Special relativity — Michelson-Morley experiment

2.3.25 · Physics › Modern Physics

WHY this experiment was done

Agar aether rest mein hai aur Earth $v \approx 30 km/s$ speed se Sun ke around orbit karti hai, to Earth par hume ek "aether wind" $v$ speed se past hoti mehsoos honi chahiye. Wind ke saath aur wind ke across jaane wali light measurably affect hogi — jaise ek swimmer nadi mein hoti hai.

HOW the apparatus works — the interferometer

Michelson interferometer:

Ek light beam ek half-silvered mirror (beam splitter) se takraati hai jo ise do perpendicular beams mein tod deta hai.
Har beam distance $L$ tak mirror tak jaati hai aur wapis reflect hoti hai.
Beams wapis milti hain aur interfere karti hain, bright/dark fringes banati hain.
Apparatus ko $90°$ rotate karna swap kar deta hai kaun sa arm "parallel" hai aether wind ke — fringes shift ho jaati hain agar wind exist karti hai.

Figure — Special relativity — Michelson-Morley experiment

DERIVATION — from first principles

Hum round-trip times compute karte hain yeh assume karke ki light $c$ speed se aether ke relative move karti hai, aur apparatus $v$ speed se uske through move karta hai.

Parallel arm (along the wind)

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

Yeh step kyun? Har term us leg ke liye distance $\div$ effective speed hai.

Common denominator par combine karo: $t_{∥} = \frac{L ( c + v ) + L ( c - v )}{( c - v ) ( c + v )} = \frac{2 L c}{c ^{2} - v ^{2}} = \frac{2 L}{c} \cdot \frac{1}{1 - \frac{v ^{2}}{c ^{2}}}$

Perpendicular arm (across the wind)

$t_{⊥} = \frac{2 L}{c ^{2} - v ^{2}} = \frac{2 L}{c} \cdot \frac{1}{1 - \frac{v ^{2}}{c ^{2}}}$

Yeh step kyun? Cross-current effective speed $c^{2} - v^{2}$ hai, jo full round trip $2 L$ ke liye use hoti hai.

Predicted time difference

Expand kyun karte hain? $β^{2} \approx (1 0^{- 4})$ bahut chhota hai, isliye leading non-zero term $\propto β^{2}$ dominate karta hai aur ek clean estimate deta hai.

Fringe shift mein convert karna

$90°$ rotate karne se arms swap ho jaate hain, isliye path difference mein change $c \cdot (2Δ t)$ hai... zyada precisely expected fringe shift hai: $Δ N = \frac{c ( 2Δ t )}{λ} \approx \frac{2 L v ^{2}}{λ c ^{2}}$

The RESULT and its meaning

Common mistakes (Steel-man + fix)

Flashcards

1905 se pehle light ko kaunse medium ki zaroorat maani jaati thi?

Luminiferous aether.

Michelson–Morley experiment kya detect karne ki koshish kar raha tha?

Aether ke relative Earth ki motion (velocity

v

), arms ke beech light-speed difference ke zariye.

Experimental result kya tha?

Null result — essentially koi fringe shift observe nahi hua.

Parallel arm ka round-trip time?

t_{∥} = \frac{2 L}{c} \frac{1}{1 - v ^{2} / c ^{2}}

Perpendicular arm ka round-trip time?

t_{⊥} = \frac{2 L}{c} \frac{1}{1 - v ^{2} / c ^{2}}

Cross-arm light speed

c^{2} - v^{2}

kyun hai?

Beam upstream aim karti hai; total speed

c

aur current

v

ke saath cross-component

c^{2} - v^{2}

hoti hai (Pythagoras).

Leading-order expected time difference?

Δ t \approx \frac{L v ^{2}}{c ^{3}}

(

β^{2}

ke proportional).

Parallel arm ki fast/slow legs cancel kyun nahi hoti?

Slow (

c - v

) leg mein zyada waqt lagta hai, isliye round trip no-wind se zyada lamba hota hai.

Null result ne SR ka kaun sa postulate support kiya?

Speed of light

c

sabhi observers ke liye same hai (constancy of

c

Aether bachane ke liye kaun si classical patch try ki gayi?

FitzGerald–Lorentz length contraction.

Earth ki orbit ke liye

β = v / c

ka order of magnitude?

\sim 1 0^{- 4}

, isliye

β^{2} \sim 1 0^{- 8}

— tiny lekin interferometer se detectable.

Recall Feynman: 12-saal ke bachche ko explain karo

Socho ek flowing river mein do identical swimmers hain, dono still water mein same speed se tairte hain. Ek nadi ke neeche jaata hai aur wapis; doosra seedha across jaata hai aur wapis. Equally fast hone ke bawajood, river current unke times mein thoda farq karta hai. Scientists sochte the ki space ek invisible "nadi" se bhari hai jise aether kehte hain jisme light flow karti thi, aur Earth us mein zoom kar rahi thi. Unhone ek machine banayi jisne do light beams race karayi — ek "nadi" ke along, ek across — time difference pakadne ke liye. Lekin beams hamesha tie kar leti thin. Chahe machine kitni bhi ghumi ho ya saal ka koi bhi waqt ho, perfect tie. Matlab koi invisible nadi nahi thi, aur light hamesha har kisi ke liye exactly same speed se travel karti hai — woh idea jo Einstein's relativity launch kiya.

Connections

Special Relativity — Einstein's Postulates
Lorentz Transformation
Length Contraction (FitzGerald–Lorentz contraction)
Time Dilation
Galilean Relativity & Velocity Addition
Maxwell's Equations and the Speed of Light
Interference of Light & Fringes