power. Magnification changes with object position even for the same lens. Finally, check whether the calculated side, orientation and size agree with the ray construction.

Check your understanding

Attempt all six before reading the key.

1. A ray enters glass from air along the normal. What changes?

A. Direction only B. Speed, but not direction or frequency C. Frequency only D. Neither speed nor direction

2. At nonzero angles, light travels from n₁ = 1.20 to n₂ = 1.80. What is sin i/sin r?

A. 2/3 B. 1.20 C. 1.80 D. 1.50

3. A real object lies between a convex lens and its nearer focus. Its image is:

A. Virtual, upright and enlarged B. Real, inverted and diminished C. Virtual, upright and diminished D. Real, upright and enlarged

4. A concave lens has focal length −80 cm. Its power is:

A. +1.25 D B. −0.0125 D C. −1.25 D D. −80 D

5. For a thin lens, u = −24 cm and v = +12 cm. What is magnification?

A. +0.50 B. −0.50 C. −2 D. +2

6. An oblique ray crosses a parallel glass slab with air on both sides. The emerging ray is:

A. Always perpendicular to the slab B. Unchanged because no refraction occurs C. Always aimed at a common focus D. Parallel to the incident direction and laterally displaced

Explained key

1. B — At normal incidence both angles are zero. Speed decreases and wavelength changes; frequency stays the same. A visible bend is not required.

2. D — The ratio is n₂/n₁ = 1.80/1.20 = 1.50. Using 1.80 alone incorrectly treats the first medium as having index 1.

3. A — Outgoing rays diverge; backward extensions meet on the object side. The inside-focus case gives an enlarged virtual image, not a screen image.

4. C — Convert first: f = −0.80 m. P = 1/(−0.80) = −1.25 D. The negative sign identifies divergence; −0.0125 comes from using centimetres incorrectly.

5. B — m = v/u = 12/(−24) = −0.50: inverted and half-sized. Adding the mirror formula’s extra minus sign would give the wrong orientation.

6. D — Refraction occurs at both faces. Parallel faces and the same outside medium restore the direction, while oblique travel through the slab shifts the ray sideways.

Sources

Primary: NCERT Science, §§9.3.1–9.3.8, printed pp. 146–158. Supporting: NIOS Light Energy, §§15.9–15.14, pp. 339–347. Technical cross-check only: NCERT Ray Optics, §9.3 and §§9.5.2–9.5.3.

Analogy

A straight straw viewed obliquely across the water surface can look bent. Rays from its submerged part refract as they leave water. The eye traces the outgoing rays backwards and locates that part at an apparent position. The straw itself has not bent: the light path changed at the boundary.

Quick reference

Refraction and lenses quick reference

• Measure i and r from the normal; normal incidence gives no bend
• n = c/s; higher optical index means lower light speed
• n₁ sin i = n₂ sin r; frequency is unchanged at a stationary boundary
• Parallel slab, same outside medium: emergent direction parallel; oblique incidence gives lateral shift
• Thin glass lenses in air: convex converges, concave diverges
• Convex plus inside-focus real object: virtual, upright, enlarged image
• Concave plus real object: virtual, upright, diminished image
• Left-to-right light: real object u < 0; convex f > 0; concave f < 0
• Lens: 1/f = 1/v − 1/u; m = v/u = hᵢ/hₒ
• m < 0 means inverted; m > 0 means upright; |m| gives size ratio
• P = 1/f with f in metres; unit dioptre (D)
• Solid lines show light travel; dashed backward extensions locate virtual images