Mirrors — plane, concave, convex; mirror equation 1 - v + 1 - u = 1 - f

2.5.2 · Physics › Optics

1. Teen mirrors (KYA HAIN)

2. Focus aadhe mein KYON hota hai: $f = R /2$ derive karna (Scratch se Derivation)

Maano incoming ray axis ke parallel hai. Yeh $M$ se takra ke axis ko $F$ par cross karti hai.

Incident ray axis ke parallel hai, toh normal $M C$ se angle $θ$ banati hai.
Law of reflection se, reflected ray bhi $M C$ se $θ$ banayegi.
Line $M C$ axis ko $C$ par kaatti hai, aur $θ$ angle banati hai (alternate angles, kyunki ray ∥ axis hai).

Toh triangle $M F C$ mein, angles $∠ M C F = θ$ aur $∠ F M C = θ$ hain ⇒ isosceles ⇒ $F M = F C$ .

Paraxial ray ke liye (axis ke paas, chota $θ$ ), $M$ pole $P$ ke paas hota hai, toh $F M \approx F P$ . Isliye: $F P = F C \Rightarrow F is the midpoint of P C \Rightarrow P F = \frac{1}{2} P C$

3. Sign convention (HOW — signs ko honest rakhna)

4. Mirror equation $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ derive karna (scratch se)

Hum ek concave mirror use karte hain jisme object C se aage hai, aur ek real, inverted image banti hai. Object ki height $h$ , image ki height $h^{'}$ .

Step 1 — Pole ray se magnification. Object ke top se pole $P$ tak ek ray jaati hai jo axis ke baare mein symmetrically reflect hoti hai (axis, $P$ par normal hai). Do right triangles (object aur image axis ke saath) similar hain: $\frac{h ^{'}}{h} = \frac{image distance}{object distance}$ Yeh step kyon? Pole ray axis se equal angles banati hai, toh triangles ke equal angles share hote hain ⇒ similar ⇒ sides ke ratios equal hain. Signs ke saath, $h^{'}$ aur $h$ opposite signs ke hain (image inverted), jo magnification deta hai: $m = \frac{h ^{'}}{h} = - \frac{v}{u}$

Step 2 — Focus ray aur similar triangles use karo. Axis ke parallel ray lo (height $h$ ); yeh $F$ se reflect hoti hai. Pole aur $F$ par bane triangles consider karo. Parallel ray triangle (height $h$ , $P$ se) aur image triangle ( $F$ se, height $h^{'}$ ) compare karo: $\frac{h ^{'}}{h} = \frac{P F - (image side)}{P F} \dots$ Ise proper signed lengths $u, v, f$ ke saath karne par (poora algebra, koi naya physics nahi) milta hai: $\frac{v - f}{f} = - \frac{h ^{'}}{h} = \frac{v}{u}$

Step 3 — Solve karo. $\frac{v - f}{f} = \frac{v}{u} \Rightarrow uv - u f = v f \Rightarrow uv = v f + u f$ $uv f$ se divide karo: $\frac{1}{f} = \frac{1}{u} + \frac{1}{v}$

Figure — Mirrors — plane, concave, convex; mirror equation 1 - v + 1 - u = 1 - f

5. Worked examples

6. Common mistakes (Steel-man + fix)

7. Active recall

Recall Forecast-then-verify checkpoints

Concave, object at C: image kahan hai? → C par bhi, real, inverted, same size.
Convex ke liye $f$ ka sign? → positive.
Plane mirror image size ke baare mein kya constant rehta hai? → hamesha object ke barabar.
Agar $m = + 2$ hai, toh image real hai ya virtual? → virtual (positive $m$ ⇒ erect ⇒ single mirror ke liye mirror ke peeche).

Ek mirror sirf kaunsa ek physical law follow karta hai?

Law of reflection: angle of incidence = angle of reflection (normal ke baare mein).

Focal length aur radius of curvature mein kya relation hai?

f = R /2

(paraxial rays).

Spherical mirror ke kisi bhi point par normal C ki taraf kyon directed hota hai?

Kyunki sphere ki radius hamesha uski surface ke perpendicular hoti hai, isliye woh centre of curvature ki taraf point karti hai.

Mirror equation batao.

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

with New Cartesian signs.

Mirrors ke liye magnification formula?

m = h^{'} / h = - v / u

f

ka sign: concave vs convex?

Concave

f < 0

, convex

f > 0

Plane mirror ki image properties?

Virtual, erect, same size, laterally inverted, equal distance behind.

Kisi bhi real object ke liye convex mirror kaisi image banata hai?

Hamesha virtual, erect, diminished.

Concave mirror, object focus ke andar (

∣ u ∣ < ∣ f ∣

): image kaisi?

Virtual, erect, magnified (mirror ke peeche).

m < 0

kya batata hai?

Image inverted hai (aur isliye single mirror ke liye real hai).

Mirror equation mein distances ke signs kyon zaroori hain?

Equation signed similar-triangle geometry se derive hoti hai; sirf magnitudes se galat/contradictory results milte hain.

Recall Feynman: ek 12-saal ke bachche ko samjhao

Ek mirror ek aisi wall hai jo light ko aise wapas phenkta hai jaise ball floor se bounce karti hai — woh utne hi angle par nikalta hai jis angle par aaya tha. Ek flat mirror bas tumhari same size ki copy banata hai. Chamach ka andar (concave) light ko curl karke ek focus naam ke point par squeeze karta hai — isliye yeh shaving ke liye bada chehra bana sakta hai. Chamach ki peeth (convex) light ko bahar faila deta hai, toh tum ek chota, chauda picture dekhte ho — car mirror ke liye perfect taaki zyada road dikh sake. Choti equation $\frac{1}{v} + \frac{1}{u} = \frac{1}{f}$ sirf yeh batane ka ek saaf tarika hai ki picture kahan landing karti hai, aur hum ise triangles draw karke aur yeh notice karke laaye ki woh same shape ke hain.

Connections

Reflection of Light — Laws & Normal
Refraction & Lenses — Lens Maker's Equation (same algebra, different signs)
Spherical Aberration (why $f = R /2$ is only paraxial)
Magnification & Image Formation
Optical Instruments — Telescopes & Microscopes
Total Internal Reflection