Optical instruments — human eye, simple microscope, compound microscope, telescope

2.5.8 · Physics › Optics

1. Human Eye — kyun hume instruments chahiye

$D$ kyun matter karta hai: ek unaided aankh kisi $h$ height ke object se jo sabse bada visual angle le sakti hai woh tab hota hai jab object near point par ho: $θ_{0} \approx tan θ_{0} = \frac{h}{D}$ Yahi $θ_{0}$ woh reference hai jiske against saare "magnifying powers" compare kiye jaate hain.

2. Angular magnification — master definition

3. Simple Microscope (magnifying glass)

Derivation — image at near point $D$ (maximum magnification)

Lens tumhe object ko paas rakhne deta hai (bada angle) aur ek magnified virtual image $D$ tak phenk deta hai jahaan aankh focus kar sakti hai.

Lens ke saath angle: image (ya object, paraxially) $θ = h / u$ subtend karta hai.
Naked-eye angle: $θ_{0} = h / D$ .

$M = \frac{θ}{θ _{0}} = \frac{h / u}{h / D} = \frac{D}{u}$

Lens equation $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ use karo. Image ko near point par rakho: $v = - D$ (virtual, same side). Tab $\frac{1}{- D} - \frac{1}{u} = \frac{1}{f} \Rightarrow \frac{1}{u} = - \frac{1}{D} - \frac{1}{f} \Rightarrow \frac{D}{u} = - (1 + \frac{D}{f}) .$ Magnitude lete hain: $M_{D} = 1 + \frac{D}{f}$

Yeh step kyun? $v = - D$ set karne se sabse strained lekin sabse sharp image milti hai, jo sabse bada $M$ deti hai.

Image at infinity (relaxed eye)

Object ko exactly focus par rakho, $u = - f$ , image infinity par jaati hai, aankh relaxed: $M_{\infty} = \frac{D}{f}$ Chhota kyun? Comfort ki ek cost hoti hai: $M_{D} - M_{\infty} = 1$ .

4. Compound Microscope

Figure — Optical instruments — human eye, simple microscope, compound microscope, telescope

Derivation

Objective (real intermediate image ki linear magnification): $m_{o} = \frac{v _{o}}{u _{o}} \approx \frac{L}{f _{o}}$ jahaan $L$ = tube length ≈ do foci ke beech ki distance (intermediate image eyepiece focus ke thoda andar banti hai). $\approx L / f_{o}$ kyun? Object $f_{o}$ ke thoda bahar hota hai, isliye $u_{o} \approx f_{o}$ aur $v_{o} \approx L$ .

Eyepiece intermediate image par simple microscope ki tarah kaam karta hai: $m_{e} = \frac{D}{f _{e}} (relaxed eye, image at \infty)$

Total: $M = m_{o} m_{e} = \frac{L}{f _{o}} \cdot \frac{D}{f _{e}}$ (Image at near point ke liye $m_{e} = 1 + D / f_{e}$ use karo.)

5. Telescope (astronomical, refracting)

Derivation (normal adjustment, final image at infinity)

Door ke object se angle $α$ par parallel rays objective ke focal plane mein height $h$ ki image banaati hain: $α \approx \frac{h}{f _{o}}$ Eyepiece, is image ko apne focus par rakhte hue, rays ko angle $β$ par parallel baahar bhejta hai: $β \approx \frac{h}{f _{e}}$ Angular magnification: $M = \frac{β}{α} = \frac{h / f _{e}}{h / f _{o}} = \frac{f _{o}}{f _{e}}$ Lamba objective, short eyepiece kyun? Bada $f_{o} / f_{e}$ = badi magnification.

Tube length (normal adjustment): $L = f_{o} + f_{e}$ (do foci coincide karte hain).

Final image at near point ke liye: $M = \frac{f _{o}}{f _{e}} (1 + \frac{f _{e}}{D})$ .

6. Big-picture comparison (80/20 table)

Instrument	$M$ (relaxed eye)	$M$ (near point)	Tube length
Simple microscope	$D / f$	$1 + D / f$	—
Compound microscope	$\frac{L}{f _{o}} \frac{D}{f _{e}}$	$\frac{L}{f _{o}} (1 + \frac{D}{f _{e}})$	$\approx v_{o} + f_{e}$
Telescope	$f_{o} / f_{e}$	$\frac{f _{o}}{f _{e}} (1 + \frac{f _{e}}{D})$	$f_{o} + f_{e}$

Woh 20% jo 80% deta hai: microscopes mein short focal lengths chahiye aur woh multiply karte hain; telescopes mein lamba objective / short eyepiece chahiye aur woh angular magnification use karte hain.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek chhoti si chinti pakdo. Uski taangein dekhne ke liye use aankh ke paas laate ho — lekin bahut paas laane par blur ho jaati hai. Magnifying glass ek "cheat" hai: yeh chinti ko super paas rakhne deti hai (taaki badi lage) jabki ek blurry-door image tumhari aankh tak bhejti hai taaki aankh focus kar sake. Microscope sirf do magnifying glasses ki ek line hai — pehla midair mein ek badi picture banata hai, doosra us picture ko magnify karta hai, toh bigness multiply ho jaati hai. Telescope ulta trick hai: Chaand bahut bada hai lekin door hai, toh ek bada front lens "distance ko shrink" karke ek chhoti clear image banata hai, aur ek chhota back lens us image ka angle tumhari aankh mein mota kar deta hai.

Flashcards

Instruments ke liye angular (linear nahi) magnification kyun use karte hain?

Retinal image ka size aankh par subtend hone wale angle par depend karta hai, object ki real size par nahi.

Normal aankh ke liye least distance of distinct vision

D

25 cm (near point).

Simple microscope magnification, image at near point?

M = 1 + D / f

Simple microscope magnification, relaxed eye (image at ∞)?

M = D / f

Near-point magnification relaxed se exactly 1 zyada kyun hoti hai?

M_{D} - M_{\infty} = (1 + D / f) - D / f = 1

Compound microscope total magnification (relaxed eye)?

M = \frac{L}{f _{o}} \cdot \frac{D}{f _{e}}

(objective × eyepiece).

Compound microscope mein magnifications multiply kyun hoti hain, add kyun nahi?

Eyepiece pehle se magnified intermediate image ko magnify karta hai, isliye effects compound hote hain.

Telescope angular magnification (normal adjustment)?

M = f_{o} / f_{e}

Normal adjustment mein telescope ki tube length?

L = f_{o} + f_{e}

Telescope

f_{o} / f_{e}

use karta hai

D / f

nahi, kyun?

Uska object infinity par hota hai; reference angle real angle

α = h / f_{o}

hota hai,

h / D

nahi.

High microscope magnification ke liye focal lengths kaisi honi chahiye?

Short (

f_{o}

aur

f_{e}

chhote).

High telescope magnification ke liye lenses kaise hone chahiye?

Lamba objective

f_{o}

, short eyepiece

f_{e}

Connections

Lens equation and sign conventions
Linear vs angular magnification
Resolving power and diffraction limit
Defects of vision — myopia, hypermetropia
Reflecting telescope (Cassegrain) vs refracting
Power of a lens and dioptres