Photoelectric effect — Einstein's explanation, work function: Visual walkthrough

2.3.2 · D2 · Physics › Modern Physics › Photoelectric effect — Einstein's explanation, work function

Step 1 — Ek photon, light ka ek coin hai

KYA. Waves ko ek pal ke liye bhool jaao. Light ko tiny identical coins ki ek stream ki tarah imagine karo. Har coin ko photon kehte hain. Ek given colour ke har coin mein bilkul same amount of energy hoti hai — kabhi thodi zyada nahi, kabhi thodi kam nahi.

KYUN. Experiments (parent ke table mein dekho) ne dikhaya ki electrons par light ka effect colour par depend karta hai, light ki quantity par nahi. Identical coins ki stream yeh perfectly capture karti hai: ek coin ki value colour se fix hoti hai; brightness bas kitne coins per second hain — yeh hai. Yeh idea Planck's quantum hypothesis & blackbody radiation se liya gaya hai.

PICTURE. Figure mein, light ka colour left → right change hota hai (red se blue). Har coin par likha number — uski energy — jaise-jaise hum bluer jaate hain, badhta jaata hai.

Figure — Photoelectric effect — Einstein's explanation, work function

Toh: bluer light ⇒ bigger $ν$ ⇒ har coin zyada worth ka.

Step 2 — Electron metal mein chipka hua hai

KYA. Metal ke andar electrons hain. Surface ke sabse paas wala electron sabse kamzor glue se pakda hua hai. Us sabse aasaan electron ko empty space mein nikaalने mein ek fixed amount of energy lagti hai jise hum ==work function $ϕ$ == (Greek "phi") kehte hain.

KYUN. Ek electron free float nahi karta — kuch cheez use bind karti hai. Koi bhi energy motion mein convert hone se pehle, escape toll chukana zaroori hai. Us toll ko $ϕ$ naam dene se hum honest bookkeeping kar paate hain.

PICTURE. Socho ki ek ball $ϕ$ depth ke ek well ke bottom mein hai. Ball tab tak ground par roll nahi kar sakti jab tak use well se uthaya na jaaye.

Step 3 — Ek coin, ek electron, aur receipt

KYA. Ek single photon ek single electron dwara absorb hota hai. Coin ki saari energy $h ν$ us ek hi electron mein jaati hai. Phir woh energy do cheezon par kharch hoti hai, order mein:

Pehle, escape toll $ϕ$ chukao.
Jo bhi bacha woh electron ki kinetic energy (uski speed) ban jaata hai.

KYUN. Yeh Energy conservation hai — receipt balance honi chahiye. Andar aaya paisa ( $h ν$ ) barabar hai kharch hua toll ( $ϕ$ ) aur bacha hua change (leftover motion) ke. Ek coin ek escape khareedta hai; aap do weak coins pool nahi kar sakte, kyunki doosra coin tab aata hai jab bahut der ho jaati hai — electron already relax ho chuka hota hai. Isliye ek below-toll coin kabhi bhi kuch nahi karta, kabhi nahi.

PICTURE. $h ν$ energy ka ek jar baahar girata hai. Pehla portion toll box $ϕ$ bhar deta hai; overflow "speed" cup mein spill hota hai.

Subscript "max" kyun? Humne receipt surface electron ke liye likhi — jo sabse sasta free hone wala hai. Ek deeper electron zyada bada toll deta hai, toh use kam change milta hai. Isliye surface electron sabse fast nikalta hai. Uski kinetic energy woh maximum hai jo koi bhi ejected electron rakh sakta hai — isliye $K E_{m a x}$ .

Step 4 — Einstein's equation mein rearrange karo

KYA. Toll ko receipt ki doosri side par le jaao.

KYUN. Hum ek aisa formula chahte hain jo practical sawaal ka jawaab de: "given colour ke liye, fastest electron kitna fast hai?" Isliye hum $K E_{m a x}$ isolate karte hain.

PICTURE. Receipt ek see-saw ki tarah tip karti hai: energy-in ek side par, toll subtract hoti hai, motion bacha rehta hai.

Step 5 — Threshold aur "kuch nahi hota" case

KYA. Pucho: sabse kamzor coin kaunsi hai jo phir bhi electron free karti hai? Jawaab: wo jिसकी energy exactly toll ke barabar ho, zero change chhode. Us colour ki frequency ==threshold frequency $ν_{0}$ == hai.

KYUN. Yeh degenerate, break-even case hai — "kaam karta hai" aur "kuch nahi karta" ke beech ki boundary. Iske neeche coin toll bhi nahi chuka sakta, toh $K E_{m a x}$ negative aayega, jo impossible hai. Negative answer nature ka tarika hai yeh kehne ka ki koi bhi electron nahi niklega.

PICTURE. Teen coins well mein girti hain:

bahut weak ( $h ν < ϕ$ ): coin toll box nahi bhar sakta → electron ruk jaata hai (crossed out).
exactly threshold ( $h ν = ϕ$ ): toll box bilkul bhara, zero overflow → electron barely escape karta hai, speed = 0.
strong ( $h ν > ϕ$ ): overflow spill hota hai → electron baahir nikalta hai.

Step 6 — Voltage se electron ko measure karo (stopping potential)

KYA. Hum $K E_{m a x}$ ruler se nahi padh sakte. Iske bajaye hum escape hue electron ko ek electrical hill par chadhaate hain aur exact hill height $V_{0}$ dhundhte hain jo sabse fast wale ko bhi rok de. Woh $V_{0}$ stopping potential hai.

KYUN. $K E_{m a x}$ invisible hai; voltage dial aur read karna aasaan hai. Ek charge $e$ (electron ka charge) ko voltage $V_{0}$ ke through "uphill" push karne mein $e V_{0}$ energy lagti hai. Jab hill bilkul itni tall ho ki fastest electron freeze ho jaaye, tab uski saari kinetic energy climbing mein kharch ho gayi. Toh $e V_{0}$ equals $K E_{m a x}$ — ek invisible quantity ka measurable stand-in.

PICTURE. Fastest electron $V_{0}$ height ka ramp chadh ke bilkul top par ruk jaata hai; uski saari motion-energy ab "climb-energy" ban gayi.

Step 7 — Woh straight line jisne Planck's constant measure kiya

KYA. Measured $V_{0}$ (upar) ko frequency $ν$ (across) ke against plot karo. Tumhe ek perfectly straight line milegi.

KYUN. Kyunki $V_{0} = \frac{h}{e} ν - \frac{ϕ}{e}$ ki shape (value) = (slope)(input) − (offset) hai. Magic yeh hai: slope $h / e$ mein sirf universal constants hain — toh har metal ek same steepness ki line deta hai. Alag-alag metals bas line ko sideways slide karte hain (alag toll $ϕ$ ⇒ alag $ν_{0}$ ). Us slope ko measure karne se tumhare haath $h$ aata hai — exactly aise hi Millikan ne Einstein ko confirm kiya (dekho Millikan oil drop & measurement of h).

PICTURE. Do metals, do lines, same slope, alag x-intercepts $ν_{0}$ . Steepness Planck's constant ko visible banaa rahi hai.

Recall Graph ka har feature kahan rehta hai?

$V_{0}$ vs $ν$ ka slope ::: $h / e$ — har metal ke liye identical. Line ka x-intercept ::: threshold frequency $ν_{0} = ϕ / h$ . Line ka y-intercept ::: $- ϕ / e$ . Bade $ϕ$ ka effect ::: line right shift hoti hai (bada $ν_{0}$ ), same slope.

Ek-picture summary

Upar sab kuch ek single image mein compress kiya gaya hai: coin ( $h ν$ ) andar aata hai, toll ( $ϕ$ ) subtract hota hai, change ( $K E_{m a x}$ ) speed ban jaata hai, voltage ramp ( $e V_{0}$ ) use measure karta hai, aur kai colours ka plot slope $h / e$ ki straight line trace karta hai.

Recall Poore walkthrough ki Feynman-style retelling

Light coins ki tarah aati hai. Har coin ki value uske colour se set hoti hai: bluer coins zyada worth ke hote hain ( $E = h ν$ ). Ek electron metal mein glue se chipka hai; use peelne ka sabse sasta wala ek fixed toll $ϕ$ leta hai. Jab ek coin hit karta hai, ek electron ek coin grab karta hai. Pehle toll deta hai; jo change bacha woh speed ban jaata hai ( $K E_{m a x} = h ν - ϕ$ ). Agar coin toll se kam worth ka hai (colour $ν_{0}$ se redder hai), toh koi nahi niklega — weak coins pool karna help nahi karta, kyunki har electron sirf ek hi coin grab karta hai. Fastest escapee kitna fast hai yeh jaanne ke liye hum use electrical hill chadhaate hain jab tak woh ruk na jaaye; hill height $V_{0}$ times charge $e$ uski energy ke barabar hai ( $e V_{0} = h ν - ϕ$ ). Us hill height ko colour ke against plot karo aur tumhe ek straight line milegi jिसकी steepness $h / e$ hai — dharte par har metal ke liye same. Us steepness se Millikan ne literally Planck's constant padhा.

Connections

Planck's quantum hypothesis & blackbody radiation — jahan $E = h ν$ (Step 1) janam leti hai.
Wave-particle duality of light — "coin" light ka particle face hai.
Energy conservation — Step 3 mein receipt.
Millikan oil drop & measurement of h — Step 7 line se $h$ padhna.
Compton effect · de Broglie wavelength — jahan particle picture aage jaati hai.