Thin lenses — lens equation, lens maker's equation: Question bank

2.5.6 · D5 · Physics › Optics › Thin lenses — lens equation, lens maker's equation

Poora page teen facts pe tikaa hai jo tumne parent note mein pehle se build kiye hain. Main unhe yahan number karta hoon taaki neeche ke traps precisely unhe point kar sakein:

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

Surface 1 par (air $\to$ glass, radius $R_{1}$ ) $u$ par rakha object ek intermediate image $v_{1}$ distance par banata hai:

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

Surface 2 par (glass $\to$ air, radius $R_{2}$ ) wahi $v_{1}$ object ban jaata hai, aur final image $v$ par milti hai:

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

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

$\frac{1}{v} - \frac{1}{u} = \frac{1}{f} (Gaussian lens equation)$

Neeche sab kuch inhi facts ke edges ko probe karta hai. Char schematics sabse tricky ideas ko anchor karti hain — reveals padhne se pehle inhe zaroor dekho.

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

Recall Picture ke saath edge case: infinity par rakha object

$f$ par focus hota hai Parallel rays ( $u \to - \infty$ , isliye $\frac{1}{u} \to 0$ ) lens equation ko $\frac{1}{v} = \frac{1}{f}$ tak collapse kar dete hain: sab focal point par milte hain. Yahi $f$ ko define karta hai — figure dekho.

True ya false — justify karo

"Converging lens hamesha light ko real image par converge karti hai."

False — object ko focus ke andar rakho (

∣ u ∣ < f

, yaani

- f < u < 0

), toh

\frac{1}{v} = \frac{1}{f} + \frac{1}{u}

hai (positive) + (larger-magnitude negative)

< 0

, isliye

v < 0

: ek virtual, erect, magnified image (magnifying-glass mode, figure s04 ki left branch).

"Diverging lens kabhi bhi real object ka real image nahi bana sakti."

True — real object ke liye

u < 0

aur

f < 0

, isliye

\frac{1}{v} = \frac{1}{f} + \frac{1}{u}

do negatives ka sum hai, jo hamesha

v < 0

deta hai: image virtual hoti hai, incoming side par.

"Lens maker's equation alag

f

deta hai depending on kaunsa face pehle light enter kare."

False — lens ko flip karne par

R_{1} \to - R_{2}

aur

R_{2} \to - R_{1}

ho jaata hai; substitute karne par,

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

, isliye

f

unchanged rehta hai. Thin lens ki ek hi focal length hoti hai.

"Symmetric biconvex lens mein

R_{1} = R_{2}

hota hai."

Sign mein False — magnitudes equal hote hain lekin

R_{1} = + R

(centre outgoing side par) aur

R_{2} = - R

(centre incoming side par, s01 dekho); unhe equal likhne se crucial minus drop ho jaata hai.

"Agar glass converging lens ko paani mein duba do, toh woh converging rehti hai same

f

ke saath."

Partly — woh usually converging rehti hai (glass ab bhi paani se slower hai) lekin

f

badh jaata hai kyunki

(n_{glass} - 1)

ki jagah chhota

(n_{glass} / n_{water} - 1)

aa jaata hai; kam relative slowing matlab weaker bending.

"Flat glass plate ka

f = 0

hota hai."

False —

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

ke saath dono curvature terms vanish ho jaate hain, isliye

\frac{1}{f} = 0

, matlab

f = \infty

(infinite focal length, zero power). Ye kuch bhi focus nahi karta.

"Magnification

m = v / u

ka negative hona matlab image chhoti hai."

False —

m

ka sign orientation encode karta hai (negative = inverted), jabki magnitude

∣ m ∣

size encode karta hai. Negative

m

phir bhi

∣ m ∣ > 1

ho sakta hai (bada).

"Lens derivation ke andar ka intermediate image ek real image hai jo glass mein floating hai."

False —

v_{1}

ek bookkeeping construct hai; thin-lens limit mein do surfaces coincide karte hain (figure s02), isliye

v_{1}

physically kabhi materialize nahi hota. Ye precisely isliye cancel hota hai kyunki ye kabhi real waypoint tha hi nahi.

Error dhundho

Koi lens equation ko

\frac{1}{v} + \frac{1}{u} = \frac{1}{f}

likhta hai, mirror form copy karke.

Galat sign — Cartesian lenses mein minus use hota hai:

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

. Ye mirror wala "+" sirf tab dikhta hai jab tum real object ka already-negative

u

substitute karo.

Ek student

\frac{1}{f} = (n - 1) (\frac{1}{R _{1}} + \frac{1}{R _{2}})

compute karta hai biconvex lens ke liye kyunki "dono faces same taraf curve hote hain."

Error — formula mein minus hai,

(\frac{1}{R _{1}} - \frac{1}{R _{2}})

, aur sign convention already

R_{2} < 0

banata hai; terms sirf us minus ke through add hote hain, plus likhne se nahi.

Ek student kehta hai "single-surface law mein main

n_{1} / u

aur

n_{2} / v

use karta hoon, lekin dono sides par

n = 1.5

hi lunga."

Error — do media alag hote hain (

n_{1}

= incoming medium,

n_{2}

= outgoing); wahi difference bending ka poora source hai. Har jagah ek

n

use karne se

\frac{n _{2} - n _{1}}{R} = 0

milta hai (koi refraction nahi).

Ek student lens ke surface 2 ko fresh scratch se object distance set karta hai,

v_{1}

ke equal nahi rakha.

Error — thin-lens assumption yahi hai ki dono surfaces ek plane par baithe hain (s02), isliye surface 1 ki image (

v_{1}

) hi surface 2 ka object hai usi same distance

v_{1}

par; gap introduce karne se cancellation trick toot jaati hai.

Ek student

i = α + ϕ

derive karta hai lekin

tan α = h / u

likhta hai

u

ko real object ke liye positive maanke.

Error — yahan

α

incoming ray ka small tilt hai aur

h

surface par ray height hai; real object ke liye

u < 0

, isliye

α = h / (- u)

, aur

u

ka sign bhoolne se us tilt ki direction ulat jaati hai.

Ek student claim karta hai ki paraxial rays aur marginal (edge) rays same point par focus karte hain, isliye aberrations exist nahi karti.

Error — derivation ne

sin θ \approx θ

use kiya, jo sirf axis ke paas valid hai; edge rays ise violate karte hain aur pehle focus karte hain, jo exactly spherical aberration hai (marginal rays paraxial focus se pehle axis cross karte hain).

Why questions

Equations (1) aur (2) add karne par

n / v_{1}

term kyun cancel hota hai?

(1) mein

v_{1}

par intermediate image (glass mein)

+ n / v_{1}

ke roop mein aati hai; (2) mein wahi

v_{1}

(same point, same medium — figure s02) object hai aur

- n / v_{1}

ke roop mein aata hai, isliye (1)+(2) add karne par dono terms subtract hokar zero ban jaate hain.

Light ka glass mein slow down karna zaruri kyun hai lens ke liye kuch bhi bend karne ke liye?

Bending

(n - 1)

factor se drive hoti hai; agar

n = 1

toh glass ki speed air ke barabar hai, isliye

(n - 1) = 0

aur

\frac{1}{f} = 0

— koi speed change nahi matlab koi refraction nahi (ye Snell's law hai equal indices ke saath).

Zyada curved surface (chhota

∣ R ∣

) light ko zyada strongly kyun bend karta hai?

Kyunki

1/ R

maker's equation mein appear karta hai: jaise

∣ R ∣

shrink hota hai,

1/∣ R ∣

badhta hai, isliye same

(n - 1)

se bada

1/ f

milta hai — tighter curves rays ko zyada steep normal dete hain bend karne ke liye.

Poore lens par directly attack karne ki jagah pehle single-surface law kyun derive karte hain?

Ek lens do refracting surfaces hain; ek curved face par bending ek baar derive karne se hum use dobara use kar sakte hain, jo exactly parent note ki building-block strategy hai.

Derivation mein

u \to - \infty

wala special case focal length kyun hai?

Parallel incoming rays infinitely door rakhe object ke correspond karti hain, isliye

\frac{1}{u} \to 0

aur lens equation

\frac{1}{v} = \frac{1}{f}

par collapse ho jaati hai — by definition

f

wahan hai jahan parallel light milti hai (figure s03).

Optical centre se guzarne wali ray undeviated kyun rehti hai,

m = v / u

dete hue?

Centre par dono lens faces locally parallel hain (thin flat plate ki tarah), isliye ray apni entry ke parallel exit karti hai negligible offset ke saath; similar triangles (

h

aur

h^{'}

heights over distances

u

aur

v

) phir

h^{'} / h = v / u

force karte hain.

Edge cases

Converging lens ke focal point par exactly rakha object (

u = - f

): image kahan hai?

\frac{1}{v} = \frac{1}{f} + \frac{1}{- f} = 0

, isliye koi finite image nahi — emerging rays exactly parallel hain aur image infinity par banti hai (figure s03 ka reverse).

Optical centre par rakha object (

u \to 0^{-}

): kya hota hai?

\frac{1}{v} = \frac{1}{f} + \frac{1}{u} \to - \infty

(

1/ u

dominate karta hai), isliye

v \to 0

: image object ke saath lens par coincide karti hai,

m \to 1

— degenerate, koi real separation nahi banta.

Virtual object (

u > 0

): converging rays lens se peeche ek point ki taraf ja rahi hain lens se hit hone se pehle — kya isse real image mil sakti hai?

Haan —

u > 0

aur converging

f > 0

ke saath,

\frac{1}{v} = \frac{1}{f} + \frac{1}{u}

do positives ka sum hai, isliye

v > 0

: outgoing side par ek real image,

f

se bhi paas banti hai. Ye figure s04 ki right branch hai aur lens combinations ke liye relevant regime hai, jahan ek lens converging rays doosre ko pass karta hai.

Plano-convex lens ka ek flat face hai (

R_{2} \to \infty

): kya woh phir bhi converge karti hai?

Haan — flat face

1/ R_{2} \to 0

contribute karta hai, isliye

\frac{1}{f} = (n - 1) / R_{1}

; ek curved face akela bhi finite positive

f

deta hai. Kisi bhi face par curvature kaafi hai.

Ek concavo-convex (meniscus) lens jahan

∣ R_{1} ∣ < ∣ R_{2} ∣

: converging hai ya diverging?

R > 0

-if-centre-on-outgoing-side rule (s01) use karke, yahan dono faces ka

R

same sign ka hai, isliye yeh

(\frac{1}{R _{1}} - \frac{1}{R _{2}})

ke sign par depend karta hai: zyada sharply curved face (chhota

∣ R ∣

, bada

1/∣ R ∣

) jeetta hai. Sharper convex

\to

converging; sharper concave

\to

diverging — net curvature decide karta hai, shape ka naam nahi.

Jaise

n \to 1

(glass index air ke paas jaaye), lens kya karta hai?

(n - 1) \to 0

isliye

\frac{1}{f} \to 0

f \to \infty

: lens optically invisible ho jaata hai, light seedha through transmit karta hai — woh limit jahan glass aur air indistinguishable hain.

Real object ke saath converging lens mein, image exactly kab real se virtual mein flip hoti hai?

Crossover

u = - f

par (object focus par):

∣ u ∣ > f

ke liye image real aur inverted hai;

∣ u ∣ < f

ke liye woh virtual aur erect ho jaati hai. Focal point do regimes ke beech boundary hai (figure s04 mein vertical asymptote).

Connections

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)

Question bank — Thin lenses — lens equation, lens maker's equation

True ya false — justify karo

Error dhundho

Why questions

Edge cases

Connections

Concept Map