Power of a lens, combination of lenses: Question bank

2.5.7 · D5 · Physics › Optics › Power of a lens, combination of lenses

Shuru karne se pehle, ek reminder — sirf teen facts hain jinpar har answer tika hua hai:

Pehle char pictures samjho

Is page ke har trap ka connection in char pictures mein se kisi ek se hai. Inhe abhi anchor karo, phir reveals obvious ho jaayenge.

Picture 1 — $P = 1/ f$ kyun hota hai (definition ke peeche ka triangle). Ek ray axis ke parallel height $h$ par aati hai. Thin lens ke baad wo focus ki taraf jaati hai, jo $f$ door hai. Red triangle dekho: ray ne horizontal run $f$ mein height $h$ itni drop ki. Bending angle $δ$ wo angle hai jo lens par banta hai.

Figure — Power of a lens, combination of lenses

Picture 2 — $R_{1}$ aur $R_{2}$ ka matlab (lensmaker formula ke liye sign convention). Lensmaker formula $P = (n - 1) (\frac{1}{R _{1}} - \frac{1}{R _{2}})$ mein do signed radii chhupe hote hain. Yahan $R_{1}$ pehli surface ka radius hai jis par light padti hai aur $R_{2}$ doosri ka. Convention yeh hai (light left→right travel karti hai): jo surface incoming light ki taraf bulge karti hai wo convex hai aur uska radius positive hota hai; jo surface door curve karti hai uska radius negative hota hai.

Picture 3 — Contact lenses parallel resistors ki tarah add hote hain (conductance analogy). Parallel mein resistors apne reciprocals (conductances) add karte hain: $\frac{1}{R} = \frac{1}{R _{1}} + \frac{1}{R _{2}}$ . Contact mein lenses apne reciprocals (powers) add karte hain: $\frac{1}{F} = \frac{1}{f _{1}} + \frac{1}{f _{2}}$ . Same shape ki equation — side-by-side dekho.

Picture 4 — $- d P_{1} P_{2}$ term kahan se aata hai. Axis ke parallel ek ray ko height $h_{1}$ par lens 1 mein jaate hue follow karo. Lens 1 use $δ_{1} = P_{1} h_{1}$ se bend karta hai. Gap $d$ ke baad ray lens 2 se takraane se pehle chhoti height $h_{2} = h_{1} - d δ_{1}$ tak drop ho jaati hai. Lens 2 us chhoti height ko bend karta hai: $δ_{2} = P_{2} h_{2}$ .

True or false — justify karo

Jo lens bada focal length rakhta hai wo zyada powerful lens hota hai.

False. Power hai

P = 1/ f

, toh power aur

f

ka inverse relation hai — lamba focal length matlab weaker lens jo rays ko dheere bend karta hai (Picture 1).

Power hamesha positive quantity hoti hai kyunki ye "strength" measure karta hai.

False. Power focal length ka sign carry karta hai: convex

f > 0

deta hai

P > 0

, concave

f < 0

deta hai

P < 0

. Sign bending ki direction encode karta hai (converging vs diverging), sirf magnitude nahi.

Contact mein do lenses hamesha dono mein se kisi bhi ek akele lens se zyada strong (higher-magnitude) combination dete hain.

False. Powers sign ke saath add hote hain:

+ 5 D

aur

- 2 D

sirf

+ 3 D

denge, jo

+ 5 D

lens se weaker hai. Ek concave lens convex ko partly cancel kar deta hai.

Agar

f

double ho jaaye, toh power half ho jaati hai.

True.

P = 1/ f

ki wajah se,

f

double karne par

P

1/2

ho jaata hai. Yahi "reciprocal relationship" ka poora matlab hai.

Contact mein lenses ke liye, focal lengths series mein resistors ki resistances ki tarah add hoti hain.

False. Powers (reciprocals) add hote hain, toh lenses in contact Resistors in Parallel ki tarah behave karte hain, jahan reciprocals add hote hain:

\frac{1}{F} = \frac{1}{f _{1}} + \frac{1}{f _{2}}

(Picture 3).

+ 5 D

power wale lens ki focal length

5 m

hoti hai.

False.

f = 1/ P = 1/5 = 0.2 m = 20 cm

, na ki

5 m

. Power, focal length in metres ka reciprocal hai.

Separation formula

P = P_{1} + P_{2} - d P_{1} P_{2}

kabhi bhi

P_{1} + P_{2}

se chhoti power nahi de sakta.

False (do converging lenses ke liye).

P_{1}, P_{2} > 0

ke saath term

- d P_{1} P_{2}

negative hai, toh do convex lenses ko alag karne par combined power

P_{1} + P_{2}

se kam ho jaati hai (Picture 4).

Lens combination ki magnifications, powers ki tarah add hoti hain.

False. Powers add hote hain (

P = P_{1} + P_{2}

) lekin magnifications multiply hoti hain:

m = m_{1} \times m_{2} \times \dots

, kyunki har stage pichli image ko scale karta hai, jaisa Magnification of Lenses mein dekha jaata hai.

"Dioptre" unit "per centimetre" ke barabar hai.

False.

1 D = 1 m^{- 1}

, metres mein focal length ke liye define kiya gaya hai. Centimetres mein focal length ko reciprocal lene se pehle metres mein convert karna zaroori hai.

Error dhundho

f = 20 cm

, toh

P = 1/20 = 0.05 D

Error centimetres use karna hai. Pehle convert karo:

20 cm = 0.2 m

, toh

P = 1/0.2 = 5 D

. Dioptre sirf metres ke liye define hai.

f = - 40 cm

wale concave lens ki

P = 1/40 = 2.5 D

hai."

Sign drop ho gaya.

f = - 0.4 m

, toh

P = 1/ (- 0.4) = - 2.5 D

. Concave lenses ki power negative hoti hai — minus hi unhe converging power cancel karne deta hai.

"Lenses in contact:

P_{1} = + 4 D

P_{2} = - 1 D

, toh

F = f_{1} + f_{2}

Tum focal lengths add nahi kar sakte. Powers add karo:

P = 4 + (- 1) = + 3 D

, phir

F = 1/3 m \approx + 33.3 cm

. Focal-length addition sirf resistor-series problems mein sahi hai, lenses mein nahi.

d

se separated do lenses:

P = P_{1} + P_{2} - d P_{1} P_{2}

mein

d = 10

diya."

Yahan

d

metres mein hona chahiye taaki dioptres ke saath match kare.

d = 0.10 m

use karo;

d = 10

daalne se correction term absurdly bada ho jaata hai.

"Contact mein lenses ke liye maine

P = P_{1} + P_{2} - d P_{1} P_{2}

mein

d = 0

use kiya, aur mujhe dar hai answer badal gaya."

Koi error nahi actually —

d = 0

par term

- d P_{1} P_{2}

vanish ho jaati hai aur formula

P_{1} + P_{2}

ban jaata hai. Yahi built-in sanity check hai jo confirm karta hai ki dono formulas agree karte hain.

"Ek biconvex lens ke liye maine lensmaker formula mein

R_{1} < 0

aur

R_{2} > 0

set kiya."

Signs ulte hain. Left→right enter karne wali light ke liye biconvex lens mein, pehli surface light ki taraf bulge karti hai toh

R_{1} > 0

, aur doosri surface door curve karti hai toh

R_{2} < 0

; dono terms tab add hote hain (Picture 2).

P = (n - 1) (1/ R_{1} - 1/ R_{2})

ki wajah se, lens ko paani mein use karne par (jahan surrounding

n

badhta hai) uski power same rehti hai."

Factor actually

(n_{lens} / n_{medium} - 1)

hai; paani mein ye zero ki taraf shrink ho jaata hai, toh lens bahut weaker ho jaata hai. Power relative refractive index par depend karti hai, Lensmaker's Equation se.

Why questions

Power

1/ f

kyun define ki jaati hai, sirf

f

directly use kyun nahi karte?

Height

h

par ek ray Picture 1 ka red triangle banati hai sides

h

aur

f

ke saath, toh

tan δ = h / f

; small angles ke liye

tan δ \approx δ

, jo bending-per-height

δ / h = 1/ f

deta hai. Wo deviation strength "power" ka naam hai, aur stacked lenses ki strengths add hoti hain.

Contact mein do lenses ki powers kyun add hoti hain, lekin unki focal lengths nahi?

Do thin-lens equations add karne par intermediate image distance

v_{1}

cancel ho jaata hai, aur

1/ F = 1/ f_{1} + 1/ f_{2}

bachta hai — reciprocals ka sum, yaani powers ka, focal lengths ka nahi (Picture 3).

Do convex lenses ko alag karne par unki combined power kyun kam hoti hai?

Lens 1 se bend hua ray gap

d

mein chhoti height tak drop ho jaata hai, toh lens 2 chhoti height bend karta hai aur kam contribute karta hai; correction

- d P_{1} P_{2}

(do positives ke liye negative) total se subtract hota hai (Picture 4).

Zyada dense glass ya zyada sharply curved surface zyada power kyun deta hai?

P = (n - 1) (1/ R_{1} - 1/ R_{2})

mein, bada

n

(n - 1)

ko bada banata hai aur chhota

∣ R ∣

curvature term ko bada banata hai — dono bending aur isliye power ko increase karte hain.

Opticians spectacles ke liye dioptres mein prescription kyun likhte hain, centimetres mein kyun nahi?

Dioptres directly add hote hain (eye power + spectacle power), toh ek defect correct karna simple addition hai, jabki focal lengths ke saath awkward reciprocal arithmetic karni padti — Defects of Vision dekho.

Microscope ya telescope objective aur eyepiece ke liye powers alag-alag kyun quote karte hain?

Kyunki unka behaviour power/magnification rules ke through combine hota hai; design inhi combination laws se banta hai, jaisa Microscope and Telescope mein use hota hai.

Separation formula mein

d

metres mein kyun hona chahiye, jabki wo "sirf ek distance" lagta hai?

Term

d P_{1} P_{2}

ki units hain

m \times D \times D = m \cdot m^{- 2} = m^{- 1} = D

; sirf metres hi units ko power jaisi banate hain jo

P_{1} + P_{2}

mein add ho sake.

Edge cases

Ek flat glass plate (dono surfaces flat,

R_{1}, R_{2} \to \infty

) ki power kya hoti hai?

1/ R_{1} - 1/ R_{2} \to 0

, toh

P = 0

aur

f \to \infty

. Ye kuch bhi bend nahi karta — zero power ka "lens", isliye windows light focus nahi karti.

Agar ek convex lens (

+ P

) ko equal magnitude ke concave lens (

- P

) ke contact mein rakha jaaye toh combined power kya hogi?

P_{net} = P + (- P) = 0

, effective flat plate milta hai: pair equal amounts converge aur diverge karta hai aur net bending zero hoti hai.

Separation formula mein, do convex lenses ke liye

d

ki kaun si value combined power zero kar deti hai?

P_{1} + P_{2} - d P_{1} P_{2} = 0

set karo,

d = (P_{1} + P_{2}) / (P_{1} P_{2}) = f_{1} + f_{2}

milta hai. Us spacing par pair zero-power (telescopic) system ki tarah act karta hai.

Jab kisi lens ki focal length zero approach karti hai (ideal ultra-strong lens) toh power kya hoti hai?

P = 1/ f \to \infty

. Practically koi real lens yahan nahi pahunchta, lekin formula dikhata hai ki arbitrarily short focal length ka matlab arbitrarily large power hai.

Agar kisi lens ki power exactly

0 D

ho, toh kya wo convex hai ya concave?

Dono nahi — ye effectively ek plane plate hai. Zero power positive (convex) aur negative (concave) ke beech boundary par hota hai, koi rays bend nahi karta.

Contact mein lenses ke liye, kya order (pehle kaun sa lens) net power change karta hai?

Nahi.

P_{1} + P_{2} = P_{2} + P_{1}

— addition commutative hai, toh lenses swap karne par same combined power (aur same

F

) milta hai.

Do lenses

f_{1} + f_{2}

ke barabar

d

se alag kiye gaye hon: is arrangement mein kya khaas hai?

Combined power zero hai (upar wale edge case se), toh parallel rays parallel hi enter aur exit karti hain — ye exactly afocal, telescope-jaisi configuration hai jo Microscope and Telescope mein use hoti hai.

Recall Ek-line self-test

Upar ke har answer ko cover karo; jo "trap" tum ek sentence mein justify nahi kar sakte, wo ek aisa concept hai jise exam se pehle dobara dekhna chahiye.

Connections

Thin Lens Equation — wo equation jo hum combined power derive karne ke liye add karte hain.
Lensmaker's Equation — $(n - 1) (1/ R_{1} - 1/ R_{2})$ traps ka source.
Magnification of Lenses — magnifications multiply hoti hain, powers add hoti hain.
Microscope and Telescope — afocal aur combination arrangements.
Resistors in Parallel — same reciprocal-addition mathematics.
Defects of Vision — prescriptions dioptres mein kyun hote hain.