Photoelectric effect — Einstein's explanation, work function

2.3.2 · Physics › Modern Physics

YEH topic exist kyun karta hai?

Jab light kisi metal se takrati hai, toh electrons bahar nikal sakte hain. Classical wave theory ne aise cheezein predict ki thi jo experiments se bilkul match nahi karin:

Classical wave prediction	Actually kya hota hai
Zyada bright light ⇒ zyada energetic electrons	Zyada bright light ⇒ zyada electrons, same max energy
Kaafi time tak wait karo toh koi bhi frequency kaam karegi	Threshold frequency se neeche, kuch nahi hoga, kabhi bhi
Energy "accumulate" karne ke liye kuch time delay hona chahiye	Emission instantaneous hoti hai (< $1 0^{- 9}$ s)

Waves ki yeh failure dekh ke Einstein (1905) ne Planck ke quanta ko seriously liya.

KEY TERMS kya hain?

Einstein ne equation kaise derive ki (scratch se)

Step 1 — Ek photon ki energy. Planck/Einstein: frequency $ν$ ki light energy ke packets mein aati hai $E_{photon} = h ν .$ Yeh step kyun? Hum maan rahe hain ki light quantized hai; yeh woh bold postulate hai jise hum experiment ke against test karte hain.

Step 2 — Ek electron ke liye energy bookkeeping. Ek photon ek electron absorb karta hai. Woh energy do "bills" pay karti hai:

Escape cost = work function $ϕ$ .
Jo bacha woh kinetic energy ban jaata hai.

Conservation of energy se: $h ν = ϕ + K E_{m a x}$ $K E_{m a x}$ kyun, sirf $K E$ kyun nahi? Metal ke andar gehre electrons ko surface tak pahunchne ke liye extra energy chahiye, toh woh slower bahar nikalte hain. Seedha surface par wala electron (binding $= ϕ$ ) maximum KE leta hai.

Step 3 — Measurable stopping potential se connect karo. Hum sabse fast electron ko potential $V_{0}$ se rokते hain, toh electrical work $e V_{0}$ uski KE ke barabar hai: $e V_{0} = K E_{m a x} = h ν - ϕ .$ Rearrange karo: $V_{0} = \frac{h}{e} ν - \frac{ϕ}{e}$ Yeh itna elegant kyun hai: $V_{0}$ vs $ν$ plot karo ⇒ ek straight line milegi. Slope $= h / e$ (sabhi metals ke liye same!), x-intercept $= ν_{0}$ , y-intercept $= - ϕ / e$ . Isse Millikan ne $h$ measure karke Einstein ko confirm kiya.

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

Graphs padhna (Dual Coding)

$V_{0}$ vs $ν$ : slope metal se independent hai; metals sirf intercept shift karte hain (bada $ϕ$ = line right shift hoti hai).
Photocurrent vs intensity (fixed $ν > ν_{0}$ ): origin se seedhi line — zyada photons, zyada electrons.
$K E_{m a x}$ vs intensity: flat horizontal line — intensity electron ki energy change nahi karti.

Worked examples

Steel-manned mistakes

Recall Feynman: 12-saal ke bachche ko samjhao

Socho ek wall (metal) hai aur chote balls (electrons) uspe glue se chipke hain. Light ek aisi vending machine ki tarah hai jo sirf ek-coin candies bechti hai — har candy (photon) ki fixed energy uski color (frequency) se set hoti hai, na ki kitni candies kharido usse. Ek ball ko kheenchne ke liye tumhe glue cost ( $ϕ$ ) pay karni padegi. Agar ek candy glue se zyada worth nahi hai, toh ball waheen rahegi — chahe tum kitni bhi weak candies phenko. Ek zyada rich candy (bluer light) use karo aur ball ud jaayegi, leftover energy uski speed ban jaati hai.

Flashcards

Intensity badhane se

K E_{m a x}

kyun nahi badti?

Intensity = photons ki sankhya; har electron sirf ek photon absorb karta hai, toh uski energy

h ν

se set hoti hai, photon count se nahi.

Einstein's photoelectric equation batao.

K E_{m a x} = h ν - ϕ

Work function

ϕ

define karo.

Metal surface se sabse kam tightly bound electron ko free karne ke liye minimum energy chahiye.

Threshold frequency

ν_{0}

kya hai?

Emission ke liye minimum frequency;

ν_{0} = ϕ / h

(isse

K E_{m a x} = 0

milta hai).

Stopping potential aur KE ka relation?

e V_{0} = K E_{m a x} = h ν - ϕ

V_{0}

ν

graph ka slope?

h / e

— har metal ke liye same.

Emission instantaneous hona classically explain karna impossible kyun hai?

Waves ko energy accumulate karne mein time lagta; photons ek single absorption event mein

h ν

deliver karte hain.

Threshold wavelength formula?

λ_{0} = h c / ϕ

eV–nm problems ke liye convenient constant?

h c = 1240

eV·nm.

Bada

ϕ

hone par

V_{0}

–

ν

line shift kyun hoti hai?

Yeh

ν_{0}

(x-intercept) badhata hai aur y-intercept

- ϕ / e

ko neeche le jaata hai; slope

h / e

same rehta hai.

Connections

Planck's quantum hypothesis & blackbody radiation — $E = h ν$ ka origin.
Wave-particle duality of light — photon as particle aspect.
de Broglie wavelength — duality matter tak extend ki.
Compton effect — photon momentum, aur particle evidence.
Energy conservation — derivation ki bookkeeping backbone.
Millikan oil drop & measurement of h — experimental confirmation.