2.5.6
Intuition Bada picture (WHY lenses light ko modte hain)Ek lens bas do refracting surfaces ko jod kar banaya hota hai. Light har ek surface par isliye modti hai kyunki glass aur air ke beech light ki speed badal jaati hai (Snell's law). "Thin" lens ka matlab hai ki sheesha itna patla hai ki hum light ke andar travel karne ki distance ignore kar dete hain — to hum dono surfaces ki bending ko ek hi point par add kar dete hain. Neeche sab kuch usi ek idea ka bookkeeping hai.
Definition Thin lens aur sign convention (Cartesian, real-is-positive light goes →)
Thin lens : thickness ≪ Distances lens se measure ki jaati hain, light direction ke along.
Object distance u ⇒ u < 0
Image distance v ⇒ v > 0
Focal length f f > 0 f < 0
Kisi surface ka radius R R > 0 outgoing side par ho.
HUM KYA CHAHTE HAIN: object position aur lens shape diye hue, find karna hai ki image kahan banti hai. Do equations ye kaam karti hain:
Lens equation u , v , f Lens maker's equation f glass ki shape (R 1 , R 2 , n
Intuition YE YAHAAN SE KYUN SHURU KAREIN?Agar main ek curved glass surface par bending describe kar sakta hoon, to main ise bas do baar apply kar deta hoon. Ek baar derive karo, baar baar use karo.
Radius R n 1 n 2 single-surface refraction formula :
v n 2 − u n 1 = R n 2 − n 1
Worked example HUM ISSE KAISE DERIVE KARTE HAIN (paraxial, first principles)Ye step kyun? Chhote angles ke liye Snell's law use karo taaki sin θ ≈ θ O h i C n 1 i = n 2 r
Exterior-angle geometry use karke (angles chhote, isliye tan θ ≈ θ ≈ h / distance
i = α + ϕ α = − u h ϕ = R h r = ϕ − β β = v h
Ye step kyun? n 1 i = n 2 r h n 1 ( − u h + R h ) = n 2 ( R h − v h ) − u n 1 + R n 1 = R n 2 − v n 2 v n 2 − u n 1 = R n 2 − n 1
Definition ke anusaar f u → − ∞ v 1 = f 1
f 1 = ( n − 1 ) ( R 1 1 − R 2 1 ) Lens maker’s equation
Aur compare karne par, thin-lens (Gaussian) equation :
v 1 − u 1 = f 1
f ( n − 1 ) n ( 1/ R 1 − 1/ R 2 ) R → ∞ 1/ f = 0
Worked example (A) Biconvex lens,f n = 1.5 R 1 = + 20 + R 2 = − 20 Ye signs kyun? Doosri surface ulti taraf curve karti hai, centre incoming side par ⇒ negative.
f 1 = ( 1.5 − 1 ) ( 20 1 − − 20 1 ) = 0.5 ⋅ 20 2 = 10 0.5 = 0.05 cm − 1 f = + 20
Worked example (B) Lens equation se image positionSame lens f = 20 u = − 30 Ye step kyun? v 1 − u 1 = f 1 v 1 = f 1 + u 1 = 20 1 + − 30 1 = 60 3 − 2 = 60 1 v = + 60 m = v / u = 60/ ( − 30 ) = − 2
Worked example (C) Diverging lensf = − 15 u = − 10 v 1 = − 15 1 + − 10 1 = 30 − 2 − 3 = − 30 5 = − 6 1 ⇒ v = − 6 cm m = v / u = ( − 6 ) / ( − 10 ) = + 0.6
v 1 + u 1 = f 1 Ye sahi kyun lagta hai: mirrors mein "+" use hota hai aur equations ek jaisi lagti hain.
Fix: Lenses ke liye (Cartesian convention) ye ek minus hai: v 1 − u 1 = f 1 u
Common mistake Maker's equation meinR 2 Ye sahi kyun lagta hai: dono faces "same taraf curve hote lagte hain", isliye tum 1/ R 1 + 1/ R 2 Fix: Har face ke liye centre-of-curvature rule alag se use karo. Symmetric biconvex lens ke liye R 1 = + R R 2 = − R add hote hain: f 1 = ( n − 1 ) R 2
Common mistake Ye sochna kif Ye sahi kyun lagta hai: dono surfaces alag hain.
Fix: Sides swap karne par R 1 ↔ R 2 same f deta hai. Ek thin lens ka ek hi focal length hota hai.
Single spherical surface formula v n 2 − u n 1 = R n 2 − n 1
Lens maker's equation f 1 = ( n − 1 ) ( R 1 1 − R 2 1 )
Thin-lens (Gaussian) equation v 1 − u 1 = f 1
Derivation mein intermediate image kyun gayab ho jaata hai? Do single-surface equations add karne par n / v 1
R R > 0
Magnification formula m = h h ′ = u v
Flat plate (R → ∞ f Infinite power zero — 1/ f = 0
Converging lens mein f Positive (f > 0
f = 20 v v = + 60 m = − 2
Light ka glass mein "slow" hona bending ke liye kyun zaroori hai? ( n − 1 ) n = 1
Recall Feynman: 12 saal ke bacche ko samjhaoSocho ek marching band road se hata kar keechar wali grass par angle se chal rahi hai. Jo side pehle keechar mein aati hai wo slow ho jaati hai, isliye poori line swing karti hai. Glass light ke liye keechar hai. Ek lens ke do curved keechar wale edges hote hain; light har ek par thodi swing karti hai. Agar dono swings light ko ek jagah push karti hain, to saari rays wahin milti hain — wohi image hai! Mota, zyada curved glass = zyada swing = milne ka point aur paas.
Mnemonic Maker's equation yaad karo"n-minus-one times curve-difference" → ( n − 1 ) ( R 1 1 − R 2 1 ) "v-minus-u equals f" (sab reciprocals). Lens = minus , Mirror = plus .
Refraction at a single spherical surface Snell's law & paraxial approximation Spherical mirrors — mirror equation Magnification and image formation Power of a lens (dioptres) Lens combinations & equivalent focal length Lens aberrations (chromatic, spherical)
Single-surface refraction
Sign convention Cartesian
