Photoelectric effect — Einstein's explanation, work function: Foundations

2.3.2 · D1 · Physics › Modern Physics › Photoelectric effect — Einstein's explanation, work function

Yeh page ek toolbox hai. Parent note mein symbols freely use hote hain jaise $ν$ , $h$ , $ϕ$ , $E = h ν$ , "eV", $V_{0}$ . Agar tumne pehle inhe kabhi nahi dekha, toh parent note ek foreign language jaisi lagegi. Yahan hum inhe zero se banate hain, ek dusre par depend karne ke order mein, har ek ke saath ek picture aur ek reason ki yeh kyun exist karta hai.

0 — Metal andar se actually kaisa dikhta hai

Koi bhi light aane se pehle, metal ko khud imagine karo.

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

Figure dekho. Blue shaded region metal ka andar hai — electrons (orange dots) ke liye ek aaram-deh, low-energy ghar. Dashed line ke upar baahir ka vacuum hai. Vertical gap jo electron ko jump karna padta hai, woh escape ki cost hai. Hum us cost ko thodi der mein $ϕ$ naam denge. Dhyan do ki electrons alag-alag gehraai par baithe hain: jo sabse upar hai (sabse kam tightly held), use sabse chhota jump karna padta hai.

Yeh picture kyun chahiye: poora topic yahi hai ki "light kisi electron ko us jump ke liye enough energy kaise deti hai?" Well dekhe bina, baad ke symbols kuch bhi matlab nahi rakhte.

1 — Frequency $ν$ : light wave ka "colour"

Figure mein, dono waves same speed se daayein travel kar rahi hain. Blue wave stretched out hai — ek second mein kam wobbles — low $ν$ (redder light). Orange wave squashed hai — ek second mein zyada wobbles — high $ν$ (bluer light).

Wavelength $λ$ (Greek "lambda") $ν$ ka partner hai: do wave crests ke beech ki doori, metres mein. Figure mein marked arrow ek $λ$ dikhata hai. Speed of light $c$ inhe aapas mein jodata hai: $c = ν λ \Rightarrow ν = \frac{c}{λ} .$ Yeh relation kyun? Ek second mein wave ek distance $c$ aage badhti hai. Agar $ν$ crests gujarte hain aur har ek $λ$ apart hai, toh total distance $ν \times λ$ hai. Woh distance $c$ ke barabar honi chahiye. Toh high frequency ⇔ short wavelength — squashed waves. (Yeh Wave-particle duality of light se connect hota hai, jahan wohi light bhi particles ki tarah treat ki jaati hai.)

2 — Energy, aur woh tiny unit jisme hum ise measure karte hain

eV introduce kyun karen? Kyunki work functions aur photon energies "kuch eV" par aati hain. $2.3$ eV likhna $3.68 \times 1 0^{- 19}$ J se zyada friendly hai. Yahi unit exactly woh reason hai ki parent note mein shortcut $h c = 1240$ eV·nm exist karta hai.

3 — Planck's constant $h$ : ek energy bundle ka size

Yahan hai poore chapter ka pivot.

$E = h ν$ ki shape notice karo: energy frequency ke proportional hai. Frequency double karo (bluer light) ⇒ har bundle ki energy double. Isliye blue light electrons ko red se zyada kick karti hai.

4 — Photon : light ka ek bundle

Figure do dials visible banata hai. Top row: dim blue light — kam photons, lekin har ek ek bada (high-energy) packet hai. Bottom row: bright red light — zyada photons, lekin har ek ek chhota packet hai. Yeh single picture poore topic ki jaan hai: ek electron ek photon se milta hai aur poochta hai "kya tum akele mujhe free karne ke liye kaafi bade ho?" — tumhare padosiyon ki sankhya irrelevant hai.

Photon idea kyun chahiye: ek continuous wave electron ko dheere-dheere bahut saari weak waves se energy accumulate karne deti. Photons yeh forbid karte hain — ek below-size packet sirf fail hota hai, chahe kitne bhi aayein.

5 — Work function $ϕ$ : escape ki toll

Alag metals ka alag $ϕ$ hota hai (sodium ≈ 2.3 eV, harder metals zyada). Yeh ek toll hai: ek photon ko kuch escape hone se pehle kam se kam $ϕ$ pay karna hi padega.

6 — Threshold frequency $ν_{0}$ aur threshold wavelength $λ_{0}$

Pichle do ideas combine karo. $h ν$ worth ka ek photon $ϕ$ toll pay karta hai. Sabse chhoti frequency jo toll just afford kar sake — kuch bhi bacha ke bina — threshold define karti hai.

$ν_{0}$ se neeche (light bahut red), koi bhi electron kabhi escape nahi karta, chahe brightness kitni bhi ho. $ν_{0}$ par ya uske upar, escape hota hai. Yeh sharp cutoff exactly wahi hai jo classical waves explain nahi kar sakti thi.

7 — Kinetic energy $K E$ aur $K E_{m a x}$ : bacha hua

Photon ke $ϕ$ toll pay karne ke baad, jo bhi energy bachti hai woh electron ko motion ke roop mein milti hai: $photon in h ν = toll ϕ + leftover K E .$ Yeh sirf energy ka book-keeping hai — kuch create ya lose nahi hota (dekho Energy conservation).

Subscript "max" kyun? Well mein gehre electrons surface tak pahunchne ke liye $ϕ$ se zyada pay karte hain, isliye woh dheere nikalte hain. Surface electron (exactly $ϕ$ pay karta hua) sabse zyada leftover rakhta hai — maximum: $K E_{m a x} = h ν - ϕ .$

8 — Charge $e$ , potential $V$ , aur stopping potential $V_{0}$

Aakhri symbols invisible KE ko kisi cheez se connect karte hain jo voltmeter read kar sake.

Stopping point par, electron ki saari kinetic energy hill climb karne mein kharach ho jaati hai: $e V_{0} = K E_{m a x} = h ν - ϕ \Rightarrow V_{0} = \frac{h}{e} ν - \frac{ϕ}{e} .$ Yeh $V_{0}$ versus $ν$ ki ek straight line hai: slope $h / e$ (har metal ke liye same!), aur line zero cross karti hai $ν_{0}$ par. Parent note exactly yahi plot karta hai. eV bookkeeping mein, $K E_{m a x}$ eV mein numerically $V_{0}$ volts ke barabar hota hai, kyunki $e$ cancel ho jaata hai — isliye " $0.80$ eV" se " $0.80$ V" milta hai.

Prerequisite map

Equipment checklist

Main bata sakta hoon ki frequency

ν

ka kya matlab hai aur iska unit kya hai

Har second mein wave wobbles ki sankhya, unit hertz (Hz) =

s^{- 1}

Main jaanta hoon ki

ν

λ

aur

c

kaise relate karte hain

c = ν λ

, isliye

ν = c / λ

— zyada frequency matlab chhota wavelength.

Main joules aur electron-volts ke beech convert kar sakta hoon

1 eV = 1.6 \times 1 0^{- 19}

Main Planck's constant jaanta hoon aur woh kya karta hai

h = 6.63 \times 1 0^{- 34}

J·s; yeh frequency ko ek photon ki energy mein

E = h ν

ke zariye convert karta hai.

Main explain kar sakta hoon ki photon kya hai

Light energy ka ek indivisible bundle worth

h ν

; brightness = photons ki sankhya, colour = har photon ki energy.

Main work function

ϕ

define kar sakta hoon

Metal surface se least tightly bound electron ko free karne ke liye minimum energy.

Main threshold formulas jaanta hoon

ν_{0} = ϕ / h

aur

λ_{0} = h c / ϕ

;

ν_{0}

se neeche kabhi emission nahi.

Main samajhta hoon ki

K E_{m a x}

kyun hai

K E

nahi

Surface electrons sabse zyada leftover energy rakhte hain; gehre wale

ϕ

se zyada pay karte hain aur dheere nikalte hain.

Main stopping potential ko KE se relate kar sakta hoon

e V_{0} = K E_{m a x} = h ν - ϕ

;

V_{0}

ν

plot karne par slope

h / e

ki line milti hai.

Main electron ke charge ka size jaanta hoon

e = 1.6 \times 1 0^{- 19}