Young's double slit — fringe width derivation

2.5.11 · Physics › Optics

HUM KYA DERIVE KAR RAHE HAIN?

Hum $β$ ke liye ek formula chahte hain in terms mein:

$λ$ = wavelength of light,
$d$ = do slits ke beech separation,
$D$ = slits se screen tak ki distance.

YEH TEEN KYO? Kyunki "do sources waves ek door screen tak bhej rahe hain" ki geometry sirf teen cheezoon pe depend karti hai — sources kitne door hain ( $d$ ), screen kitni door hai ( $D$ ), aur wave ka natural ruler ( $λ$ ).

GEOMETRY KAISE KAAM KARTI HAI

Figure — Young's double slit — fringe width derivation

Setup: do slits $S_{1}$ aur $S_{2}$ , $d$ se separate. Ek screen $D$ distance pe rakhi hai. Hum screen pe ek point $P$ ko measure karte hain jo central point $O$ se $y$ distance pe hai (woh point jo slits ke midpoint ke directly saamne hai).

Step 1 — Path difference nikalo. $S_{2}$ se wave ko $S_{1}$ ke mukable zyada travel karna padta hai $P$ tak pahunchne ke liye (jab $P$ centre se upar ho).

Step 2 — Subtract karo path difference $Δ$ pane ke liye. $S_{2} P^{2} - S_{1} P^{2} = (y + \frac{d}{2})^{2} - (y - \frac{d}{2})^{2} = 2 y d$

Saath hi $S_{2} P^{2} - S_{1} P^{2} = (S_{2} P - S_{1} P) (S_{2} P + S_{1} P)$ .

Kyun? Yeh factoring hume difference $Δ = S_{2} P - S_{1} P$ isolate karne deta hai, jo exactly woh path difference hai jo humein chahiye.

Kyunki $D ≫ d$ aur $D ≫ y$ hai, dono distances $\approx D$ hain, isliye $S_{2} P + S_{1} P \approx 2 D$ : $Δ = S_{2} P - S_{1} P = \frac{2 y d}{S _{2} P + S _{1} P} \approx \frac{2 y d}{2 D} = \frac{y d}{D}$

BRIGHT AUR DARK FRINGES KAISE MILTE HAIN

$n$ -ve bright fringe ki position: $\frac{y _{n} d}{D} = nλ$ set karo: $y_{n} = \frac{nλ D}{d}$

Kyun? Humne bas bright condition ko master path-difference formula mein plug kiya aur $y$ ke liye solve kiya.

FRINGE WIDTH $β$ DERIVE KARNA

Fringe width consecutive bright fringes, $n$ aur $n + 1$ , ke beech ka gap hai: $β = y_{n + 1} - y_{n} = \frac{( n + 1 ) λ D}{d} - \frac{nλ D}{d}$

$n$ cancel ho jaata hai — yahi deep reason hai ki fringes equally spaced hote hain!

Angular fringe width

Kyunki $y = D tan θ \approx D θ$ small angles ke liye, angular spacing hai: $θ_{fringe} = \frac{β}{D} = \frac{λ}{d}$

Kyun useful hai? Yeh $D$ hata deta hai — angular pattern sirf source geometry $λ / d$ pe depend karta hai.

Worked examples

Forecast-then-Verify

Recall Forecast: agar

$d$ double karo toh $β$ ka kya hoga? Forecast: Halve ho jaayega. Verify: $β \propto 1/ d$ , toh $d \to 2 d$ se $β \to β /2$ hota hai. ✓ Fringes paas aa jaate hain.

Recall Forecast: red (700 nm) se violet (400 nm) karo. Wider ya narrower?

Forecast: Narrower. Verify: $β \propto λ$ , chhota $λ$ → chhota $β$ . ✓

Common mistakes (Steel-manned)

Active Recall

YDSE mein fringe width

β

kya hai?

β = \frac{λ D}{d}

, consecutive bright (ya dark) fringes ke beech spacing.

YDSE mein master path difference formula?

Δ = \frac{y d}{D}

Bright fringe ke liye condition?

Δ = nλ

(

n = 0, \pm 1, \pm 2, \dots

Dark fringe ke liye condition?

Δ = (n + \frac{1}{2}) λ

, yaani odd multiples of

λ /2

n

-ve bright fringe ki position?

y_{n} = \frac{nλ D}{d}

Fringes equally spaced kyun hote hain?

Kyunki

y_{n} \propto n

linearly hai, isliye

y_{n + 1} - y_{n}

constant hai (

n

cancel ho jaata hai).

Slit separation

d

double karne par

β

par kya asar?

β

halve ho jaata hai (kyunki

β \propto 1/ d

Index

n

ke medium mein setup daalne par

β

par kya asar?

β \to β / n

(

λ \to λ / n

kyunki).

Angular fringe width?

θ = β / D = λ / d

Δ = y d / D

derive karne mein kaun si approximation use hoti hai?

D ≫ d, y

, isliye

S_{1} P + S_{2} P \approx 2 D

(small-angle).

Recall Feynman: ek 12-saal ke bachche ko samjhao

Socho do bachche ek talab mein bilkul same rhythm se thappad maar rahe hain, ripples bana rahe hain. Jahan do ripple-crests milte hain, paani bahut zyada uchhalata hai (bright). Jahan ek crest aur ek dip milte hain, woh cancel ho jaate hain aur paani flat rehta hai (dark). Door ek deewar pe tum bright-dark-bright-dark stripes dekhoge. Yeh stripes kitni dur hain yeh teen cheezoon pe depend karta hai: ripples kitne bade hain ( $λ$ ), deewar kitni door hai ( $D$ ), aur do thappad maarne wale kitne door hain ( $d$ ). Thappadne walon ko paas rakho aur ek door deewar pe badi ripples banao → mote stripes. Recipe hai "ripple-size times wall-distance, divided by tapper-gap."

Connections

Interference of light — YDSE textbook ka classic case hai.
Coherence and coherent sources — isliye ek single source + do slits chahiye.
Path difference and phase difference — $Δ ϕ = \frac{2 π}{λ} Δ$ .
Diffraction grating — bahut saare slits → same idea ka sharper version.
Refractive index — water-immersion shrink explain karta hai.
Small angle approximation — yahan key simplifying tool hai.