Geometrical Optics

🟢 Lite — Quick Review (1h–1d)

Rapid summary for last-minute revision before your exam.

Geometrical Optics — Key Facts

Geometrical optics deals with light travelling in straight lines (rays) and how it interacts with mirrors and lenses. The key principle is the law of reflection: the angle of incidence equals the angle of reflection, measured from the normal (perpendicular to the surface).

Reflection from Plane Mirror:

• Image is virtual (formed behind the mirror), upright, same size as object
• Image distance from mirror = object distance from mirror
• Lateral inversion: left and right appear swapped

Spherical Mirrors:

Type	Shape	Focal Length	Image Nature
Concave	Curves inward	Negative (f < 0)	Real or virtual depending on object position
Convex	Curves outward	Positive (f > 0)	Always virtual, diminished

Mirror Formula (sign convention — New Cartesian Sign Convention): $$\frac{1}{f} = \frac{1}{v} + \frac{1}{u}$$

where f = focal length, v = image distance, u = object distance (all measured from pole/centre of mirror).

Magnification: $$m = \frac{h_i}{h_o} = \frac{-v}{u}$$

⚡ ECAT Exam Tip: For concave mirrors — when object is beyond C (centre of curvature), image is real and inverted. When object is between F and mirror, image is virtual and magnified.

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🟡 Standard — Regular Study (2d–2mo)

For students who want genuine understanding…

Refraction — Snell’s Law:

When light passes from one medium to another, it bends. The relationship between angles: $$n_1 \sin\theta_1 = n_2 \sin\theta_2$$

where n = refractive index (ratio of speed of light in vacuum to speed in medium), θ_1 = angle of incidence, θ_2 = angle of refraction.

Critical Angle and Total Internal Reflection (TIR): When light goes from denser to rarer medium: $$\sin\theta_c = \frac{n_2}{n_1}$$

TIR occurs when θ_i > θ_c inside the denser medium. This is how fibre optic cables work!

Refractive Index: $$n = \frac{c}{v} = \frac{\sin i}{\sin r}$$

For light going from air to medium: $n = \frac{\sin\theta_{air}}{\sin\theta_{medium}}$

Lens Formula (Thin Lenses): $$\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$$

Lens Maker’s Formula: $$\frac{1}{f} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

where R_1 and R_2 are radii of curvature of the two surfaces.

Power of Lens: $$P = \frac{1}{f} \text{ (in metres)} \Rightarrow \text{Power in dioptres (D)}$$

Converging lens (convex): positive power; Diverging lens (concave): negative power.

⚡ ECAT Exam Tip: For a prism, minimum deviation δ_min occurs when the angle of incidence equals the angle of emergence. The refractive index $n = \frac{\sin(A+\delta_m)/2}{\sin(A/2)}$ where A = prism angle.

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🔴 Extended — Deep Study (3mo+)

Comprehensive theory for complete mastery…

Derivation of Lens Maker’s Formula:

Consider a thin lens with two spherical surfaces. For the first surface (air to glass): $$\frac{n_2}{v_1} - \frac{n_1}{u} = \frac{n_2 - n_1}{R_1}$$

For the second surface (glass to air): $$\frac{n_1}{v} - \frac{n_2}{v_1} = \frac{n_1 - n_2}{R_2}$$

Adding both equations for thin lens (u and v measured from lens): $$\frac{1}{v} - \frac{1}{u} = (n-1)\left(\frac{1}{R_1} - \frac{1}{R_2}\right)$$

Since $\frac{1}{f} = \frac{1}{v} - \frac{1}{u}$, we get the lens maker’s formula.

Combined Lens System:

For two thin lenses in contact: $\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2}$

For two lenses separated by distance d: $$\frac{1}{F} = \frac{1}{f_1} + \frac{1}{f_2} - \frac{d}{f_1 f_2}$$

Refraction through a Prism:

Deviation angle: $\delta = (i_1 + i_2) - (r_1 + r_2) = i_1 + i_2 - A$

At minimum deviation: $i_1 = i_2$ and $r_1 = r_2 = A/2$ $$\therefore \delta_m = 2i - A \Rightarrow i = \frac{A + \delta_m}{2}$$ $$n = \frac{\sin i}{\sin r} = \frac{\sin\left(\frac{A+\delta_m}{2}\right)}{\sin(A/2)}$$

Optical Fibres:

Step-index fibre: core (high n) surrounded by cladding (low n). Light undergoes TIR at core-cladding boundary. Numerical aperture: $$NA = \sqrt{n_1^2 - n_2^2}$$

Maximum acceptance angle: $\sin\theta_a = NA$

Dispersion:

White light splits into colours because refractive index varies with wavelength (n decreases as wavelength increases). Violet bends most, red bends least. This gives the familiar spectrum.

Scattering of Light (Rayleigh Scattering): $$I \propto \frac{1}{\lambda^4}$$

This explains why the sky is blue (shorter wavelengths scatter more) and why sunsets are red/orange (longer wavelengths survive).

⚡ ECAT 2024 Analysis: Questions on optical fibres and Rayleigh scattering appeared in recent papers. The formula $I \propto 1/\lambda^4$ is frequently tested. Focus on sign conventions — many students lose marks due to inconsistent conventions.

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