What this chapter is about
Ray optics treats light as straight lines called rays, valid when the objects and apertures involved are much larger than the wavelength of light (about 400–700 nm). Under this approximation you can explain reflection, refraction, image formation by mirrors and lenses, and the working of common optical instruments without invoking wave effects such as diffraction or interference.
The chapter builds on the laws of reflection and refraction you met earlier and applies them systematically: plane and spherical mirrors, refraction at single surfaces and through prisms, thin lenses, combinations of lenses, and total internal reflection. It then uses these results to analyse the human eye, the simple microscope, the compound microscope, and telescopes. Understanding how these instruments magnify, resolve detail, and limit the field of view is both conceptually rich and practically useful.
After working through this material you should be able to locate images formed by mirrors and lenses using ray diagrams and formulas, calculate magnification and power, explain phenomena like the sparkling of diamonds and mirages, and describe the optical design of microscopes and telescopes in quantitative terms.
Key ideas
- Laws of reflection: The incident ray, the reflected ray and the normal lie in one plane; the angle of incidence equals the angle of reflection.
- Laws of refraction (Snell's law): n₁ sin θ₁ = n₂ sin θ₂, where n is the refractive index and θ is measured from the normal.
- Mirror formula: 1/v + 1/u = 1/f, with the sign convention that distances measured against the incident light are negative.
- Lens-maker's equation: 1/f = (n − 1)(1/R₁ − 1/R₂), connecting the focal length of a thin lens to its radii of curvature and refractive index.
- Total internal reflection occurs when light travels from a denser to a rarer medium and the angle of incidence exceeds the critical angle θ_c given by sin θ_c = n₂/n₁.
- Power of a lens P = 1/f (in metres), expressed in dioptres (D); powers add for thin lenses in contact.
- Magnifying power of an optical instrument is the ratio of the angle subtended at the eye with the instrument to the angle subtended at the unaided eye at the near point (usually 25 cm).
- Angular magnification of a telescope equals the ratio of the focal length of the objective to that of the eyepiece: m = f_o / f_e.
Formulas and facts to remember
- Mirror formula: 1/v + 1/u = 1/f; magnification m = −v/u.
- Refraction at a single spherical surface: n₂/v − n₁/u = (n₂ − n₁)/R.
- Thin-lens formula: 1/v − 1/u = 1/f; magnification m = v/u.
- Lens-maker's equation: 1/f = (n − 1)(1/R₁ − 1/R₂).
- Power: P = 1/f (f in metres, P in dioptres). For lenses in contact, P_net = P₁ + P₂ + …
- Critical angle: sin θ_c = 1/n when light goes from medium of index n to air.
- Prism deviation: At minimum deviation D_m, the ray inside the prism is parallel to the base; n = sin[(A + D_m)/2] / sin(A/2), where A is the prism angle.
- Simple microscope magnifying power: m = 1 + D/f (image at near point) or m = D/f (image at infinity), where D = 25 cm.
- Compound microscope magnifying power: m ≈ (L/f_o)(D/f_e), with L the tube length.
- Astronomical telescope magnifying power: m = f_o/f_e; tube length ≈ f_o + f_e for normal adjustment.
Worked examples
Example 1: Image by a concave mirror
A concave mirror has a focal length of 15 cm. An object 4 cm tall stands 30 cm in front of the mirror. Find the image position, nature and size.
Solution
Using the sign convention (object on the left, distances measured from the pole):
- u = −30 cm, f = −15 cm (concave mirror, focus in front).
Mirror formula: 1/v + 1/u = 1/f 1/v = 1/f − 1/u = 1/(−15) − 1/(−30) = −1/15 + 1/30 = −2/30 + 1/30 = −1/30 v = −30 cm.
Image is 30 cm in front of the mirror, real and inverted.
Magnification m = −v/u = −(−30)/(−30) = −1. Image height = m × object height = −1 × 4 cm = −4 cm (inverted, same size).
Example 2: Refraction through a glass slab
A ray of light in air strikes a glass slab (n = 1.5) at 45°. Find the angle of refraction and the critical angle for the glass-air boundary.
Solution
Snell's law: n₁ sin θ₁ = n₂ sin θ₂ 1 × sin 45° = 1.5 × sin θ₂ sin θ₂ = (√2/2)/1.5 = 0.707/1.5 ≈ 0.471 θ₂ ≈ 28.1°.
Critical angle (glass to air): sin θ_c = 1/1.5 ≈ 0.667 θ_c ≈ 41.8°.
Example 3: Magnifying power of a compound microscope
A compound microscope has an objective of focal length 1 cm and an eyepiece of focal length 5 cm. The tube length (distance between the two lenses) is 20 cm. Estimate the magnifying power when the final image is at the near point (25 cm).
Solution
Magnifying power m ≈ (L/f_o)(1 + D/f_e)
Here L ≈ 20 cm (approximate separation minus focal lengths, but for estimation we take 20 cm as the effective tube length), f_o = 1 cm, f_e = 5 cm, D = 25 cm.
m ≈ (20/1) × (1 + 25/5) = 20 × 6 = 120.
The microscope provides a magnification of about 120.
Common mistakes
- Forgetting the sign convention and mixing up real/virtual image positions → always assign signs before substituting in mirror or lens formulas.
- Using the thin-lens formula for a single refracting surface → the single-surface formula has n₁ and n₂ explicitly; the thin-lens formula is derived for a lens in air.
- Confusing magnification (ratio of image to object size) with magnifying power (ratio of angles subtended at the eye) → magnifying power is used for instruments like microscopes; linear magnification for simple mirror/lens problems.
- Applying total internal reflection when light goes from rarer to denser medium → TIR only occurs when light travels from higher to lower refractive index.
- Adding focal lengths instead of powers when combining lenses → powers (in dioptres) add; focal lengths do not add directly.
Quick revision
- Light rays reflect and refract obeying angle rules; use sign conventions consistently.
- Mirror formula: 1/v + 1/u = 1/f; lens formula: 1/v − 1/u = 1/f.
- Total internal reflection needs light going from denser to rarer and angle > critical angle.
- Power of a lens P = 1/f (dioptres); powers add for thin lenses in contact.
- Microscope magnifying power ≈ (L/f_o)(D/f_e); telescope magnifying power = f_o/f_e.
- Critical angle sin θ_c = 1/n links refractive index to the onset of TIR.