Sign convention for mirrors and lenses: Foundations

2.5.3 · D1 · Physics › Optics › Mirrors aur lenses ke liye sign convention

Ye page ye assume karta hai ki aap kuch bhi nahin jaante. Parent note Sign convention padhne se pehle, aapko us mein aane wale har symbol par complete command chahiye. Hum unhe ek-ek karke build karte hain, ek dusre ke upar.

0. Woh number line jis par sab kuch rehta hai

Optics mein sab kuch ek horizontal line ke saath measure kiya jaata hai jise principal axis kehte hain. Isse bilkul waise sochein jaise school mein aapne $x$ -axis draw ki thi, lekin ye mirror ya lens ke centre se hokar flat lete hue jaati hai.

Figure — Sign convention for mirrors and lenses

Figure s01 — Ek horizontal arrow (principal axis) jiske saath lal rang ka pole origin $x = 0$ ke roop mein marked hai. Baayein taraf ke tick marks $- 3, - 1$ padhte hain; daayein taraf ke ticks $+ 1, + 3$ padhte hain. Ye dikhata hai ki optics mein "distance" asliyat mein ek shared ruler par ek signed position hai.

Isko kyon zaroori hai? Kyunki akeli "distance" ambiguous hai — 30 cm kis taraf? Ek axis aur ek origin fix karke, "30 cm baayein" aur "30 cm daayein" do alag numbers ban jaate hain. Yahi poora khel hai.

1. Origin — pole aur optical centre

Har ruler ko ek zero chahiye. Hamara zero mirror/lens par hi hota hai.

2. Positive direction kaunsa hai? Incident light

Ab hume decide karna hai ki kaunsa direction $+$ count hoga. Convention us direction ko choose karta hai jis mein incident light travel kar rahi hai, jise hum hamesha left → right draw karte hain.

Figure s02 — Teen lal arrows (incident light) left→right travel karte hue ek black mirror/lens line ki taraf. Object dot upstream left par baitha hai ( $u < 0$ ); right ki taraf downstream region ko positive label kiya gaya hai. Ise aise padhein: "jis taraf light jaati hai woh $+$ hai; object ke peeche $-$ hai."

Ye kyun, aur "mere daayein = positive" kyun nahin? Kyunki mirror ke liye light wapas bounce hoti hai, isliye "downstream" actually mirror ke peeche hoti hai. $+$ ko light se jodhna, kisi fixed left/right se nahin, yahi hai jo ek rule ko mirrors aur lenses dono ko cover karne deta hai.

3. Object distance $u$ — real AUR virtual objects

Normal ("real") object ke liye hum ise light ke baayein, upstream rakhhte hain. Left = light ke ulta = negative.

4. Image distance $v$ — answer coordinate

Khoobsurti ye hai: $v$ ka sign aapko image ki nature free mein bata deta hai.

Figure s03 — Do side-by-side panels. LEFT (mirror): baayein object arrow, uski real inverted image (laal) bhi baayein, isliye $v < 0$ . RIGHT (lens): baayein object arrow, uski real inverted image (laal) daayein, isliye $v > 0$ . Tasveer dikhati hai ki ek hi word "real" mirror aur lens ke liye opposite signs kyun deta hai — kyunki mirror light ko wapas reflect karta hai, lens use through bhejta hai.

Mirror: rays reflect hokar wapas baayein jaati hain. Baayein image (real) $\Rightarrow v < 0$ ; mirror ke peeche image (virtual) $\Rightarrow v > 0$ .
Lens: light daayein pass through hoti hai. Daayein image (real) $\Rightarrow v > 0$ ; baayein image (virtual, object ki hi side) $\Rightarrow v < 0$ .

Real vs virtual images ki language ke liye, dekhein Real and virtual images.

5. Radius of curvature $R$ aur focal length $f$

Curved mirror/lens ek sphere ka ek tukda hota hai. Us sphere ka centre curvature ka centre $C$ hai; pole se $C$ tak ki distance radius of curvature $R$ hai.

Figure s04 — Ek concave mirror jisme curvature ka centre $C$ aur pole $P$ hai. Ek laal ray axis ke parallel aati hai, mirror par point $M$ par hit karti hai, aur focus $F$ se reflect hoti hai. Line $C M$ ek radius (ek "normal") hai, aur do marked equal angles dikhate hain ki reflected ray axis ko exactly $P$ aur $C$ ke beech mein kyun cross karti hai — yaani $P F = \frac{1}{2} P C$ , matlab $f = R /2$ .

Kyunki $F$ aur $C$ bhi usi number line par positions hain, unhe bhi signs milte hain:

6. Heights $h$ aur $h^{'}$ — upar-neeche ki axis

Objects aur images sirf points nahin hain; unki height hoti hai. Height ko axis ke perpendicular measure karte hain.

7. Magnification $m$ — aur mirror ka minus sign kahan se aata hai

Figure s05 — Ek concave mirror jisme object arrow (height $h$ , upar) distance $u$ par baayein hai aur inverted image arrow (height $h^{'}$ , neeche, laal) distance $v$ par hai. Chief ray object ki tip se pole $P$ tak jaati hai aur reflect hoti hai. Do shaded right triangles — object-tip–axis–pole aur image-tip–axis–pole — similar hain (ye $P$ par angle share karte hain aur dono mein right angle hai). Yahi similarity ratio produce karti hai, aur axis ke neeche image ka flip hi minus sign produce karta hai.

$m < 0$ → image inverted (ulti).
$m > 0$ → image erect (seedhi).
$∣ m ∣ > 1$ magnified, $∣ m ∣ < 1$ diminished.

Zyada detail ke liye Magnification in optics dekhein.

8. Reciprocal notation $\frac{1}{v}$ — kyun fractions, aur lens mein minus kyun

Parent ke formulas $\frac{1}{v}$ , $\frac{1}{u}$ , $\frac{1}{f}$ use karte hain — distances ke reciprocals (one-over).

Intuition Mirror ko

$+$ kyun milta hai lekin lens ko $-$ Mirror light ko wapas reflect karta hai jis taraf se wo aayi thi, isliye object side aur image side same side hoti hai — object aur (real) image distances ek hi sign share karte hain, aur signed formula $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ ban jaata hai. Lens light ko seedha through transmit karta hai, isliye real image object ke opposite side par land karti hai — object aur image distances opposite signs carry karte hain, aur woh flip plus ko minus mein badal deta hai: $\frac{1}{v} - \frac{1}{u} = \frac{1}{f}$ . Ye same physics hai (reciprocals balancing); sirf image kaunsi side land karta hai ye alag hai. Poore step-by-step derivations Mirror formula derivation aur Lens formula and lensmaker's equation mein hain; lens case Refraction at spherical surfaces mein bhi grounded hai.

Is balance ko step-by-step Mirror formula derivation aur Lens formula and lensmaker's equation mein build hote dekhoge.

Ye sab topic mein kaise feed karta hai

Equipment checklist

Cover the right side; can you say each before revealing?

Sabhi optical distances kis line ke saath measure ki jaati hain?

Principal axis,

x

-axis ki tarah use ki gayi.

Us number line ka zero (origin) kahan hai?

Pole par (mirror) ya optical centre par (lens).

Positive direction kaunsa choose kiya gaya hai?

Jis direction mein incident light travel karti hai — left se right.

Real object vs virtual object ke liye

u

ka sign?

Real object

u < 0

(left/upstream); virtual object

u > 0

(right/downstream, converging rays se).

Virtual object kya hota hai?

Woh point element ke peeche ki taraf jis taraf light pahunche se pehle hi converge ho rahi hoti hai — ye agale element ke liye object ki tarah kaam karta hai, jisme

u > 0

hota hai.

v

ka sign aapko kya batata hai?

Image ki nature/side (real vs virtual, front vs behind).

Concave mirror vs convex lens ka focal length sign?

Concave mirror

f < 0

; convex lens

f > 0

— geometry, "converging" nahin.

f = R /2

kyun hai?

Reflection point tak radius normal hai; reflection ka law triangle

C M F

ko isosceles banata hai, isliye

F

P

C

ke exactly halfway baitha hai.

Mirror ke

m

mein minus kyun hai lekin lens ke

m

mein nahin?

Similar triangles ratio dete hain; mirror ke liye

u, v

ek hi sign share karte hain, lens ke liye opposite hain, isliye mirror ko

m = - v / u

chahiye aur lens ko

m = v / u

Lens formula mein

-

kyun hai jahan mirror mein

+

hai?

Mirror reflect karta hai (image object ki same side), lens transmit karta hai (image opposite side), jo

u

ke relative

v

ka sign flip kar deta hai.

Mirror ke liye

f

aur

R

ka relation?

f = R /2