Special relativity — Michelson-Morley experiment: Foundations

2.3.25 · D1 · Physics › Modern Physics › Special relativity — Michelson-Morley experiment

Parent note ki ek bhi line pe trust karne se pehle, tumhe har woh symbol apna banana hoga jo woh tumhare saamne phenkta hai. Neeche hum unhe ek ek karke, bilkul zero se build karte hain. Har ek ko milta hai: plain words mein kya matlab hai → jis picture mein yeh rehta hai → topic ko iski zaroorat kyun hai. Upar se neeche padho; ladder ki har rung apne neeche wali par khadi hai.

1. Speed — distance over time

Topic ko iski zaroorat kyun hai: poori derivation kuch nahi hai sirf light ke har leg ke liye chhote chhote "distance ÷ speed" times jodne ke alawa. Agar tum $t = d / speed$ par solid ho, toh aadhi algebra pehle se tumhari hai.

2. $c$ — speed of light

Topic ko iski zaroorat kyun hai: poora experiment yeh poochhta hai "kya $c$ sach mein har direction mein same hai, ya Earth ki motion isko tilt karti hai?" Jis cheez ko measure kiya ja raha hai uske naam ke bina yeh sawaal nahi poochha ja sakta. $c$ ki value kahan se aati hai, iske liye Maxwell's Equations and the Speed of Light dekho.

3. $v$ — Earth ki speed aur "aether wind"

Topic ko iski zaroorat kyun hai: bina wind ke detect karne ke liye koi speed difference nahi hai. $v$ woh villain hai jise experiment hunt karta hai. Poori prediction isi par jeeti marti hai ki is $v$ ka koi effect hai ya nahi.

4. Speeds add aur subtract karna (Galilean intuition)

Topic ko iski zaroorat kyun hai: parallel-arm times $\frac{L}{c - v}$ aur $\frac{L}{c + v}$ poori tarah is rule se bane hain. Jab experiment fail hota hai, toh yahi rule (light ke liye) abandon karna padta hai — Galilean Relativity & Velocity Addition dekho.

5. $L$ — arm length

Topic ko iski zaroorat kyun hai: $L$ effect ka scale set karta hai. Lambe arms matlab tiny wind effect accumulate hone ke liye zyada time — isliye real apparatus ne beam ko kai mirrors se fold kiya taaki $L$ effectively $\sim 11$ m ho jaye.

6. Right triangle aur Pythagorean theorem

Yeh woh ek geometric piece hai jo perpendicular arm demand karta hai, toh hum ise fully build karte hain.

Topic ko iski zaroorat kyun hai: yahi woh jagah hai jahan se perpendicular arm ki speed $c_{⊥} = c^{2} - v^{2}$ aati hai. Bina Pythagoras ke cross-arm formula nahi, aur dono arms ke beech comparison nahi.

7. $β$ — speed ratio $v / c$

Topic ko iski zaroorat kyun hai: poori prediction $\propto β^{2}$ tak shrink ho jaati hai. Ratio ko naam dene se hum keh sakte hain "effect $v / c$ mein second order hai" — chhota, par zero nahi. $β$ Lorentz Transformation, Length Contraction aur Time Dilation har jagah dobara aata hai.

8. Do arm times $t_{∥}$ aur $t_{⊥}$

Ab jab hum $c$ , $v$ , $L$ aur $c_{⊥}$ ke malik hain, hum do round-trip times build kar sakte hain jo experiment compare karta hai.

Topic ko iski zaroorat kyun hai: yeh do formulas poori prediction ka dil hain. Inme har symbol — $L$ , $c$ , $v$ , $c_{⊥}$ — upar build kiya gaya hai, toh tum ab exactly dekh sakte ho ki yeh kaise combine hote hain. Experiment dono ke beech ka difference measure karta hai.

9. $β$ aur approximation $\frac{1}{1 - x} \approx 1 + x$

Topic ko iski zaroorat kyun hai: exact bracket unreadable hai. Yeh tool us sawaal ka jawab deta hai "yeh messy formula kaisa dikhta hai jab $v$ tiny ho?" — aur tiny exactly humara case hai, ise clean $Δ t \approx \frac{L v ^{2}}{c ^{3}}$ mein melt karta hai.

10. Waves: wavelength $λ$ , aur interference fringes

Topic ko iski zaroorat kyun hai: time difference $Δ t$ clock se time karne ke liye bahut chota hai. Lekin fringe shift mein convert kiya — ek visible stripe movement $λ$ ki units mein measured — yeh kuch aisa ban jaata hai jo aankhein aur apparatus pakad sakta hai. Yeh poori measurement ka engine hai; ise Interference of Light & Fringes mein deepen karo.

11. $Δ$ (delta), $Δ t$ , aur fringe-shift formula $Δ N$

Topic ko iski zaroorat kyun hai: yeh bridge hai theory (ek time difference) se observation (ek visible stripe shift) tak. Experiment jo kuch bhi report karta hai woh is $Δ N$ ke baare mein ek statement hai — aur headline result yeh hai ki yeh zero nikla.

Prerequisite ladder (text)

Neeche har rung apne se pehle wali par khadi hai — yeh wahi chain hai jo parent note chalti hai, ab har symbol ke saath tumhara apna:

Speed = distance ÷ time — woh ek arithmetic rule jo har arm ke peeche hai.
$c$ (speed of light) aur $v$ (Earth ki speed) — race mein do speeds.
Speeds add karna ( $c \pm v$ ) — parallel-arm effective speeds deta hai.
$L$ — sets karta hai ki light kitni door travel karti hai, isliye effect ka size.
Right triangle + Pythagoras — cross speed $c_{⊥} = c^{2} - v^{2}$ produce karta hai.
$t_{∥}$ aur $t_{⊥}$ — do round-trip times, steps 1–5 se build ki gayi.
$β = v / c$ aur small- $x$ approximation — difference $Δ t$ ko $\frac{L v ^{2}}{c ^{3}}$ tak melt karte hain.
$λ$ , fringes, aur $Δ N = \frac{c ( 2Δ t )}{λ}$ — time difference ko ek visible stripe shift mein convert karte hain.
Null result — $Δ N$ zero measure hua, aether ko khatam karta hai.

Equipment checklist

Self-test: right side cover karo aur dekho ki kya tum har jawab zor se bol sakte ho.

Time, distance, speed ko link karne wala formula likho.

t = \frac{distance}{speed}

c

kya hai aur iska rough value kya hai?

Speed of light,

\approx 3 \times 1 0^{8}

m/s.

Is experiment mein

v

kya hai aur iska value?

(Supposed) aether mein Earth ki orbital speed,

\approx 3 \times 1 0^{4}

m/s.

Aether wind ke against jaate hue, light ki effective speed kya hai?

c - v

; wind ke saath yeh

c + v

hai.

L

kya hai?

Interferometer arm ki one-way length; round trip

2 L

hai.

Pythagorean theorem state karo.

(hypotenuse)² = (leg)² + (leg)², yaani

h^{2} = a^{2} + b^{2}

c_{⊥}

kya hai aur yeh

c^{2} - v^{2}

kyun hai?

Cross-arm speed; light upstream aim karti hai,

c

hypotenuse hai aur

v

ek leg, toh cross leg

c^{2} - v^{2}

hai.

Parallel-arm time

t_{∥}

likho.

t_{∥} = \frac{L}{c - v} + \frac{L}{c + v}

Perpendicular-arm time

t_{⊥}

likho.

t_{⊥} = \frac{2 L}{c ^{2} - v ^{2}}

β

define karo aur Earth ke liye iska value do.

β = v / c \approx 1 0^{- 4}

, toh

β^{2} \approx 1 0^{- 8}

Tiny

x

ke liye

\frac{1}{1 - x}

approximate karo.

\approx 1 + x

Tiny

x

ke liye

\frac{1}{1 - x}

approximate karo.

\approx 1 + \frac{x}{2}

λ

kya hai aur yahan iska value kya hai?

Wavelength (crest-to-crest),

\approx 500

= 5 \times 1 0^{- 7}

Fringe shift

Δ N

kya hai aur iska formula?

Stripe pattern kitna slide hota hai, wavelengths mein;

Δ N = \frac{c Δ t}{λ}

(aur

90°

rotation par

\frac{c ( 2Δ t )}{λ}

Δ

ka kya matlab hai?

"Mein change" ya "ke beech difference."

Δ N

ki measured value kya thi?

Essentially zero — woh null result jisne aether ko khatam kiya.

Jab upar ki har line instantly wapas aaye, tum Michelson–Morley derivation aur Special Relativity — Einstein's Postulates mein uske consequences ke liye ready ho.