2.5.11 · D1Young's double slit — fringe width derivation
Do identical light-waves ek saath chalna shuru karti hain, lekin ek ko screen pe kisi jagah pahunchne ke liye thoda zyada door chalna padta hai — aur woh tiny extra walk wavelengths mein measure hoti hai. Jab extra walk wavelengths ki puri sankhya ho, waves milke ek bright band banati hain; aadha wavelength ka fark ho toh woh cancel hokar darkness ban jaati hain.
Yeh page parent derivation mein jo bhi letter aur symbol quietly assume kiye gaye hain, unhe absolute zero se build karta hai — ek wave-picture ek waqt mein. Last section tak aap fringe spacing ka final formula dekhenge aur samjhenge ki uska har ek mark asal mein kis cheez ki picture hai . Hum sirf ek ripple se shuru karte hain.
Kisi bhi letter se pehle, dekho ki yahan light kya hai : ek repeating upar-neeche ripple jo aage travel karti hai.
Figure 1 neeche aisi ek wave dikhata hai.
λ λ wave ke ek crest se agli crest tak ki distance hai. Yeh wave ka apna built-in ruler hai.
Picture: Figure 1 mein, λ Topic ko yeh kyun chahiye: poora bright/dark game kisi travel-distance ko ek wave kitni lambi hai se compare karne ke baare mein hai. Bina compare karne ke kisi length ke, "path difference" ek naked number hoga jiska koi matlab nahi hoga.
Definition Crest aur troughCrest ripple ka sabse ऊँcha point hota hai (wave upar push hoti hai); trough sabse neecha hota hai (wave neeche push hoti hai).
Picture: Figure 1 mein, lavender dot ek crest mark karta hai (ek red hump ka top), mint dot ek trough mark karta hai (ek dip ka bottom).
Kyun: "crest meets crest" (bright) aur "crest meets trough" (dark) — yahi poora mechanism hai — dekho Interference of light .
Definition Coherent sourcesDo sources coherent hote hain jab woh ek doosre ke relative ek fixed rhythm maintain karte hain — source 1 se har crest source 2 se crest ke relative ek predictable moment pe nikalta hai.
Picture: do log ek talab ko bilkul same beat pe tap karte hain hamesha ke liye, kabhi drift nahi karte.
Topic ko yeh kyun chahiye: agar do slits randomly out of step ho jaate, toh bright aur dark spots flicker karke grey mein smear ho jaate. Ek steady pattern ke liye coherence zaroori hai — isliye hum do alag lamps ki jagah do slits ke peeche ek lamp use karte hain. Poori kahani Coherence and coherent sources mein.
Figure 2 neeche poora apparatus set up karta hai — agli chaar definitions ke liye ise refer karo.
S 1 S 2 d S 1 S 2 do narrow openings hain jinse light pass hoti hai; yeh do tiny coherent sources ki tarah behave karte hain. d beech ki distance hai.
Picture: Figure 2 mein, left barrier pe do mint dots; d Kyun: do waves yahan paida hoti hain. Kyunki S 1 + 2 d S 2 − 2 d d
d Yeh sahi kyun lagta hai: dono slits pe chhoti distances hain.
Yeh galat kyun hai: d do slits ke centres ke beech ka gap hai, na ki ek single slit kitni moti hai. Slit width diffraction envelope control karti hai (dekho Diffraction grating ); slit separation d Fix: d S 1 S 2
Abhi bhi Figure 2 pe:
Definition Screen distanceD D slit barrier se screen tak ki seedhi-across distance hai.
Picture: Figure 2 mein, slits se wall tak lamba horizontal lavender arrow.
Kyun: screen jitna door, do paths screen pe utna zyada spread hoti hain, isliye D
O O do slits ke midpoint ke directly opposite hai.
Picture: Figure 2 mein, jahan dashed central axis wall se milti hai.
Kyun: yeh hamara "zero" hai — sab screen heights yahan se measure hoti hain.
y P P screen pe koi bhi chosen point hai; y O Picture: Figure 2 mein, coral dot P O y Kyun: hum chahte hain ki ek formula ho ki bright fringes kahan land karti hain. Woh "kahan" y
Yeh dil hai, isliye ise apna figure milta hai. Figure 3 neeche P
θ θ P Picture: Figure 3 mein, dashed central axis aur ek ray ke beech chhota butter-coloured wedge. Jab P θ P = O θ = 0 Hum ise abhi kyun introduce karte hain: screen pe distances (y θ
Definition Path differenceΔ Δ door wale slit se wave kitna zyada travel karti hai P Δ = S 2 P − S 1 P Picture: Figure 3 mein, S 1 S 2 P S 2 coral stub extra walk Δ Kyun: sab kuch — bright, dark, spacing — Δ λ Path difference and phase difference mein.
Δ λ Δ ratio Δ/ λ Δ λ
Definition Pythagoras' theoremEk right triangle ke liye jisme legs a b c c = a 2 + b 2 Picture: Figure 3 mein bada right triangle jo horizontal D P S 1 P S 2 P Yeh tool kyun, koi aur kyun nahi? Har ray across (D upar (ek height) jaati hai. Yeh literally ek right triangle hai, aur Pythagoras woh ek rule hai jo do perpendicular distances ko straight-line distance mein convert karta hai. Isliye parent likhta hai S 1 P = D 2 + ( y − 2 d ) 2
≫ D ≫ d D d bahut bada hai." Jab screen bahut door ho, do rays almost parallel hoti hain aur angle θ Picture: Figure 3 mein do rays almost parallel lagti hain — aankhein sikaudo toh woh merge ho jaati hain; wedge θ Hum ise kyun chahiye: yeh hume sin θ tan θ θ S 1 P + S 2 P 2 D Δ = D y d Small angle approximation mein.
Δ = d sin θ Δ = y d / D Yeh sahi kyun lagta hai: woh alag dikhte hain.
Yeh galat kyun hai: yeh same statement hain. Kyunki sin θ ≈ tan θ = y / D d sin θ ≈ y d / D Fix: small-angle approximation ke under identical — ek angle form hai, doosra screen-position form.
Extra walk Δ sab kuch decide karta hai. Exactly do special cases hain, aur hum dono banate hain.
Definition Bright-fringe conditionCrests aligned hona matlab extra walk wavelengths ka pura number hai:
Δ = nλ , n = 0 , ± 1 , ± 2 , … n kaunsa bright band hai.
Definition Dark-fringe conditionCrest-on-trough matlab extra walk wavelengths ka pura number plus aadha wavelength hai:
Δ = ( n + 2 1 ) λ , n = 0 , ± 1 , ± 2 , … Δ = ( 2 n + 1 ) 2 λ odd multiple.
Common mistake "Dark matlabΔ = nλ /2 Yeh sahi kyun lagta hai: "aadha wavelength = cancel" clean lagta hai, aur n = 1 λ /2 Yeh galat kyun hai: n = 2 Δ = λ puri wave hai → bright , dark nahi!
Fix: dark = ( n + 2 1 ) λ λ /2
Ab har condition ko master formula Δ = D y d y
n n integer label hai (n = 0 , ± 1 , ± 2 , … n = 0 n = 1 Picture: bright stripes ko centre se bahar number karo: 0 , 1 , 2 , … Kyun: ek specific fringe ko locate karne ke liye hume use naam dena hoga; n
β β do neighbouring bright bands ke beech ka gap hai (equivalently, do neighbouring dark bands ke beech): β = y n + 1 − y n Picture: adjacent stripes ke beech constant spacing.
Kyun: yeh poori derivation ka final target hai — woh ek number jo batata hai ki pattern kitna spread-out hai. Consecutive bright positions subtract karne pe, n β = d ( n + 1 ) λ D − d nλ D = d λ D . Ise ab sab kuch ke saath pado: λ D d
Neeche ki chain dikhati hai ki is page ka har foundation kaise agli ko feed karta hai, ending at fringe-width result. Figure 5 wahi chain ek picture ke roop mein draw karta hai agar diagram tumhare liye render na ho.
Wave and wavelength lambda
Interference bright and dark
Screen distance D and height y
Pythagoras right triangle
Small angle approx D much bigger
Fringe order n and positions y_n
Fringe width beta equals lambda D over d
Khud test karo — har answer reveal karne se pehle zyaah se bolo.
λ Ek wave crest se dusre tak ki distance — wave ka apna ruler.
d Do slit centres ke beech ka gap; ek single slit ki width NAHI.
D Slit barrier se screen tak ki seedhi-across distance.
Point O Screen pe, do slits ke midpoint ke directly opposite — y
y Chosen point P O
Fringe angle θ Woh angle jo P
Path difference Δ Ek slit ki wave P Δ = S 2 P − S 1 P
Δ Δ = d sin θ ≈ D y d
Bright-fringe condition batao. Δ = nλ
Dark-fringe condition batao. Δ = ( n + 2 1 ) λ
Dark fringes bright ones ke relative kahan hote hain? Bilkul neighbouring bright fringes ke beech mein.
sin θ ≈ tan θ ≈ θ Corrections θ 3
Sources coherent kyun hone chahiye? Taaki pattern grey mein flicker hone ki jagah stable rahe.
"Across D y S 1 P + S 2 P ≈ 2 D Har ray ≈ D D ≫ y , d ( d /2 ) 2 y 2
tan θ = y / D Tangent opposite over adjacent hota hai; opposite side y D
Integer n y n n y n = nλ D / d y n = ( n + 2 1 ) λ D / d
β Do neighbouring bright (ya dark) fringes ke beech ki spacing, y n + 1 − y n = λ D / d
Young's double slit — fringe width derivation — woh parent jiske liye yeh page tumhe prepare karta hai.Interference of light — crest-meets-crest mechanism.Coherence and coherent sources — kyun ek lamp, do slits.Path difference and phase difference — Δ Small angle approximation — woh tool jo geometry ko clean karta hai.Refractive index — "in water" example ke liye zaroori hai jo aage aata hai.Diffraction grating — jahan slit width vs. separation matter karta hai. 
