Geometric optics — rectilinear propagation, reflection, refraction: Foundations

2.5.1 · D1 · Physics › Optics › Geometric optics — rectilinear propagation, reflection, refr

Isse pehle ki tum parent note Geometric optics follow kar sako, tumhe ideas ka ek chhota toolbox apna banana hoga. Hum inhe order mein build karte hain — har ek sirf wohi use karta hai jo pehle aa chuka hai.

1. Ek point aur uske coordinates $(x, y)$

Pehla figure dekho. Neeche ki line $x$ -axis hai, side wali line $y$ -axis hai, aur jahan woh cross karti hain woh corner origin $(0, 0)$ hai. Lavender mein marked point $A$ $(0, a)$ par baitha hai — zero steps right, $a$ steps up.

Figure — Geometric optics — rectilinear propagation, reflection, refraction

Is topic ko yeh kyun chahiye. Parent note mein har derivation (reflection, refraction) light-sources aur screen-points ko $A = (0, a)$ aur $B = (d, b)$ jaise spots par rakhti hai. Numbers ke woh pairs sirf is map par points hain. Agar tum $(x, y)$ nahi padh sakte, to algebra ka kuch bhi matlab nahi.

2. Do points ke beech distance — square root kahan se aati hai

Square root kyun, aur yeh tool kyun? Kyunki right triangles ke liye Pythagoras rule hai: sideways gap aur upar-neeche gap ek right triangle ki do choti sides hain, aur seedhi distance lambi slanted side hai (hypotenuse). Pythagoras kehta hai $(long side)^{2} = (side_{1})^{2} + (side_{2})^{2}$ , isliye square ko undo karne ke liye hum square root lete hain. Figure 2 dekho: mint dashed lines do choti sides hain, coral line woh hypotenuse hai jo hum chahte hain.

Is topic ko yeh kyun chahiye. Ek ray jo path travel karta hai woh seedhe segments ka sum hai; har segment ki length inhi square roots mein se ek hai. "Least-time" idea total length (ya time) ko jitna ho sake chhota karne se kaam karta hai — isliye hume pehle har length likh paana zaroori hai.

3. Right triangle, aur uske andar ka angle

Figure 3 mein angle $θ$ top corner par hai. Opposite side (butter colour) uske saamne hai; adjacent side (mint) use touch karti hai; coral hypotenuse ray khud hai.

Is topic ko yeh kyun chahiye. Reflection aur refraction dono mein ray, boundary, aur normal ek right triangle banate hain. Ray ka normal se jo angle banta hai woh hamara $θ$ hai. "Ray kitni slanted hai" yeh measure karne ke liye, hume ek aisa number chahiye jo woh slant capture kare — woh number hai sine, agla.

4. Sine — ek number jo "slant" measure karta hai

Yeh tool kyun aur koi doosra kyun nahi? Hum ek aisa number chahte hain jo bataaye ray kitni tilted hai, jo is baat par depend na kare ki ray kitni lambi khinchi. Opposite side ko hypotenuse se divide karna exactly wahi deta hai: ray ko do baar lambi karo aur dono lengths double ho jaati hain, isliye ratio unchanged rehta hai. Woh ratio $0$ (ray reference line ke saath flat) se $1$ (ray right angles par) tak badhta hai, isliye yeh "slant ki matra" perfectly encode karta hai.

5. Normal — woh line jisse hum angles measure karte hain

Figure 4 mein ek mirror (flat surface) hai aur slate normal seedha upar khada hai. Incoming ray normal se angle $θ_{i}$ banati hai, mirror se nahi. Optics mein har angle — incidence, reflection, refraction — is line se measure hota hai.

Is topic ko yeh kyun chahiye. Teeno laws (reflection, Snell) angles-from-the-normal ke saath state hain. Normal ke bina, "angle of incidence" ka koi fixed meaning nahi hai.

6. Light ki speed $c$ , medium mein speed $v$ , aur ratio $n$

Kyunki $v$ kabhi $c$ ko beat nahi kar sakta, fraction $c / v$ hamesha kam se kam $1$ hota hai, isliye $n \geq 1$ hamesha.

Yeh ratio kyun aur subtraction kyun nahi? Hum care karte hain kitni times slower ke baare mein, na ki km/s mein kitna slower — ek ratio materials mein fairly compare karta hai aur units drop karta hai. Woh ratio exactly woh quantity hai jo control karti hai ki ray kitni sharply bend hogi, isliye parent note Snell's law us par build karta hai.

Recall Speeds par quick check

Agar diamond ke liye $n = 2$ hai, to kya light uske andar faster hai ya slower, aur kitne factor se? ::: Slower — exactly $c$ ka aadha, kyunki $v = c / n = c /2$ .

7. Derivative ka chhota idea (sirf utna jo chahiye)

Yeh tool kyun? "Least time" ek minimisation question hai, aur minimum dhundhne ka ek universal trick hai: dhundho jahan slope zero hai. Tumhe yahaan calculus master karne ki zaroorat nahi — bas accept karo ki "flat slope = valley bottom = least time," aur yahi woh jagah hai jahan sines pop out hote hain.

Prerequisite map

Ise top-down padho: coordinates aur Pythagoras tumhe lengths dete hain; triangles aur normal tumhe angles as sines dete hain; speeds tumhe index dete hain; least-time slope-is-zero idea unhe do bade laws mein bandh karta hai.

Equipment checklist

Recall Kya main ready hoon? (answers cover karo)

$A = (0, a)$ words mein likho. ::: Woh point jo origin se zero steps right aur $a$ steps up hai. $(x, 0)$ se $(0, a)$ tak distance? ::: $x^{2} + a^{2}$ , seedha Pythagoras se. Us distance mein square root kyun? ::: Pythagoras hypotenuse ka square deta hai, isliye hum use undo karne ke liye root lete hain. $sin θ$ words mein? ::: Right triangle ki hypotenuse se divide ki hui opposite side. $sin θ$ unit-free kyun hai? ::: Yeh ek length divided by ek length hai, isliye units cancel ho jaate hain. Normal kya hai? ::: Contact ke point par surface ke perpendicular line — angles usi se measure hote hain. $n = c / v$ ka matlab? ::: Light material mein vacuum ki tulna mein kitni times slower travel karti hai; hamesha $\geq 1$ . Agar $n$ bada hai, to kya $v$ bada hai ya chhota? ::: Chhota — bada $n$ matlab slow light. $d T / d x = 0$ kya dhundhta hai? ::: $x$ ki woh value jahan total time sabse chhota hota hai — time-valley ka bottom. Woh kaun sa single principle woh flat-slope idea use karta hai? ::: Fermat's principle — light least-time path leti hai.

Connections

Parent: Geometric optics — woh note jise yeh foundations unlock karte hain.
Fermat's principle — "slope = zero" ke peeche ka least-time idea.
Total internal reflection aur Optical fibres — jahan index $n$ aur Snell's law aage le jaate hain.
Wavefronts and Huygens' principle — ray ke neeche wave-level picture.