Power of a lens, combination of lenses: Worked examples

2.5.7 · D3 · Physics › Optics › Power of a lens, combination of lenses

Kuch bhi calculate karne se pehle, ek reminder jo hum baar baar use karenge:

Scenario matrix

Lens power ke har problem ka ghar in cells mein se kisi ek mein hota hai. Table ke neeche, examples E1–E9 mein se har ek announce karta hai ki woh kaunsa cell clear karta hai.

#	Cell (kya cheez ise alag banati hai)	Bada khatara	Kaunsa example
C1	Single convex lens, $f > 0$	cm→m bhool jaana	E1
C2	Single concave lens, $f < 0$	minus sign chhhod dena	E2
C3	Do lenses contact mein, dono converging	koi nahi — seedha addition	E3
C4	Contact: convex + concave (opposite signs)	sign cancellation	E4
C5	Contact se net zero power (degenerate)	$F \to \infty$ , $1/ P$ undefined	E5
C6	Do lenses $d$ se alag	$- d P_{1} P_{2}$ use karna zaroori	E6
C7	Limiting check: gap formula mein $d \to 0$ karo	C3/C4 pe collapse hona chahiye	E7
C8	Word problem (real world: eyeglasses / corrective)	words ko signs mein translate karo	E8
C9	Exam twist: combined $F$ diya hai, missing lens dhundho	arithmetic ulti karo	E9

E1 — Single convex lens (cell C1)

Worked example Statement

Ek convex lens ki focal length $f = 25 cm$ hai. Uski power nikalein.

Forecast: convex matlab converging, toh ek positive power expect karo. Quarter-metre focal length kaafi chhoti hai, toh kuch dioptres expect karo, koi tiny fraction nahi.

Step 1 — Metres mein convert karo. $f = 25 cm = 0.25 m$ Yeh step kyun? $P = 1/ f$ sirf metres mein valid hai; pehle convert karna non-negotiable hai.

Step 2 — Definition apply karo. $P = \frac{1}{f} = \frac{1}{0.25} = + 4 D$ Yeh step kyun? Power hai hi focal length ka reciprocal; yeh Thin Lens Equation world ka master rule hai.

Step 3 — Sign fix karo. Convex $\Rightarrow f > 0 \Rightarrow P > 0$ , toh $P = + 4 D$ . Yeh step kyun? Power wahi sign carry karta hai jo $f$ ka hai; ek converging lens ki power positive hoti hai.

Verify: $4 D = 4 m^{- 1}$ , aur $1/4 m^{- 1} = 0.25 m = 25 cm$ ✓ wapis shuru par. Sign positive, "converging" se match karta hai.

E2 — Single concave lens (cell C2)

E3 — Do converging lenses contact mein (cell C3)

Figure — Power of a lens, combination of lenses

Worked example Statement

Ek $+ 5 D$ convex lens ko ek convex lens jiska $f_{2} = 20 cm$ hai, ke contact mein rakha gaya hai. Combined power $P$ aur focal length $F$ nikalein.

Forecast: do converging lenses ek doosre ke peechhe light ko do baar usi taraf modte hain — combo dono mein se kisi se bhi zyada strong hona chahiye (higher power, shorter focal length).

Step 1 — Dono powers ek hi unit mein lao. $P_{1} = + 5 D, P_{2} = \frac{1}{0.20} = + 5 D$ Yeh step kyun? Hum sirf powers add kar sakte hain, focal lengths nahi — isliye pehle $f_{2}$ ko $P_{2}$ mein convert karo.

Step 2 — Add karo (contact mein hain). $P = P_{1} + P_{2} = 5 + 5 = + 10 D$ Yeh step kyun? Contact matlab $d = 0$ : lens 1 se modi hui ray lens 2 se same height par milti hai, toh bending cleanly add hoti hai (figure mein dono stacked lenses dekho — koi gap nahi, koi spreading nahi).

Step 3 — Focal length wapis nikalein. $F = \frac{1}{P} = \frac{1}{10} = 0.10 m = 10 cm$ Yeh step kyun? Focal length report karne ke liye power ko ek baar aur invert karo.

Verify: Combined $F = 10 cm$ , $f_{1} = 20 cm$ ya $f_{2} = 20 cm$ dono se chhota hai ✓ — do lenses ek se zyada converge karte hain, bilkul forecast jaisa. Yeh Resistors in Parallel pattern hai: do equal "resistances" half ho jaati hain.

E4 — Contact: convex + concave (cell C4)

Worked example Statement

$P_{1} = + 5 D$ ka ek convex lens $f_{2} = - 50 cm$ wale concave lens ke contact mein hai. $P$ aur $F$ nikalein.

Forecast: concave lens bending power ghataata hai. Toh net positive hona chahiye lekin $+ 5 D$ se kamzor — combo phir bhi converge karega, bas aaram se.

Step 1 — Concave lens convert karo. $P_{2} = \frac{1}{- 0.50} = - 2 D$ Yeh step kyun? Concave $\Rightarrow$ negative power; sign le chalo.

Step 2 — Signed powers add karo. $P = P_{1} + P_{2} = 5 + (- 2) = + 3 D$ Yeh step kyun? Contact $\Rightarrow$ add karo; $P_{2}$ ka minus sign automatically "cancelling" kar deta hai.

Step 3 — Focal length. $F = \frac{1}{3} \approx + 0.333 m = + 33.3 cm$ Yeh step kyun? Positive $F$ confirm karta hai ki combination converging hai.

Verify: $+ 3 D < + 5 D$ ✓ (concave partner ne kamzor kiya) aur $F = 33.3 cm > 20 cm$ ✓ (kamzor lens = zyada lambi focal length). Sab ek hi direction mein point kar rahe hain.

E5 — Degenerate case: net zero power (cell C5)

E6 — Alag-alag lenses (cell C6)

Worked example Statement

Do convex lenses, $f_{1} = 20 cm$ aur $f_{2} = 30 cm$ , $d = 10 cm$ se alag hain. Combined power $P$ aur focal length $F$ nikalein.

Forecast: ab woh touch nahi kar rahe, toh clean sum $P_{1} + P_{2}$ gap term se kharab ho jaata hai. Combined power $P_{1} + P_{2}$ se thodi kam hogi.

Step 1 — Sab kuch metres mein, sab kuch power mein. $P_{1} = \frac{1}{0.20} = 5 D, P_{2} = \frac{1}{0.30} = 3.33 D, d = 0.10 m$ Yeh step kyun? Gap formula $d$ ko do powers se multiply karta hai, isliye teeno SI units mein saath hone chahiye.

Step 2 — Separation formula use karo. $P = P_{1} + P_{2} - d P_{1} P_{2}$ Yeh step kyun? Jab $d \neq = 0$ toh ray lens 2 se alag height par milti hai (figure dekho: lens 1 se red ray gap ke upar drift karti hai phir lens 2 se milti hai). Woh drift exactly $- d P_{1} P_{2}$ correct karta hai.

Step 3 — Values daalein. $P = 5 + 3.33 - (0.10) (5) (3.33) = 8.33 - 1.67 = 6.67 D$ Yeh step kyun? Gap term $- 1.67 D$ naive sum $8.33 D$ ko reduce karta hai.

Step 4 — Focal length. $F = \frac{1}{6.67} \approx 0.15 m = 15 cm$

Verify: $6.67 D < 8.33 D$ ✓ (gap hamesha do convex lenses ke liye power churaata hai). Exact form: $P = 5 + \frac{10}{3} - \frac{1}{10} \cdot 5 \cdot \frac{10}{3} = \frac{20}{3} D$ , aur $\frac{20}{3} = 6.67$ ✓. Yeh principle har Microscope and Telescope ke dil mein hai, jahan objective aur eyepiece ek fixed distance par hote hain.

E7 — Limiting check: gap ko gayab karo (cell C7)

Worked example Statement

Separation formula lo aur $d \to 0$ karo. Dikhao ki yeh contact formula reproduce karta hai.

Forecast: koi gap nahi toh correction term mar jaana chahiye, sirf addition bachega — sanity check jo C6 ko C3/C4 se jodta hai.

Step 1 — General form likho. $P = P_{1} + P_{2} - d P_{1} P_{2}$ Yeh step kyun? Yeh ek master formula hai jo dono cases contain karta hai; contact sirf iska $d = 0$ slice hai.

Step 2 — $d = 0$ substitute karo. $P = P_{1} + P_{2} - (0) P_{1} P_{2} = P_{1} + P_{2}$ Yeh step kyun? Gap term mein $d$ factor hai, toh gap ko zero karne se woh puri tarah delete ho jaata hai.

Verify: E6 ke numbers $d = 0$ ke saath dobara chalao: $P = 5 + 3.33 - 0 = 8.33 D$ , pure-contact sum se match karta hai. Aur E3 ke numbers ( $5$ aur $5$ ) ke saath: $5 + 5 - 0 = 10 D$ ✓. Ek formula sab pe rule karta hai; contact koi alag rule nahi, bas $d = 0$ special case hai.

E8 — Real-world word problem: reading glasses (cell C8)

Worked example Statement

Ek optician ek far-sighted person ke liye $+ 2.5 D$ power ke reading glasses prescribe karta hai. (a) Woh focal length kya hogi, aur lens kis type ka hai? (b) Agar ek mild $- 0.5 D$ scratch-coating lens iske contact mein cement kar di jaaye, toh net prescription kya hogi?

Forecast: positive prescription $\Rightarrow$ converging (convex) lens, kyunki far-sighted aankhon ko converging mein madad chahiye. $- 0.5 D$ add-on power thodi si ghataani chahiye.

Step 1 — Power se focal length nikalein. $f = \frac{1}{P} = \frac{1}{2.5} = 0.40 m = 40 cm$ Yeh step kyun? Opticians dioptres mein bolte hain; lens ko imagine karne ke liye focal length mein invert karo.

Step 2 — Classify karo. Positive power $\Rightarrow$ convex/converging. Far-sight (hypermetropia) converging lens se correct hota hai, toh yeh physically theek hai (dekho Defects of Vision).

Step 3 — Contact mein combine karo. $P_{net} = 2.5 + (- 0.5) = + 2.0 D$ Yeh step kyun? Cemented = contact mein = powers add hote hain, signs ke saath.

Step 4 — Nayi focal length. $F = \frac{1}{2.0} = 0.50 m = 50 cm$

Verify: $+ 2.0 D < + 2.5 D$ ✓ (negative coating ne kamzor kiya) aur $50 cm > 40 cm$ ✓ (kamzor $\Rightarrow$ lambi focal length). Abhi bhi positive, abhi bhi converging — glasses kaam kar rahe hain, bas thode se mild.

E9 — Exam twist: missing lens dhundho (cell C9)

Worked example Statement

Do thin lenses contact mein $F = + 15 cm$ focal length ki combination banate hain. Unme se ek $- 4 D$ power ka concave lens hai. Doosre lens ki power aur focal length nikalein.

Forecast: combination converging hai ( $F > 0$ ), phir bhi ek member diverging hai ( $- 4 D$ ). Toh unknown lens ek strong convex hona chahiye — itna strong ki $- 4 D$ ko beat kare aur phir bhi net positive rahe.

Step 1 — Pehle combined power nikalein. $P = \frac{1}{F} = \frac{1}{0.15} = 6.67 D$ Yeh step kyun? Humein result pata hai; ek part reverse-engineer karne ke liye poore ko power mein express karo.

Step 2 — Addition rule ulta karo. Contact matlab $P = P_{1} + P_{2}$ , toh $P_{2} = - 4 D$ ke saath: $P_{1} = P - P_{2} = 6.67 - (- 4) = 6.67 + 4 = 10.67 D$ Yeh step kyun? $P_{1} + P_{2} = P$ mein unknown ke liye solve karna simple subtraction hai — isliye powers (focal lengths nahi) friendly quantity hain.

Step 3 — Uski focal length. $f_{1} = \frac{1}{10.67} \approx 0.0938 m \approx 9.4 cm$ Yeh step kyun? Mili hui power ko physical answer ke liye focal length mein convert karo.

Verify: Wapis add karo: $10.67 + (- 4) = 6.67 D$ , jo $F = 1/6.67 = 0.15 m = 15 cm$ deta hai ✓. Positive, strongly convex, exactly forecast jaisa — yeh concave lens ko outmuscle karta hai.

Recall Har example kaunsa cell hai?

E1 :: C1 single convex E2 :: C2 single concave E3 :: C3 do convex contact mein E4 :: C4 convex + concave contact mein E5 :: C5 net-zero degenerate E6 :: C6 d se alag E7 :: C7 limiting d→0 check E8 :: C8 real-world eyeglasses E9 :: C9 exam reverse-engineering

Connections

Thin Lens Equation — woh equation jo hum combinations ke liye add karte hain.
Lensmaker's Equation — $P$ tak pahunchne ka doosra raasta.
Magnification of Lenses — in combos ke liye net $m = m_{1} m_{2} \dots$ .
Microscope and Telescope — separated-lens case (E6) action mein.
Resistors in Parallel — contact lenses jaisi hi reciprocal arithmetic.
Defects of Vision — jahan se E8 ke dioptre prescriptions aate hain.