Young's double slit — fringe width derivation: Question bank

2.5.11 · D5 · Physics › Optics › Young's double slit — fringe width derivation

Derivation mein yahan symbols ka use hoga, pehle yaad kar lo:

$λ$ = wavelength (wave ka apna built-in ruler),
$d$ = slit separation, $D$ = slit-to-screen distance,
$y$ = screen par centre $O$ se upar kisi point ki height,
$Δ = \frac{y d}{D}$ = path difference, $β = \frac{λ D}{d}$ = fringe width.

True or false — justify karo

True or false: Central fringe (at

y = 0

) hamesha bright hoti hai.

True — centre par dono waves equal distances travel karti hain, isliye

Δ = 0

, jo

n = 0

ke saath

nλ

hai: bilkul in phase, maximum brightness.

True or false: 10th fringe ki fringe width 1st fringe se zyada hoti hai.

False — neighbours ke beech spacing

λ D / d

hoti hai, jo

n

par depend nahi karti;

y_{n + 1} - y_{n}

mein

n

's cancel ho jaate hain, isliye saari gaps identical hoti hain.

True or false: Agar dono slits ko paas laao, toh fringes spread out ho jaati hain.

True —

β \propto 1/ d

, isliye chhota

d

wider pattern deta hai; sources ke paas hone se unka overlap zyada fan out hota hai.

True or false: Red light use karne se blue ki jagah fringes narrow ho jaati hain.

False —

β \propto λ

aur red ki wavelength zyada hoti hai, isliye red wider fringes deta hai.

True or false: Poore setup ko water mein duba dene se fringes wide ho jaati hain.

False — water mein

λ^{'} = λ / n_{w}

shrink ho jaata hai, isliye

β^{'} = β / n_{w}

; pattern contract karta hai (fringes paas aa jaati hain).

True or false: Angular fringe width

λ / d

change ho jaati hai agar screen ko aur door le jaao.

False — angular spacing

θ = β / D = λ / d

mein

D

nahi hai; screen ko door karne se

β

aur

D

dono saath stretch hote hain, angle fixed rehta hai.

True or false: Exactly

λ /2

ka path difference dark fringe deta hai.

True — aadhi wavelength ek crest ko ek trough ke saamne rakh deti hai (out of phase), isliye waves cancel ho jaati hain; yahi

n = 0

dark condition

(n + \frac{1}{2}) λ

hai.

True or false: Agar dono slits se thodi alag frequencies ki light nikle, toh phir bhi stable fringe pattern dikhega.

False — alag frequencies ka matlab hai phase relationship baar baar change hoti rehti hai, isliye bright aur dark spots wash out ho jaate hain; tumhe coherent sources chahiye jo same frequency ke hon.

True or false: Do alag light bulbs ko do slits ke peeche rakh kar saaf YDSE fringes milenge.

False — independent bulbs mein phases randomly jump karte rehte hain, isliye koi bhi interference pattern flicker karta hai aur uniform brightness mein average ho jaata hai; isliye ek hi source ko do slits mein split kiya jaata hai.

Spot the error

Error hunt: "Dark fringes wahan hoti hain jahan

Δ = nλ /2

kisi bhi integer

n

ke liye."

Galat —

n = 2

par ye

Δ = λ

deta hai, jo bright hai. Sahi dark condition hai

Δ = (n + \frac{1}{2}) λ

, yaani

λ /2

ke odd multiples.

Error hunt: "

Δ = d sin θ

aur

Δ = y d / D

do alag unrelated formulas hain."

Galat — ye dono ek hi cheez hain; kyunki Small angle approximation mein

sin θ \approx tan θ = y / D

hota hai, isliye

d sin θ \approx y d / D

Error hunt: "Kyunki

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

, dono path lengths equal hain, isliye

Δ = 0

har jagah hai."

Galat — sum approximately

2 D

hota hai, lekin woh tiny difference

Δ = 2 y d / (S_{1} P + S_{2} P) \approx y d / D

hi asli kaam ka hai; sum ko approximate karne se difference khatam nahi hota.

Error hunt: "Formula

y_{n} = nλ D / d

tab bhi kaam karta hai jab

y

aur

D

comparable hon."

Galat — ye

D ≫ y

assume karta hai taaki

sin θ \approx tan θ

ho sake; bade angles par small-angle approximation fail ho jaati hai aur fringes evenly spaced nahi rehte.

Error hunt: "Light ki frequency water mein enter karne par change hoti hai, isliye

β

change hota hai."

Galat — frequency source set karta hai aur fixed rehti hai; speed (aur isliye wavelength) refractive index

n_{w}

se ghatti hai.

Error hunt: "Path difference screen par point ki height

y

hoti hai."

Galat —

y

ek screen distance hai; path difference

Δ = y d / D

hai, jisme

d

aur

D

milke screen position ko extra path length mein convert karte hain.

Why questions

Fringe width compute karte waqt

n

kyun cancel ho jaata hai?

Kyunki

y_{n} = nλ D / d

n

mein linear hai, isliye consecutive positions ka difference ek constant

λ D / d

hota hai chahe koi bhi fringe lo — exactly isliye fringes equally spaced hoti hain.

Derivation mein

D ≫ d

kyun chahiye hota hai?

Taaki dono path lengths

\approx D

ho sakein aur

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

; yahi exact geometry ko clean result

Δ = y d / D

mein badalta hai.

Fringes dekhne ke liye dono sources coherent kyun hone chahiye?

Steady pattern ke liye waves ke beech ek constant phase relationship chahiye hoti hai; incoherent sources ke phases randomly jump karte hain, bright aur dark uniform grey mein smear ho jaate hain — dekho Coherence and coherent sources.

Bada

λ

wider fringes kyun deta hai?

Badi wavelength ka matlab hai screen par aur door jaana padega taaki

Δ

agle poore

λ

tak build up ho sake, isliye consecutive bright spots zyada door hote hain.

Water mein pattern same rehne ki jagah shrink kyun ho jaata hai?

Water mein wavelength

λ / n_{w}

ho jaati hai, aur kyunki

β \propto λ

hai, fringe spacing bhi usi factor

n_{w}

se shrink hoti hai — dekho Refractive index.

Hum ek source ko do slits mein split karte hain do alag lamps ki jagah, kyun?

Ek wavefront split karne se dono slit-waves ka same phase history guaranteed hota hai (wo coherent hain), jo do independent lamps kabhi provide nahi kar sakte.

Angular fringe width

λ / d

aksar

β

se zyada useful kyun hoti hai?

Isme

D

nahi aata, isliye ye two-source system ki intrinsic geometry describe karta hai, chahe screen kitni bhi door rakhon.

Edge cases

Edge case: Agar

d \to 0

ho jaaye (slits merge ho jaayein) toh fringe pattern ka kya hoga?

β = λ D / d \to \infty

— fringes infinitely wide spread ho jaati hain, matlab ek single slit mein koi two-source interference nahi hoti.

Edge case: Agar

d

λ

ke comparison mein bahut bada ho jaaye toh kya hoga?

β \to 0

, isliye fringes itni paas aa jaati hain ki blur hokar uniform illumination ban jaata hai — pattern effectively disappear ho jaata hai.

Edge case: Central point

O

(

y = 0

) par

Δ

aur brightness exactly kya hogi?

Δ = 0

har wavelength ke liye hota hai, isliye ye bright central maximum hai; white light mein yahi ek pure-white fringe hoti hai kyunki har colour wahan agree karta hai.

Edge case: White light mein sirf central fringe white kyun hoti hai jabki sides par colours dikhte hain?

Centre se door, har colour ka apna

β \propto λ

hota hai, isliye alag colours alag

y

par peak karte hain aur separate ho jaate hain; sirf

y = 0

par saare colours coincide karte hain.

Edge case: Extreme

y

par kya hoga jahan

y

≪ D

nahi raha?

Small-angle approximation fail ho jaati hai, isliye

sin θ \neq = tan θ

; fringes unequally spaced ho jaati hain aur paraxial region se bahar jaane par धीरे धीरे faint hoti jaati hain.

Edge case: Agar ek thin transparent sheet sirf slit

S_{1}

ko cover kare, toh central fringe ka kya hoga?

Sheet ek arm mein extra optical path add karti hai, isliye zero total path difference wala point sideways shift ho jaata hai — poora pattern (central bright fringe samait) centre se hat jaata hai.