Newton's rings — derivation — Hinglish notes

2.5.13 · Physics › Optics

KYA ho raha hai?

Rings (circles) KYUN? Air film ki thickness $t$ contact point se equidistant sabhi points par same hoti hai. Constant $t$ ka locus ek circle hota hai → fringes of equal thickness circles hote hain.

Centre dark KYUN hota hai? Contact ke exact point par $t = 0$ hota hai, phir bhi ek reflection par sudden phase flip hoti hai (neeche explain hai). Net effect → destructive interference → central dark spot.

Optical path difference KAISE aata hai (first principles)

Do rays matter karte hain:

Ray A glass lens ke bottom se reflect hoti hai (glass→air boundary).
Ray B transmit hoti hai, air gap (thickness $t$ ) cross karti hai, flat plate ke top se reflect hoti hai (air→glass boundary), aur wapas aati hai.

B ka geometric extra path film se neeche aur upar jaata hai. Near-normal incidence par, refractive index $μ$ ki film mein:

$geometric path difference = 2 μ t$

Woh crucial half-wavelength

Toh total optical path difference hai:

$Δ = 2 μ t \pm \frac{λ}{2}$

$\pm$ sirf bookkeeping ke liye hai; jo important hai woh hai ki extra $λ /2$ exist karta hai.

Bright aur dark rings ke conditions

Constructive (bright) ke liye $Δ = mλ$ chahiye; destructive (dark) ke liye $Δ = (m + \frac{1}{2}) λ$ chahiye. $Δ = 2 μ t + λ /2$ substitute karne par:

Geometry: $t$ ko radius $r$ se relate karna

Derivation (circle theorem / Pythagoras):

Spherical surface ka centre contact point se height $R$ par rakho. Horizontal distance $r$ par lens surface ka ek point plate se height $t$ par hota hai. Radius $R$ ke circle ke liye chord (intersecting-chords) theorem se:

$r^{2} = t (2 R - t)$

Yeh step kyun? Air gap ke through vertical chord ke segments $t$ aur $(2 R - t)$ hain; perpendicular half-chord $r$ hai, aur unka product $r^{2}$ ke equal hota hai.

Kyunki $t ≪ R$ (gap microns mein hai, $R$ ~metres mein hai), $2 R - t \approx 2 R$ :

$r^{2} \approx 2 R t ⟹ t = \frac{r ^{2}}{2 R}$

Final ring-radius formulas (the payoff)

Dark rings — $t = \frac{r ^{2}}{2 R}$ ko $2 μ t = mλ$ mein dalo:

$2 μ \cdot \frac{r _{m}^{2}}{2 R} = mλ ⟹ \frac{μ r _{m}^{2}}{R} = mλ$

Bright rings — $2 μ t = (m - \frac{1}{2}) λ$ use karke:

Diameter $D = 2 r$ ke terms mein (jo aap actually measure karte ho):

$D_{m}^{2} = 4 mλ R (dark, air)$

$λ$ ya $R$ measure karna — unknown centre ko eliminate karna

Subtract KYUN karte hain? Agar true centre offset hai, toh har $D^{2}$ mein same additive error hogi; subtract karne se woh cancel ho jaati hai aur awkward " $+ λ /2$ " constant bhi cancel ho jaata hai. Clean linear relation milta hai.

Worked examples

Common mistakes (steel-manned)

Active recall flashcards

Newton's rings ka centre reflected light mein dark KYUN hota hai?

t = 0

par geometric path 0 hai, lekin ek reflection (air→glass)

λ /2

phase flip add karta hai → net

Δ = λ /2

→ destructive.

Air film ke liye total optical path difference kya hai?

Δ = 2 μ t + λ /2

(factor 2 = down-and-up;

λ /2

= Stokes flip).

t

ke terms mein dark ring condition kya hai?

2 μ t = mλ

t

ke terms mein bright ring condition kya hai?

2 μ t = (m - \frac{1}{2}) λ

Film thickness aur radius ke beech relation kya hai?

r^{2} = t (2 R - t) \approx 2 R t

, isliye

t = r^{2} /2 R

m

-ve dark ring ki radius (air mein)?

r_{m} = mλ R

Diameter difference

D_{n}^{2} - D_{m}^{2}

KYUN use karte hain?

Uncertain centre aur constant

λ /2

term dono cancel ho jaate hain, jisse

D_{n}^{2} - D_{m}^{2} = 4 (n - m) λ R

milta hai.

Experiment se

λ

ka formula kya hai?

λ = (D_{n}^{2} - D_{m}^{2}) / [4 (n - m) R]

Agar gap mein liquid

μ

bhar dein toh ring diameters ka kya hoga?

Woh

1/ μ

se shrink ho jaati hain;

D^{2} \propto 1/ μ

Rings bahar jaate jaate crowd KYUN hoti hain?

Kyunki

r_{m} \propto m

hai, isliye consecutive rings ke beech spacing

Δ r

kam hoti jaati hai.

Recall Feynman: 12-saal ke bachche ko explain karo

Ek magnifying-glass lens ko flat kaanch ke tukde par dabao. Jahan touch hota hai wahan koi gap nahi; thoda bahar ek tiny air gap hai jo pahad ki dhaal ki tarah mota hota jaata hai. Jab light us patli air gap ke top aur bottom se bounce hoti hai, toh do bounced beams ek doosre se race karte hain. Kabhi kabhi woh line up hote hain (bright), kabhi cancel karte hain (dark), aur isse glowing circles bante hain. Ek chhaliyaa rule hai: ek bounce wave ko ulta kar deta hai, isliye theek touching point par waves cancel ho jaati hain — bullseye dark hota hai. Jitni tezi se air gap badhta hai, utne hi paas-paas circles squish hote hain.

Connections

Thin film interference — Newton's rings "wedge of air" ka ek special case hai.
Stokes' relations & phase change on reflection — $λ /2$ flip ka source.
Interference — path difference & coherence — $Δ = mλ$ vs $(m + \frac{1}{2}) λ$ .
Michelson interferometer — $λ$ find karne ke liye $D^{2}$ -difference logic bhi use karta hai.
Refractive index measurement — yahan liquid-in-gap method.
Wedge fringes / air wedge — inhi circular fringes ka straight-line analogue.