Power of a lens, combination of lenses

2.5.7 · Physics › Optics

1. Power Kya Hai?

Aisa define kyon karte hain? Socho ek ray axis ke parallel hai aur lens se height $h$ par strike kar rahi hai. Refraction ke baad yeh axis ko focus par cross karti hai, jo $f$ door hai. Bending ka angle $δ$ satisfy karta hai (small angles ke liye): $tan δ \approx δ = \frac{h}{f}$

Toh bending angle per unit height $= \frac{δ}{h} = \frac{1}{f}$ . Yahi lens ki deviation produce karne ki strength hai — aur isi ko hum power kehte hain. Isliye $P = 1/ f$ arbitrary nahi hai; yeh "deviation per unit ray height" ka natural measure hai.

2. Touch Mein Rakhe Thin Lenses ka Combination

Scratch se Derivation (do thin lenses in contact):

Do thin lenses jinka focal length $f_{1}, f_{2}$ hai, ek doosre ko touch kar rahe hain. $u$ distance par rakhi object lens 1 se image banati hai.

Step 1 — Sirf Lens 1: $\frac{1}{v _{1}} - \frac{1}{u} = \frac{1}{f _{1}}$ Yeh step kyon? Yeh thin-lens equation pehle lens par apply ki gayi hai; $v_{1}$ uski image hai, jo lens 2 ke liye object ban jaati hai.

Step 2 — Sirf Lens 2: Image $I_{1}$ ( $v_{1}$ par) lens 2 ke liye object ki tarah kaam karti hai. Kyunki lenses contact mein hain, lens 2 ke liye object distance bhi $v_{1}$ hai: $\frac{1}{v} - \frac{1}{v _{1}} = \frac{1}{f _{2}}$ Yeh step kyon? Lens 2 jo bhi lens 1 ne produce kiya tha use leta hai aur phir refract karta hai; final image $v$ par hai.

Step 3 — Equations add karo: Unhe add karo taaki $v_{1}$ cancel ho jaaye: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f _{1}} + \frac{1}{f _{2}}$ Yeh step kyon? Intermediate $v_{1}$ internal bookkeeping hai — yeh vanish hona chahiye, object/image mein sirf equation reh jaaye.

Step 4 — Equivalent lens define karo: $\frac{1}{v} - \frac{1}{u} = \frac{1}{F}$ se compare karo: $\frac{1}{F} = \frac{1}{f _{1}} + \frac{1}{f _{2}} ⟺ P = P_{1} + P_{2}$

Figure — Power of a lens, combination of lenses

3. $d$ Distance Se Alag Rakhe Lenses

Extra term kyon? Jab lenses alag hote hain, lens 1 se bend hui ray $d$ distance travel karti hai aur lens 2 par alag height par land karti hai, jis height par yeh lens 1 mein enter ki thi usse alag. Lens 2 ki bending us height par depend karti hai, isliye bends cleanly add nahi hoti — correction term $- d P_{1} P_{2}$ iske liye account karta hai. Note karo jab $d = 0$ yeh $P_{1} + P_{2}$ reduce ho jaata hai ✓ (ek accha sanity check).

4. Worked Examples

5. Common Mistakes

Recall Feynman: 12-saal ke bacche ko samjhao

Ek lens ek slide ki tarah hai jo light ko ek jagah ki taraf dhakelta hai. Ek "strong" lens light ko bahut zor se dhakelta hai, toh woh jaldi milti hai — woh zor push uski power hai. Ek "weak" lens dheere dhakelta hai, light door milti hai. Hum push ko dioptres naam ke number se measure karte hain. Agar tum do lenses ko ek doosre se bilkul sata do, toh tumhe dono pushes ek ke baad ek milti hain — toh tum bas unke power numbers add kar do. Ek concave lens light ko bahar ki taraf dhakelta hai, isliye hum use ek minus push dete hain jo kuch plus cancel kar deta hai.

Flashcards

Lens ki power kya hai, aur uski unit?

P = 1/ f

(f metres mein); unit = dioptre (D) = m⁻¹.

Ek lens ka

f = - 25

cm hai. Uski power kya hai?

P = 1/ (- 0.25) = - 4 D

(concave, negative).

Contact mein lenses ke liye powers kyon add hoti hain?

Light successively bend hoti hai; do thin-lens equations add karne par intermediate image distance cancel ho jaati hai,

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

milta hai.

d

distance se alag do lenses ki combined power?

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

Contact mein lenses kis resistor configuration ki tarah behave karte hain?

Parallel resistors (reciprocals add hote hain).

Lens combination ka net magnification?

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

+ 5

D aur

- 2

D ke do lenses contact mein hain: net power aur focal length?

+ 3

F = + 33.3

cm.

Choti focal length ka matlab high power kyon hota hai?

Bending angle per unit ray height

= 1/ f

hai, isliye chota

f

light ko zyada strongly bend karta hai.

Connections

Thin Lens Equation — woh foundation jise hum ne add kiya $1/ F$ paane ke liye.
Lensmaker's Equation — $P = (n - 1) (1/ R_{1} - 1/ R_{2})$ deta hai.
Magnification of Lenses — net $m$ multiply hota hai.
Microscope and Telescope — lens combinations use karte hain (objective + eyepiece).
Resistors in Parallel — same reciprocal-addition mathematics.
Defects of Vision — opticians dwara prescribe ki gayi spectacle power dioptres mein.