Diffraction grating — condition for maxima: Question bank

2.5.15 · D5 · Physics › Optics › Diffraction grating — condition for maxima

Intuition Is page ko kaise use karein

Ye woh conceptual traps hain jo grating equation $d sin θ = mλ$ tumhare liye set karti hai — sign traps, "main kaun si condition use kar raha hoon" wale traps, aur boundary cases jahan yeh neat formula chup-chaap kaam karna band kar deta hai. Pehle answer cover karo, ek reason commit karo (sirf verdict nahi), phir reveal karo. Agar tum sirf "true/false" guess kar sakte ho bina because ke, toh abhi idea tumhara nahi hua.

Symbols jo poore page mein use hote hain, taaki koi line surprise na kare:

$d$ = paas-paas ke slits ke beech centre-to-centre spacing (metres mein).
$θ$ = ek outgoing beam ka angle jo normal se measure hota hai (woh line jo seedha bahar jaati hai, grating ke perpendicular).
$m$ = order, ek whole number $0, \pm 1, \pm 2, \dots$ jo count karta hai ki neighbouring slits ke beech kitne wavelengths ka extra path difference hai.
$λ$ = wavelength; $N$ = slits ki total number.

Full derivation ke liye parent dekho: Diffraction grating — condition for maxima.

True or false — justify karo

True or false: "Zyada slits wali grating nayi angles par maxima produce karti hai jo two-slit setup mein nahi dikhte."

False — principal maxima ki angles sirf

d

λ

m

par depend karti hain, slits ki number par nahi; slits badhane se wahi maxima sharper aur brighter hote hain, unki jagah nahi badalti. Dekho Young's double-slit experiment.

True or false: "White light input ke liye central (

m = 0

) maximum white hota hai."

True —

m = 0

par condition

d sin θ = 0

har wavelength ke liye

θ = 0

par satisfy hoti hai, isliye sab colours overlap karke white mein recombine ho jaate hain.

True or false: "Wavelength double karne se pehle-order maximum ka angle hamesha double ho jaata hai."

False — isse

sin θ_{1}

double hota hai,

θ_{1}

khud nahi; kyunki

sin

linear nahi hai, angle proportional se zyada tezi se badhta hai aur physical limit bhi cross kar sakta hai.

True or false: "Agar light grating par normally ki jagah angle se padti hai, toh equation

d sin θ = mλ

unchanged rehti hai."

False — oblique incidence ek incoming path difference add karta hai, jisse

d (sin θ_{out} \pm sin θ_{in}) = mλ

milta hai; normal-incidence form ek special case hai jahan incoming term zero hoti hai.

True or false: "Ek grating aur ek single wide slit maxima banane ke liye same physical rule follow karte hain."

False — grating maxima deta hai jahan sab slits in phase add hoti hain (

d sin θ = mλ

); single slit ki famous condition

a sin θ = mλ

uske dark minima ko mark karti hai maxima ko nahi. Dekho Single-slit diffraction.

True or false: "

N

(slits) badhane se tum kitne orders dekh sakte ho yeh badh jaata hai."

False — orders ki number

m_{m a x} = ⌊ d / λ ⌋

se cap hoti hai, jo sirf

d

aur

λ

par depend karti hai; zyada slits peaks ko sharpen karte hain lekin naaye orders nahi laate.

True or false: "

m

aur

- m

maxima centre ke dono taraf equal angles par mirror images hote hain."

True —

sin (- θ) = - sin θ

, isliye

m

aur

- m

equation ko

\pm θ_{m}

par solve karte hain; pattern normal ke baare mein symmetric hai.

True or false: "Ek diye gaye order mein red light blue se zyada bend hoti hai grating mein."

True — bada

λ

bada

sin θ_{m}

deta hai, isliye red (lambi wavelength) har non-zero order mein blue se wider angle par hoti hai. Dekho Wavelength and the visible spectrum.

Error dhundo

"Grating par bright fringes ke liye humein

d sin θ = (m + \frac{1}{2}) λ

chahiye."

Galat condition — half-integer path differences neighbouring waves ko cancel karte hain; grating maxima ke liye whole wavelengths chahiye,

d sin θ = mλ

d = 500

kyunki grating mein 500 lines per mm hain."

d

spacing hai, count nahi; tumhe invert karna hoga,

d = 1/500 mm = 2000 nm

, aur use karne se pehle metres mein convert karo.

"Kyunki

m

koi bhi integer ho sakta hai, isliye infinitely many orders hain."

sin θ \leq 1

physically order ko

m_{m a x} = ⌊ d / λ ⌋

par cap karta hai; usse aage koi real angle equation solve nahi karta.

"Small-angle approximation

sin θ \approx θ

yahan theek hai, jaise Young's experiment mein."

Grating angles aksar tens of degrees hote hain, jahan

sin θ

aur

θ

mein bahut fark hota hai; full

sin θ

rakho.

"Maximum ka peak intense hai kyunki bahut saari light wide grating se guzarti hai."

Intensity

\propto N^{2}

isliye aati hai kyunki

N

slits in phase add hoti hain (coherent addition), sirf isliye nahi ki zyada area se zyada light guzarti hai.

"Order

m = 4

exist karta hai kyunki

4

ek integer hai, toh bas

sin θ = 4 λ / d

solve karo."

Agar

4 λ / d > 1

hai toh koi real angle nahi hai; integer

m

zaroori hai lekin kaafi nahi — usse

sin θ \leq 1

bhi rakhna hoga.

"Same

d

lekin alag

N

wali do gratings alag coloured spectra deti hain."

Colours same angles par aate hain (jo

d

se set hote hain); alag

N

sirf sharpness aur resolving power badalta hai, colour positions nahi. Dekho Resolving power and spectrometers.

m = 0

par colours ek rainbow mein spread ho jaate hain."

Zero order wahan hai jahan sab wavelengths ek saath

θ = 0

satisfy karti hain, isliye yeh white aur undispersed hota hai; dispersion sirf

m \neq = 0

ke liye dikhta hai.

Why questions

Neighbouring slits ka phase mein agree karna kyon guarantee karta hai ki sab

N

slits agree karengi?

Agar neighbours exactly

mλ

se alag hain, toh slit

k

, slit 1 se

(k - 1) mλ

aage hogi — hamesha whole number of wavelengths — isliye har slit ki crest line up ho jaati hai.

Grating maxima two-slit fringes se itne sharper kyon hote hain?

N

slits ke saath, exact angle se thoda hatne par bahut saari slightly-out-of-phase contributions ek bade group mein cancel ho jaati hain, isliye bright peak jaldi collapse ho jaati hai — jaise-jaise

N

badhta hai, sharper hoti jaati hai.

λ

badhne par maximum order kyon girta hai (fixed

d

ke liye)?

m_{m a x} = ⌊ d / λ ⌋

; bada

λ

d / λ

ko chota karta hai, isliye

sin θ

ke 1 se exceed hone se pehle kam whole-wavelength path differences fit hote hain.

Grating light ko spectrum mein kyon disperse karta hai lekin plain mirror nahi karta?

Maximum angle

λ

par depend karta hai

sin θ_{m} = mλ / d

ke through, isliye har colour thodi alag direction mein jaati hai; mirror sab colours ko same way mein bhejta hai. Dekho Interference and path difference.

Slits ko paas-paas rakhna (chota

d

) colours ko zyada kyon spread karta hai?

Chota

d

sin θ_{m} = mλ / d

ko har colour ke liye bada karta hai, orders ko wider angles par push karta hai aur wavelengths ke beech angular gap badhata hai.

Path difference

d sin θ

kyon hai aur

d cos θ

ya sirf

d

kyon nahi?

Extra path woh right triangle ki side hai jo angle

θ

ke opposite hai beam aur normal ke beech; woh side us triangle par sine ki definition se

d sin θ

hai.

Edge cases

m = 0

order ke liye outgoing angle kya hai, aur kyon?

θ = 0

—

d sin θ = 0

ka ekmaatra solution; beam seedha bina deviate hue guzar jaati hai, isliye direction ke liye centre aisa behave karta hai jaise koi grating hi nahi hai.

Kya hota hai jab

d / λ

exactly ek integer ho, jaise

d / λ = 3

Highest order

m = 3

sin θ = 1

satisfy karta hai, yaani

θ = 9 0^{\circ}

; woh maximum grating surface ke saath graze karta hai aur practically observable nahi hota.

Agar wavelength slit spacing se lambi ho,

λ > d

Tab even

m = 1

sin θ = λ / d > 1

chahiye, jo impossible hai, isliye sirf central

m = 0

maximum bachta hai — koi spectrum nahi banta.

Jaise-jaise slits ki number

N \to \infty

(spacing fixed) jaati hai, pattern ka limiting behaviour kya hai?

Maxima same angles par rehte hain lekin infinitely narrow aur background ke relative infinitely intense ho jaate hain — idealized perfectly-sharp spectral line.

Agar spacing

d

grating mein perfectly uniform nahi hai toh maxima ka kya hota hai?

Condition

d sin θ = mλ

sab slits ke liye ek clean angle par nahi rehti, isliye peaks broad aur blur ho jaate hain, resolving power degrade ho jaati hai. Dekho Resolving power and spectrometers.

Kya ek maximum exist kar sakta hai jiske liye

sin θ_{m}

negative ho, aur iska kya matlab hai?

Haan — negative

m

, negative

sin θ

deta hai, yaani angle

∣ θ_{m} ∣

par normal ke opposite side par ek maximum; sign sirf yeh batata hai kaunsi side par hai.

Jab

d / λ

ka small fractional part ho, jaise

3.05

, toh highest visible order ke baare mein kya special hai?

m_{m a x} = 3

sin θ = 3 λ / d \approx 0.98

par hai, bahut

9 0^{\circ}

ke close — ek faint, extremely oblique maximum jo experimentally miss karna aasaan hai.

Connections

Diffraction grating — condition for maxima — parent derivation jise ye traps stress-test karti hain.
Young's double-slit experiment — $N = 2$ limit; same phase condition, broader fringes.
Single-slit diffraction — contrasting rule: iska $a sin θ = mλ$ minima mark karta hai, maxima nahi.
Resolving power and spectrometers — kyon sharper (bada $N$ ) peaks matter karte hain.
Wavelength and the visible spectrum — kyon colours wavelength ke hisaab se fan out hote hain.