12.2 Ray Optics

12.2 Ray Optics

Quick Answer: Ray-optics MCQs reward Snell’s law, the mirror/lens formula with a consistent sign convention, magnification, and power in dioptres. On the PAF Aeronautical Engineering academic papers, treat geometrical optics as high-yield FSc material—cockpit instruments, canopies, and fibre ideas all rest on the same reflection/refraction rules.

Ray optics (geometrical optics) models light as rays travelling in straight lines in a uniform medium. It fails when apertures become comparable to wavelength (that is wave optics), but it solves most school-level image-formation problems.

Reflection

Laws of reflection:

1. The incident ray, reflected ray, and normal at the point of incidence lie in one plane.
2. Angle of incidence i equals angle of reflection r (both measured from the normal).

A plane mirror forms a virtual, erect image as far behind the mirror as the object is in front, with lateral inversion. Magnification magnitude is 1.

Spherical Mirrors

For a spherical mirror of radius R, focal length:

$$f = \frac{R}{2}$$

Using the common Cartesian sign convention (incident light direction positive; distances to the left of the optical device often taken negative for real objects in many FSc texts—follow the convention stated in the question stem):

$$\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$$ $$m = \frac{h'}{h} = -\frac{v}{u}$$

Worked example — concave mirror

Object at u = −30 cm from a concave mirror with f = −15 cm (sign convention: object distance negative, concave f negative).

$\frac{1}{v} = \frac{1}{f} - \frac{1}{u} = \frac{1}{-15} - \frac{1}{-30} = -\frac{1}{15} + \frac{1}{30} = -\frac{1}{30}$

So v = −30 cm: real image at the centre of curvature, same size (|m| = 1), inverted.

Refraction and Snell’s Law

When light crosses an interface:

$$n_1 \sin i = n_2 \sin r$$

Absolute refractive index n = c/v (c = speed of light in vacuum). Larger n means optically denser medium and smaller speed.

• Entering denser medium (n₂ > n₁): ray bends toward the normal (r < i).
• Entering rarer medium: ray bends away from the normal.

Worked example — Snell

Light in air (n₁ ≈ 1.00) strikes glass (n₂ = 1.50) at i = 30°. Find r.

$\sin r = \frac{n_1}{n_2}\sin i = \frac{1}{1.50}\sin 30° = \frac{0.5}{1.5} = \frac{1}{3}$

$r = \sin^{-1}(1/3) \approx 19.5°$

Apparent Depth

For small angles, apparent depth ≈ real depth / n when viewing normally into a denser medium. A pool looks shallower than it is—standard FSc recognition item.

Total Internal Reflection (TIR)

When light travels from denser to rarer medium and i exceeds the critical angle i_c:

$$\sin i_c = \frac{n_2}{n_1} \quad (n_1 > n_2)$$

For glass (1.50) to air: sin i_c = 1/1.5 ⇒ i_c ≈ 41.8°. Optical fibres and some aircraft canopy/glare ideas exploit TIR: light guided by repeated internal reflection.

Thin Lenses

The thin-lens equation has the same algebraic form as the mirror formula:

$$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$$

(Exact arrangement of signs depends on the Cartesian convention used in your FSc text; many Pakistani boards use object distance u negative and f positive for a converging lens.)

Lens maker’s formula (thin lens in air):

$$\frac{1}{f} = (n - 1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

Power of a Lens

$$P = \frac{1}{f(\text{in metres})} \quad \text{(dioptre, D)}$$

Combined thin lenses in contact: P = P₁ + P₂ (focal lengths combine as reciprocals).

Worked example — power

A converging lens has f = 50 cm = 0.50 m. Power P = 1/0.50 = +2 D.

Worked example — image by converging lens

Object 30 cm from a +10 cm focal-length lens (u = −30 cm, f = +10 cm):

$\frac{1}{v} = \frac{1}{f} + \frac{1}{u} = \frac{1}{10} + \frac{1}{-30} = \frac{3 - 1}{30} = \frac{2}{30}$

v = +15 cm → real image on the far side, inverted, |m| = v/|u| = 15/30 = 0.5 (diminished).

Ray Diagrams Worth Memorising

For a convex lens:

A concave lens with a real object always gives a virtual, erect, diminished image between F and the lens.

Aeronautical Link (Exam Motivation, Not Extra Theory)

Windscreen refraction, HUD combiner plates, camera lenses on UAVs, and fibre data links all reduce to reflection, Snell, and lens power. The initial academic paper will not ask you to design a HUD—but it will ask whether a +2 D lens converges, what TIR requires, and where a real image forms.

Drill sign-consistent algebra until mirror/lens MCQs feel mechanical.