2.5.6 · D1Thin lenses — lens equation, lens maker's equation
Ek lens light ko isliye modata hai kyunki light ki speed badal jaati hai jab woh air aur glass ke beech cross karti hai, aur yeh bending wahan zyada hoti hai jahan glass zyada sharply curved ho. Is topic ki har cheez — har symbol u v f R n "light kahan aim ki gayi thi" versus "woh un bends ke baad kahan point karti hai" .
Parent note padhne se pehle tumhe kuch ideas ka ek chhota toolbox chahiye. Hum har ek ko zero se build karte hain, use ek picture se anchor karte hain, aur batate hain kyun is topic ko uski zaroorat hai . Upar se neeche padho — har item upar wale par lean karta hai.
Definition Optical axis & light directionOptical axis ek seedhi horizontal line hai jo lens ke centre se guzarti hai, uske perpendicular. Ek drawing ki centre-line ko picture karo: har cheez isi se measure hoti hai. Hum hamesha light ko left → right travel karte draw karte hain (incoming rays ka arrow).
Picture: ek horizontal line lens ke middle se guzarti hai; "duniya" ek incoming side (left) aur ek outgoing side (right) mein split ho jaati hai.Topic ko iska kyun chahiye: is chapter ki har distance isi axis ke saath, lens se measure hoti hai. Ek fixed direction ke bina, "positive" aur "negative" ka koi matlab nahi hota.
Definition Cartesian "real-is-positive-light-goes-→" conventionLens ko axis ke saath rakhee ek number line ka origin samjho, jisme positive direction wahan point karti hai jahan light ja rahi hai (right ki taraf).
Right ki taraf (outgoing side) measure ki gayi distance positive hoti hai.
Left ki taraf (incoming side) measure ki gayi distance negative hoti hai.
Intuition Signs ki zaroorat kyun hai?Ek formula ko har situation cover karni chahiye — object saamne, image peeche, image virtual usi side par. Har case ke liye alag rule yaad karne ki jagah, hum answer ke sign ko case batane dete hain . Negative v
Picture: axis par rakhaa ek ruler, lens par zero, left ko negative, right ko positive.Topic ko iska kyun chahiye: yahi woh grammar hai jo v 1 − u 1 = f 1
u v
u object tak ki distance (woh cheez jo hum dekhte hain, light ka source).v image tak ki distance (jahan light rays dobara converge hokar picture rebuild karti hain).
Dono Section 2 ki number line ka sign carry karte hain.
Ek real object incoming (left) side par hota hai, isliye u < 0
Ek real image outgoing (right) side par banta hai, isliye v > 0
Ek virtual image (rays sirf ussi se aati lagti hain) incoming side par hoti hai, isliye v < 0
Worked example Sign padhnaAgar ek calculation v = − 6 incoming side par, aur woh virtual hai. Tumhe koi naya rule nahi chahiya tha — minus sign ne sab kuch bata diya.
Topic ko iska kyun chahiye: u v beech ek relationship hai.
Intuition Rays aur jahan woh milti hainEk object ko ek fountain socho jo saari directions mein rays (light ki patli seedhi lines) spray karta hai. Ek lens unka ek fan pakad kar har ek ko re-aim karta hai. Agar saari re-aimed rays ek single point par cross karti hain, toh woh crossing point image hai — object ki ek rebuilt copy. "Converge" = rays aapas mein aana; "diverge" = rays door hona.
Picture: lines ka ek fan ek point se nikalti hai, lens par bend hoti hai, aur far side par ek single point par wapas squeeze hoti hai (real image) — ya itni phailti hain ki unki backward extensions milti hain (virtual image).Topic ko iska kyun chahiye: poora goal ("image kahan banta hai yeh dhundho") hai "rays kahan milti hain yeh dhundho." Baaki sab us point ko compute karne ki machinery hai.
R C Centre of curvature C radius R C R bada hota hai; ek sharply bulging surface ka R chhota hota hai.
Sign rule: R > 0 C outgoing (right) side par ho; R < 0 C
Picture: compass point ko C R Topic ko iska kyun chahiye: glass ki shape focal length set karti hai. Chhota R ( R 1 1 − R 2 1 )
Common mistake Dono faces keR Kyun sahi lagta hai: ek biconvex lens symmetric hai, isliye dono faces same lagte hain.
Fix: centre-of-curvature rule ko har face par alag se apply karo. Ek biconvex lens ke liye pehle face ka C R 1 > 0 C R 2 < 0
Definition Refractive indexn Refractive index n = speed in glass speed in air kitni baar slower . Air: n ≈ 1 n ≈ 1.5
( n − 1 ) n change hota hai. Agar n = 1 koi bending nahi . Isliye lens ki strength depend karti hai ki n ( n − 1 ) ( n − 1 )
Picture: "marching band" — ek line of marchers ka ek edge mud (glass) se takraa kar pehle slow ho jaata hai, poori line ko swing karta hai. Bada n Topic ko iska kyun chahiye: n f material half hai (R shape half hain).
Definition Snell's law (paraxial form)Jab ek ray ek surface cross karti hai, toh woh bend hoti hai. Snell's law kehta hai n 1 sin i = n 2 sin r i r normal se measure kiye jaate hain (woh line jo surface se seedhi bahar jaati hai, jo yahan radius ke saath C Chhote angles ke liye axis ke paas, sin θ ≈ θ n 1 i = n 2 r
Intuition Small-angle shortcut kyun?Real spheres perfectly focus karte hain sirf axis ke kareeb rays ke liye. Angles ko chhota rakhne se hum ugly sin θ θ tan θ θ Snell's law & paraxial approximation .
Topic ko iska kyun chahiye: yahi woh physical law hai jo light ko har surface par bend karta hai. Parent ki poori single-surface derivation Snell's law + small-angle geometry hai.
Definition Angle as height-over-distanceEk ray ke liye jo axis se chhote angle α d h tan α = d h α tan α ≈ α α ≈ d h
Intuition Derivation mein yeh kyun matter karta haiParent likhta hai α = − u h ϕ = R h β = v h u R v h cancel ho jaata hai — yahi wajah hai ki final formula mein koi h
Topic ko iska kyun chahiye: yeh geometry (triangles) ko algebra (h h
f parallel rays (jaise infinitely door object se, u → − ∞ focal point . Lens se us point tak ki distance focal length f
Converging lens: f > 0
Diverging lens: f < 0 peeche ek focal point se aa rahi hoon).
Topic ko iska kyun chahiye: f ek number hai jo ek lens summarise karta hai. Lens maker's equation f compute karti hai; lens equation images locate karne ke liye f use karti hai.
m m = h h ′ = u v h h ′
m > 0 m < 0 ∣ m ∣ > 1 ∣ m ∣ < 1
Picture: lens centre se seedha guzarne wala undeviated ray do similar triangles banata hai — ek object height se, ek image height se — aur unke bases (v u Magnification and image formation .Topic ko iska kyun chahiye: image locate karna half kaam hai; m
Optical axis and light direction
Sign convention plus minus
Rays and image as meeting point
Small angle height over distance
Single surface refraction
Lens ke kis side par ek real object hota hai, aur u Incoming (left) side; u < 0
Ek calculation v = − 6 Image virtual hai, incoming side par, lens se 6 cm door
R Centre of curvature C C
Lens maker's equation mein ( n − 1 ) n Bending ke liye speed mein change chahiye; agar n = 1
Snell's law ko uske paraxial (small-angle) form mein likho n 1 i = n 2 r
Hum ek angle ko h / d Small-angle: tan θ ≈ θ = height / distance
Single-surface derivation mein height h Teeno angles ek hi ray ki height use karte hain, isliye common h
Focal length f Woh point jahan incoming parallel rays converge hoti hain
m < 0 Image inverted (ulti) hai
Virtual image kya hoti hai? Woh point jahan rays sirf aati lagti hain (unki backward extensions milti hain); wahan koi real light cross nahi karti
Parent: Thin lenses 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) 
