Geometric optics — rectilinear propagation, reflection, refraction

1. Rectilinear propagation

WHY does it go straight? In a uniform medium the speed of light is the same everywhere, so the least-time path between two points is the straight line (the shortest path). Any bend would cost extra distance = extra time.

WHAT is the evidence? Sharp shadows, pinhole cameras, eclipses.

2. Reflection

Deriving $\theta_i=\theta_r$ from Fermat's least-time principle

Formula Reflection from least time

Light from $A=(0,a)$ must hit the mirror (the $x$ -axis) at point $P=(x,0)$ and reach $B=(d,b)$ . Speed is constant $c$ , so least time = least path length. $L(x)=\sqrt{x^2+a^2}+\sqrt{(d-x)^2+b^2}$ Minimise: $\dfrac{dL}{dx}=0$ $\frac{x}{\sqrt{x^2+a^2}}-\frac{d-x}{\sqrt{(d-x)^2+b^2}}=0$ The first fraction is $\sin\theta_i$ , the second is $\sin\theta_r$ (each is opposite/hypotenuse measured from the normal). Hence $\sin\theta_i=\sin\theta_r\;\Rightarrow\;\boxed{\theta_i=\theta_r}$ Why this step? $dL/dx=0$ finds the path the light actually "chooses"; the geometry turns the derivative directly into the two sines.

Figure — Geometric optics — rectilinear propagation, reflection, refraction

3. Refraction

Formula Snell's law — derived from least time

Light goes from $A$ (in medium 1, speed $v_1$ ) to $B$ (in medium 2, speed $v_2$ ), crossing the boundary (the $x$ -axis) at $P=(x,0)$ . $A=(0,a)$ , $B=(d,-b)$ . Time: $T(x)=\frac{\sqrt{x^2+a^2}}{v_1}+\frac{\sqrt{(d-x)^2+b^2}}{v_2}$ Set $\dfrac{dT}{dx}=0$ : $\frac{x}{v_1\sqrt{x^2+a^2}}=\frac{d-x}{v_2\sqrt{(d-x)^2+b^2}}$ Recognise the sines: $\dfrac{\sin\theta_1}{v_1}=\dfrac{\sin\theta_2}{v_2}$ . Using $v=c/n$ : $\boxed{n_1\sin\theta_1=n_2\sin\theta_2}$ Why this step? Now $v_1\neq v_2$ , so least-time is NOT least-distance — the extra $1/v$ factors are what bend the ray. The derivative again converts the $x$ -terms into sines.

Common mistakes (Steel-man + fix)

Active recall

Recall Test yourself (cover the answers)

State the three laws governing rays. → straight-line propagation; $\theta_i=\theta_r$ ; $n_1\sin\theta_1=n_2\sin\theta_2$ .
From what single principle do all three follow? → Fermat's principle of least time.
Which way does a ray bend entering a denser medium? → toward the normal.
What stays constant in refraction: $f$ , $\lambda$ , or $v$ ? → only frequency $f$ .

What does rectilinear propagation state?

In a uniform medium light travels in straight lines.

From which point are all optics angles measured?

The normal (perpendicular to the surface).

State the law of reflection.

\theta_i=\theta_r

, incident, reflected and normal coplanar.

Define refractive index.

n=c/v

; ratio of speed of light in vacuum to its speed in the medium (

n\ge1

State Snell's law.

n_1\sin\theta_1=n_2\sin\theta_2

What single principle yields reflection and refraction laws?

Fermat's principle of least time.

Which way does a ray bend going into a denser (higher-n) medium?

Toward the normal (

\theta

decreases).

Formula for the critical angle (denser n1 → rarer n2)?

\sin\theta_c=n_2/n_1

In refraction, which of f, λ, v stay constant?

Frequency f stays constant; λ and v change.

Wavelength inside a medium of index n?

\lambda_{med}=\lambda_0/n

Pinhole magnification formula?

m=h/H=v/u

(by similar triangles, inverted image).

Recall Feynman: explain to a 12-year-old

Light is like a super-fast runner who always picks the route that takes the least time. On flat ground (same speed everywhere) the fastest route is a straight line — so light goes straight. If a mirror is in the way, the runner touches it and comes off at the same slant he hit it. If he has to cross from a racetrack into deep sand (where he runs slower), he doesn't go straight in — he angles his path so he spends less time in the sand. That clever angling is exactly why a straw looks bent in a glass of water!

Connections

Fermat's principle — the master principle behind all three laws.
Total internal reflection and Optical fibres — direct consequence of Snell's law.
Wavefronts and Huygens' principle — the wave-level reason refraction bends light.
Mirrors and Lenses — apply reflection/refraction to curved surfaces.
Dispersion — $n$ depends on wavelength, splitting white light into colours.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, geometric optics ka pura khel sirf teen rules par chalta hai, aur teeno ek hi master idea se nikalte hain — Fermat ka least-time principle, yaani light hamesha woh raasta choose karti hai jismein sabse kam time lage. Uniform medium mein speed same hoti hai, isliye kam time ka matlab kam distance, aur kam distance ka matlab seedhi line — yeh hai rectilinear propagation. Isi wajah se shadows sharp banti hain aur pinhole camera ulta image deta hai.

Reflection mein light mirror se takra ke aati hai, aur angle of incidence = angle of reflection, lekin yaad rakho — angle hamesha normal se measure karte hain, surface se nahi. Yeh bahut common galti hai. Least-time se prove bhi ho jaata hai: jis point par bounce hoga, wahan total path shortest hota hai jab dono angle barabar ho.

Refraction mein asli twist hai — yahan light ki speed change hoti hai naye medium mein. Denser medium (zyada $n$ ) mein light slow hoti hai, isliye ray normal ki taraf bend hoti hai. Formula: $n_1\sin\theta_1=n_2\sin\theta_2$ (Snell's law). Yaad rakho $n=c/v$ , toh bada $n$ matlab choti speed — ulta mat samajhna! Aur ek important baat: refraction mein frequency same rehti hai, sirf wavelength aur speed badalte hain. Jab dense se rare medium mein jaate ho aur angle critical angle se zyada ho jaaye, toh light bilkul bahar nikal hi nahi paati — total internal reflection ho jaata hai, jo optical fibre ka jaadu hai. Bas itna samajh lo toh geometric optics ka 80% cover ho gaya.

Geometric optics — rectilinear propagation, reflection, refraction

1. Rectilinear propagation

2. Reflection

Deriving $\theta_i=\theta_r$ from Fermat's least-time principle

3. Refraction

Common mistakes (Steel-man + fix)

Active recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Optics

Connections