Thin lenses — lens equation, lens maker's equation: Visual walkthrough

2.5.6 · D2 · Physics › Optics › Thin lenses — lens equation, lens maker's equation

Hum ek target ke peeche hain: ek lens light ko kahan focus karta hai, aur uski shape ye kyun decide karti hai?

Step 1 — Ray, normal, aur angle actually hote kya hain

KYA. Kisi bhi formula se pehle, hum neeche diye picture ke characters pe agree karte hain.

Ray wo seedha raasta hai jis par light travel karti hai (arrow).
Spherical surface glass ki boundary hai, jo ek ball ke hisse ki tarah curved hoti hai. Us ball ka centre point $C$ hai, jise centre of curvature kehte hain. Surface se $C$ tak ki distance radius $R$ hai.
Normal wo line hai jo surface se seedhi bahar nikalti hai — sphere ke liye ye hamesha $C$ ki taraf point karti hai. Ye important hai kyunki optics mein angles hamesha normal se measure hote hain, glass se nahi.
Optical axis wo seedhi horizontal centre line hai; sab kuch isi ke saath measure hota hai.

KYUN. Light us angle ke hisaab se bend karti hai jo wo normal ke saath banati hai. Agar hume pata nahi normal kahan point kar raha, to angle measure nahi kar sakte, aur aage kuch bhi nahi milega. Curved surface pe normal tilt karta hai jaise-jaise tum glass pe upar jaate ho — ye tilt hi wajah hai ki ek lens focus karta hai.

PICTURE.

Figure — Thin lenses — lens equation, lens maker's equation

Step 2 — Snell's law, sirf us form mein jis ki humein zaroorat hai

KYA. Jab light ek material se doosre mein cross karti hai, to wo bend karti hai. Har material ka ek index $n$ hota hai: light us mein vacuum se kitni zyada slow travel karti hai. Air mein $n \approx 1$ ; glass mein $n \approx 1.5$ .

Full law hai $n_{1} sin i = n_{2} sin r$ , jahan $i$ normal se andar ka angle hai aur $r$ bahar ka angle.

Ye tool kyun, koi aur kyun nahi? Humein ek aisi rule chahiye jo ek surface par incoming tilt ko outgoing tilt se connect kare — ye exactly refraction hai, aur Snell's law iska ek-maatra statement hai. Hum paraxial form use karte hain kyunki hamare angles tiny hain:

$n_{1} i = n_{2} r$

Term by term: $n_{1}$ = wo index jise ray chhod rahi hai; $i$ = andar aate waqt normal se uska angle; $n_{2}$ = wo index jis mein wo enter kar rahi hai; $r$ = bahar jaate waqt normal se angle. Bada $n_{2}$ ek chota $r$ force karta hai — ray denser glass mein enter karte waqt normal ki taraf bend hoti hai.

PICTURE.

Step 3 — Teen chote triangles teen angles dete hain

KYA. Axis par ek object point $O$ (bilkul left) se ek ray lo jo surface par height $h$ par strike karti hai. Wo refract hoti hai aur image point $I$ ki taraf jaati hai. Hum teen angles name karte hain jo ray-lines axis ke saath banati hain:

$α$ — object par incoming ray ka angle,
$β$ — image par outgoing ray ka angle,
$ϕ$ — wo angle jo normal (line to $C$ ) axis ke saath banata hai.

Paraxial " $tan θ \approx height / distance$ " use karte hue, aur distances ko sign convention ke saath padhte hue ( $u < 0$ real object ke liye left par, $v > 0$ , $R > 0$ agar $C$ outgoing side par hai):

$α = \frac{h}{- u}, β = \frac{h}{v}, ϕ = \frac{h}{R}$

Term by term: har ek usi height $h$ ko divide karta hai by relevant point ( $O$ , $I$ , $C$ ) kitni door hai. $- u$ isliye aata hai kyunki $u$ negative hai lekin physical distance positive hoti hai.

KYUN. Hum "angles" ko Snell's law mein tab tak plug nahi kar sakte jab tak unhe measurable cheezein — heights aur distances — mein express na karein. Ye teen ratios geometry ko numbers $u, v, R$ mein convert karti hain jo hum ultimately chahte hain.

PICTURE.

Step 4 — Exterior-angle trick angles ko $i$ aur $r$ mein badalta hai

KYA. Ek triangle ka exterior angle doosre do opposite interior angles ke sum ke barabar hota hai. Ise picture par do baar apply karo:

$i = α + ϕ aur r = ϕ - β$

KYUN. Snell's law $i$ aur $r$ mein likhi hai (normal se measure hoti hain), lekin Step 3 ke hamare measurable ratios $α, β, ϕ$ hain (axis se measure hote hain). Ye trick bridge hai: ye normal-angles ko axis-angles se re-express karta hai.

Padhna: $i$ ke liye, incoming ray-to-axis angle $α$ plus normal ka apna tilt $ϕ$ milke normal se full angle banta hai. $r$ ke liye, ray aage swing kar chuki hai, to hum $β$ subtract karte hain.

PICTURE.

Step 5 — Substitute karo, $h$ cancel karo, aur single-surface law saamne aata hai

KYA. Step 4 ko paraxial Snell $n_{1} i = n_{2} r$ mein daalo, phir Step 3 ke ratios insert karo:

$n_{1} (\frac{h}{- u} + \frac{h}{R}) = n_{2} (\frac{h}{R} - \frac{h}{v})$

Har term mein ek hi height $h$ hai, to divide kar do (isliye answer is par depend nahi karta ki humne kaunsi ray li — sab paraxial rays agree karti hain, aur isliye hi ek image form hoti hai):

$- \frac{n _{1}}{u} + \frac{n _{1}}{R} = \frac{n _{2}}{R} - \frac{n _{2}}{v}$

Image terms left par collect karne ke liye rearrange karo:

$\frac{n _{2}}{v} - \frac{n _{1}}{u} = \frac{n _{2} - n _{1}}{R}$

Term by term: $\frac{n _{2}}{v}$ = "image kahan hai, us glass se weighted jis mein wo hai"; $\frac{n _{1}}{u}$ = wahi object ke liye; right side $\frac{n _{2} - n _{1}}{R}$ = surface ki apni bending strength — badi jab media bahut differ karein ya curve tight ho.

KYUN. Ye single-surface formula hai — hamara reusable building block. Ek curved surface, hamesha ke liye done.

PICTURE.

Step 6 — Do baar karo: ek lens do surfaces back to back hai

KYA. Ek thin lens air mein glass ( $n$ ) hai. Light surface 1 (radius $R_{1}$ , air→glass) se milti hai, phir surface 2 (radius $R_{2}$ , glass→air) se. "Thin" ka matlab hai dono surfaces ek hi plane par hain — koi gap nahi.

Surface 1, $n_{1} = 1$ , $n_{2} = n$ ke saath, image ek intermediate point $v_{1}$ par:

$\frac{n}{v _{1}} - \frac{1}{u} = \frac{n - 1}{R _{1}} (1)$

Surface 1 ka intermediate image surface 2 ka object hai. Kyunki lens thin hai, wo object bhi $v_{1}$ distance par hai. Ab $n_{1} = n$ , $n_{2} = 1$ :

$\frac{1}{v} - \frac{n}{v _{1}} = \frac{1 - n}{R _{2}} (2)$

KYUN. Hum kuch naya derive nahi karte — Step 5 ka box reuse karte hain, har surface ke liye ek baar. Yahi approach ki puri economy hai.

PICTURE.

Step 7 — Dono equations add karo: ghost image cancel ho jaata hai

KYA. (1) aur (2) add karo. Intermediate term pehle mein $+ \frac{n}{v _{1}}$ aur doosre mein $- \frac{n}{v _{1}}$ ke roop mein aata hai — ye ek doosre ko destroy kar dete hain:

$\frac{1}{v} - \frac{1}{u} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$

Ab focal length $f$ define karo as jahan parallel incoming rays (object infinitely far, $u \to - \infty$ , to $1/ u \to 0$ ) converge karti hain, yaani $1/ v = 1/ f$ :

$\frac{1}{f} = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}}) Lens maker’s equation$

aur, dono boxed lines compare karte hue, thin-lens equation:

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$

KYUN. Cancellation hi miracle hai: humein kabhi jaanna nahi pada ki ghost image $v_{1}$ kahan thi. Do surfaces ek number $f$ mein collapse ho jaati hain jo sirf shape aur glass par depend karta hai.

Maker's equation mein term by term: $(n - 1)$ = "glass light ko kitna slow karta hai" (agar $n = 1$ to zero, koi lens hi nahi); $(\frac{1}{R _{1}} - \frac{1}{R _{2}})$ = "dono faces ka total curvature difference".

PICTURE.

Step 8 — Degenerate aur edge cases jinpe kabhi mat phisalna

KYA. Har limit check karo taaki koi scenario surprise na kare.

Case	Input	Result	Reading
Flat plate	$R_{1}, R_{2} \to \infty$	$1/ f = 0$	Zero curvature → kuch nahi karta
Symmetric biconvex	$R_{1} = + R, R_{2} = - R$	$\frac{1}{f} = (n - 1) \frac{2}{R}$	Signs faces ko add karte hain
Diverging (biconcave)	$R_{1} < 0, R_{2} > 0$	$f < 0$	Rays spread karta hai, virtual focus
No glass	$n = 1$	$1/ f = 0$	$(n - 1) = 0$ : speed change nahi, bend nahi
Object at $f$	$u = - f$	$v \to \infty$	Rays parallel nikal jaati hain

KYUN. Ek formula tab trustworthy hota hai jab tum use uski extremes par sahi behave karte dekho. Do sabse khatarnak sign slips hain — (a) ye bhool jaana ki biconvex lens ke liye $R_{2}$ negative hota hai, aur (b) mirror equation ki tarah "+" likhna.

PICTURE.

Ek-picture summary

Recall Feynman retelling — puri walkthrough simple shabdon mein

Socho ek ray left par ek dot se nikal rahi hai, ek curved piece of glass se takra rahi hai. Ye kitna bend hogi jaanne ke liye mujhe us jagah "seedhi-bahar" line chahiye (the normal) — ball ke liye ye line hamesha ball ke centre ki taraf point karti hai. Main ray ko us line se compare karta hoon: andar chota angle, bahar chota angle, aur glass use swing karata hai (Snell). Main un sab angles ko simple "height ÷ distance" fractions mein badalta hoon, plug karta hoon, aur height cancel ho jaata hai — isliye ek dot ki sab rays ek naye dot par milti hain: the image. Ye mujhe ek curved surface ke liye ek rule deta hai. Lens sirf do surfaces hai jo kiss kar rahi hain, to main woh rule do baar use karta hoon — bacha hua "middle image" ek line mein plus ke saath aata hai aur doosre mein minus ke saath aur jab main unhe add karta hoon to gayab ho jaata hai. Jo bachta hai wo ek single number $f$ hai jo glass ( $n - 1$ ) aur do curves se bana hai. Flat glass? Koi curve nahi, koi focus nahi. Fatter, tighter curves, denser glass? Zyada swing, closer focus. Poori kahani yehi hai.

Recall

Step 5 mein height $h$ kyun cancel hoti hai? ::: Har angle $h / distance$ ke roop mein likha gaya tha, to $h$ ek common factor hai — ise divide karne ka matlab hai sab paraxial rays ek hi relation follow karti hain, to sab ek point par focus hoti hain. Intermediate image $v_{1}$ kyun gayab ho jaata hai? ::: Ye do surface equations mein $+ n / v_{1}$ aur $- n / v_{1}$ ke roop mein aata hai; unhe add karne se ye exactly cancel ho jaata hai. Derivation mein focal length $f$ kaise define hoti hai? ::: Wo image distance hai jab object infinitely far ho ( $u \to - \infty$ , $1/ u \to 0$ ), to $1/ v = 1/ f$ . Symmetric biconvex lens ke liye faces "add" kyun hoti hain? ::: $R_{1} = + R$ aur $R_{2} = - R$ , to subtraction $1/ R_{1} - 1/ R_{2}$ becomes $2/ R$ .

Connections

Thin lenses — lens equation, lens maker's equation (parent)
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)