Power of a lens, combination of lenses: Visual walkthrough

2.5.7 · D2 · Physics › Optics › Power of a lens, combination of lenses

Hum ek prerequisite par rely karenge: Thin Lens Equation. Baaki sab hum yahan build karenge.

Step 1 — Ek single ray aur ek lens actually kya karte hain

WHAT. Lenses combine karne se pehle, hamein agree karna hoga ki ek lens ek ray ke saath kya karta hai. Figure dekho: ek ray central axis ke parallel travel karti hai (horizontal dotted line) usse $h$ height par. Yeh lens se takraati hai aur bend hoti hai — direction change hoti hai — aur uske baad yeh axis ko ek special point par cross karti hai jise focus kehte hain, jo $f$ door hota hai.

WHY. "Power" ke baare mein har claim ek bending ke baare mein claim hai. Toh humein ek clean number chahiye "is ray ne kitna bend kiya?" ke liye. Natural number hai bending angle $δ$ — ray ki original direction aur uski nayi direction ke beech ka angle.

PICTURE.

Figure — Power of a lens, combination of lenses

Figure mein chhota right triangle padho. Uski horizontal side $f$ hai (lens se focus tak), vertical side $h$ hai (ray ki entry height), aur $δ$ focus par angle hai.

Angle $δ$ ke liye $tan δ = h / f$ kyun satisfy hota hai? Kyunki right triangle mein angle ka $tan$ defined hota hai opposite over adjacent ke roop mein — yahan $δ$ ke opposite side $h$ hai, aur adjacent side $f$ hai. Thin lenses mein rays axis ke paas rehti hain, toh angles tiny hote hain, aur tiny angles ke liye $tan δ \approx δ$ . Isliye:

$kitna bend hua δ \approx height \div focal length \frac{h}{f}$

Bend per unit height hai $δ / h = 1/ f$ . Woh number — deviation produced per unit ray height — exactly wahi hai jise hum lens ki power kehte hain.

Step 2 — Do lenses in contact setup karna

WHAT. Ab ek doora thin lens pehle wale ke bilkul saath rakh do, unke beech zero gap ke saath. Object $O$ left mein baitha hai. Hum dono lenses se light pass hone ke baad final image chahte hain.

WHY. "In contact" yahi poora trick hai. Kyunki unke beech koi space nahi hai, ek ray jo lens 1 se nikalti hai woh lens 2 mein essentially same height par enter karti hai — sideways drift karne ki koi jagah nahi hoti. Yahi shared height hai jo bends ko cleanly add hone degi. (Jab baad mein hum gap allow karenge, woh assumption toot jaata hai — Step 8 dekho.)

PICTURE.

Figure mein key idea: intermediate image $I_{1}$ lens 2 ke liye object hai. Light ko nahi pata ki ek lens khatam ho gayi; woh bas chalti rehti hai. Lens 1 ne light ko jahan "aim" kiya woh wahi jagah hai jahan se lens 2 sochta hai light aa rahi hai.

Step 3 — Lens 1 akele act karta hai

WHAT. Thin Lens Equation ko sirf lens 1 par apply karo. Object $u$ par hai; lens 1 jo image form karta hai woh $v_{1}$ par hai.

WHY. Hum ek waqt mein ek lens handle karte hain kyunki hum ek single lens ka law pehle se jaante hain. Divide-and-conquer: lens 1 solve karo, phir uska output lens 2 mein feed karo.

PICTURE.

$lens 1 image kahan bhejtaa hai \frac{1}{v _{1}} - object kahan hai \frac{1}{u} = lens 1 ki bending power \frac{1}{f _{1}}$

Har term:

$1/ v_{1}$ — intermediate image distance ka reciprocal.
$1/ u$ — object distance ka reciprocal.
$1/ f_{1}$ — lens 1 ki "strength" (yahi hai $P_{1}$ ).

Yahan kuch naya nahi — yeh raw thin-lens law hai, bas subscript 1 ke saath label kiya gaya hai taaki hum track kar sakein. Output hai $v_{1}$ , $I_{1}$ ki location, figure mein orange point ke roop mein drawn.

Step 4 — Lens 2, lens 1 ke image par act karta hai

WHAT. Ab lens 2 par thin-lens equation apply karo. Uska object $I_{1}$ hai, jo $v_{1}$ distance par baitha hai; uska image final image hai $v$ par.

WHY. Kyunki lenses touch karte hain, lens 2 ke liye object distance wahi $v_{1}$ hai jo lens 1 se nikla — gap nahi matlab us distance mein koi change nahi. Yeh woh moment hai jahan "in contact" ka faida milta hai.

PICTURE.

$final image \frac{1}{v} - lens 2 ke liye object = I_{1} \frac{1}{v _{1}} = lens 2 ki bending power \frac{1}{f _{2}}$

Har term:

$1/ v$ — final image distance ka reciprocal (jo hum ultimately chahte hain).
$1/ v_{1}$ — $I_{1}$ ki distance ka reciprocal; wahi number jo Step 3 mein aaya tha.
$1/ f_{2}$ — lens 2 ki strength (yeh hai $P_{2}$ ).

Plot twist notice karo already: $v_{1}$ yahan minus sign ke saath appear hota hai, lekin Step 3 mein plus sign ke saath appear hua tha. Yeh mismatch accident nahi hai — exactly yahi hai jo hum aage exploit karte hain.

Step 5 — Dono equations add karna: intermediate image gayab ho jaata hai

WHAT. Step 3 aur Step 4 ki equations add karo. $v_{1}$ terms hain $+ \frac{1}{v _{1}}$ aur $- \frac{1}{v _{1}}$ — yeh cancel ho jaate hain.

WHY. Intermediate image $v_{1}$ sirf internal bookkeeping tha. Lens pair ka user usse kabhi measure nahi karta — woh sirf object $u$ aur final image $v$ dekhta hai. Toh ek correct final law mein $v_{1}$ nahi hona chahiye. Equations add karna woh algebraic act hai jo use erase karta hai.

PICTURE.

$(\frac{1}{v _{1}} - \frac{1}{u}) + (\frac{1}{v} - \frac{1}{v _{1}}) = \frac{1}{f _{1}} + \frac{1}{f _{2}}$

Do $\frac{1}{v _{1}}$ boxes annihilate ho jaate hain (figure mein orange mein crossed out dikhaye gaye hain), reh jaata hai:

$sirf object aur final image \frac{1}{v} - \frac{1}{u} = dono strengths, summed \frac{1}{f _{1}} + \frac{1}{f _{2}}$

Yeh already ek single thin-lens equation jaisa lagta hai — ek imaginary "equivalent" lens ke liye. Hum usse aage naam dete hain.

Step 6 — Equivalent lens ko naam dena: powers add hote hain

WHAT. Step 5 ke result ko focal length $F$ wale single equivalent lens ki thin-lens equation se compare karo: $\frac{1}{v} - \frac{1}{u} = \frac{1}{F} .$ Dono ko match karne se hamara result milta hai.

WHY. Agar pair exactly ek lens ki tarah behave karta hai, toh hum us ek lens ko naam $F$ aur power $P = 1/ F$ de sakte hain. Physics ko nahi pata ki glass ke ek piece hain ya do — sirf net bending matter karti hai.

PICTURE.

$\frac{1}{F} = \frac{1}{f _{1}} + \frac{1}{f _{2}} ⟺ P = P_{1} + P_{2}$

$F$ — us single lens ki focal length jo pair ko replace kar sake.
$P = 1/ F$ — uski power. Kyunki $P_{1} = 1/ f_{1}$ aur $P_{2} = 1/ f_{2}$ , focal lengths ka reciprocal relation plain addition of powers mein convert ho jaata hai. Yahi poora reason hai ki power itni beloved quantity hai: $f$ ke awkward reciprocals simple sums ban jaate hain.

Step 7 — Har sign: convex, concave, aur cancelling case

WHAT. Addition $P = P_{1} + P_{2}$ mein signs hote hain. Ab hum saare cases walkthrough karte hain taaki koi scenario aapko surprise na kare.

WHY. Ek concave lens ki negative power hoti hai, aur sum isliye shrink, vanish, ya sign flip kar sakta hai. Signs skip karna sabse common error hai, toh hum har case ko apna picture dete hain.

PICTURE.

Teen cases, sab drawn:

Dono convex ( $+ 5 D$ aur $+ 3 D$ ): $P = + 8 D$ . Zyada strong converging — shorter combined focal length. (Teal panel.)
Convex + weak concave ( $+ 5 D$ aur $- 2 D$ ): $P = + 3 D$ . Concave lens subtract karta hai, convergence ko weaken karta hai. (Orange panel.)
Exactly cancelling ( $+ 4 D$ aur $- 4 D$ ): $P = 0 \Rightarrow F = \infty$ . Pair kuch bhi bend nahi karta — parallel rays parallel hi nikalti hain. Yeh degenerate case hai: "equivalent lens" effectively ek flat window hai. (Plum panel.)

Step 8 — Gap case: drift term $d P_{1} P_{2}$ KAHAAN se aata hai

WHAT. Lenses ko ek doosre se $d$ distance par karo. Rule mein ek correction term aa jaata hai: $P = P_{1} + P_{2} - d P_{1} P_{2} .$ Hum ab us term ko derive karte hain, sirf quote nahi karte.

WHY. Hamari contact derivation is baat par rely karti thi ki ray lens 2 ko same height par hit kare jis par usne lens 1 ko chhoda tha (Step 2). Gap ke saath, ray — jo lens 1 se already bent ho chuki hai — lens 2 tak pahunchne se pehle ek nayi height par drift karti hai. Lens 2 ki bend height par depend karti hai, toh total bend change ho jaata hai. Aao Step 1 se small-angle bending law se us height ko track karein.

PICTURE.

Drift, teen short lines mein. Ek ray lo jo axis ke parallel aa rahi hai lens 1 par height $h_{1}$ par (toh uski incoming slope $0$ hai).

Lens 1 ise bend karta hai. Step 1 se, bend angle hai $δ_{1} = h_{1} / f_{1} = h_{1} P_{1}$ . Toh lens 1 ke baad ray axis ki taraf neeche slope $- h_{1} P_{1}$ ke saath slope karti hai.
Yeh gap $d$ ke paas drift karti hai. Us slope ke saath horizontal distance $d$ travel karte hue, uski height change hoti hai. Lens 2 tak pahunchte pahunchte nayi height hai $h_{2} = h_{1} - d (h_{1} P_{1}) = h_{1} (1 - d P_{1}) .$ Term $d P_{1}$ exactly woh fractional drop in height hai jo gap ki wajah se hoti hai — koi gap nahi ( $d = 0$ ) matlab $h_{2} = h_{1}$ , Step 2 recover hota hai.
Lens 2 phir bend karta hai. Lens 2 bend $δ_{2} = h_{2} P_{2}$ add karta hai. Ray ka total downward bend hai $δ_{1} + δ_{2} = h_{1} P_{1} + h_{2} P_{2} = h_{1} P_{1} + h_{1} (1 - d P_{1}) P_{2} .$

Ab total bend ko entry height $h_{1}$ se divide karo effective power padne ke liye (bend per unit height, hamari Step-1 definition of $P$ ): $P = \frac{δ _{1} + δ _{2}}{h _{1}} = P_{1} + (1 - d P_{1}) P_{2} = clean sum P_{1} + P_{2} - drift correction d P_{1} P_{2} .$

$- d P_{1} P_{2}$ precisely woh height-drop factor $d P_{1}$ hai jo second lens ki power $P_{2}$ par act kar raha hai.
$d = 0$ set karo aur yeh vanish ho jaata hai, $P = P_{1} + P_{2}$ restore hota hai. Woh sanity check right mein built in hai. ✓

Ek-picture summary

Ek figure, poori kahani: object → lens 1 (bend $P_{1}$ , $I_{1}$ banata hai) → lens 2 (bend $P_{2}$ , $I_{1}$ ko object ki tarah use karta hai) → final image. Intermediate $v_{1}$ cancel ho jaata hai jab dono thin-lens equations add ki jaati hain, ek single equivalent lens chodta hai jisme $P = P_{1} + P_{2}$ — aur, jab gap khulta hai, ray drift karti hai aur hum $d P_{1} P_{2}$ subtract karte hain.

Recall Feynman retelling — plain words mein walkthrough

Ek single light ray imagine karo jo flat aur middle line ke upar high aa rahi hai. Ek lens use pakadta hai aur ek jagah ki taraf neeche tip karta hai — focus. Yeh kitna hard tip karta hai woh lens ki power hai: strong lens, sharp tip, close focus; weak lens, gentle tip, far focus. Power bas $1/ f$ hai jisme $f$ metres mein hai.

Ab ek doosra lens bilkul pehle wale ke saath lagao. Pehla lens light ko kuch beech wali jagah par aim karta hai. Light travel bhi nahi kar paati, doosra lens use pakad leta hai aur phir tip karta hai. Kyunki dono lenses touch kar rahe hain, light dono ke liye same height par hai — toh dono tips ek doosre par pile on ho jaate hain. Do tips = ek bada tip = dono powers added. Jab hum har lens ke liye neat lens-law likhte hain aur dono lines add karte hain, woh beech wali jagah literally paper par cancel ho jaati hai, prove karta hai ki woh real answer ka kabhi part nahi thi.

Signs matter karte hain: convex lens light ko inward tip karta hai (plus power), concave lens use outward tip karta hai (minus power). Ek plus aur ek minus add karo aur woh fight karte hain — itna bada minus plus ko completely cancel kar deta hai, ek aisi window deta hai jo kuch bhi bend nahi karta. Finally, agar lenses ko gap $d$ se alag karo, toh already-tipped light lens 2 tak pahunchne se pehle thodi si lower height par drift karti hai. Lens 2 ek chhoti ray ko tip karta hai, toh uska contribution $d P_{1}$ fraction se shrink ho jaata hai. Us shrink ko lens 2 ki power se multiply karo aur tumhe exactly woh correction milta hai jo hum subtract karte hain, $d P_{1} P_{2}$ — ek correction jo politely disappear ho jaata hai jis moment lenses phir se touch karte hain.

Connections

Thin Lens Equation — woh single law jo humne do baar apply kiya aur add kiya.
Lensmaker's Equation — jahan se har individual power aati hai.
Magnification of Lenses — net magnification multiply hoti hai jab powers add hoti hain.
Resistors in Parallel — focal lengths parallel resistances ko mimic karti hain; powers conductances ko mimic karti hain.
Microscope and Telescope — separated lens pairs se bane real instruments.
Defects of Vision — corrective lens powers aankhon ki apni power ke saath combine hoti hain.