Photoelectric effect — Einstein's explanation, work function: Question bank

2.3.2 · D5 · Physics › Modern Physics › Photoelectric effect — Einstein's explanation, work function

Shuru karne se pehle do symbols, taaki kuch bhi bina explain ke use na ho:

$ν$ (Greek "nu") = frequency of the light = ek second mein kitne wave-cycles guzarte hain. Bada $ν$ = bluer light = zyada energy per photon.
$ϕ$ (Greek "phi") = work function = woh fixed "toll" (energy mein) jo har metal apne surface se electron ko escape karne ke liye leta hai.

True or false — justify

Yahan har item mein ek hidden trap hai. Justification hi mark dilata hai.

T/F: Incident light ki intensity double karne se ejected electrons ki maximum kinetic energy bhi double ho jaati hai.

False. Intensity = photons per second ki sankhya hai, energy per photon nahi; har electron exactly ek photon absorb karta hai, isliye

K E_{m a x} = h ν - ϕ

bilkul nahi badlata. Intensity double karne se electrons ki sankhya double hoti hai, unki speed nahi.

T/F: Agar light frequency threshold frequency

ν_{0}

se neeche ho, toh use kaafi lambe time tak shine karne par eventually electrons eject ho jaayenge.

False. Energy indivisible packets mein aati hai; ek sub-threshold photon (

h ν < ϕ

) toll pay nahi kar sakta aur electrons turant de-excite ho jaate hain — "energy save up" nahi hoti. Kitna bhi time lage, emission kabhi nahi hogi.

T/F: Stopping potential

V_{0}

light ki intensity par depend karta hai.

False.

e V_{0} = h ν - ϕ

mein koi intensity term nahi hai.

V_{0}

sabse tez electron ki energy se decide hota hai, jo sirf

ν

aur

ϕ

se set hoti hai. Bright light zyada current deti hai lekin

V_{0}

same rehta hai.

T/F: Stopping-potential-versus-frequency graph ka slope us metal ke liye zyada bada hoga jiska work function bada hoga.

False.

V_{0} = \frac{h}{e} ν - \frac{ϕ}{e}

se, slope

h / e

hai — ek universal constant, har metal ke liye same. Bada

ϕ

sirf line ko right (bada

ν_{0}

) aur neeche (zyada negative intercept) shift karta hai.

T/F: Jis photon ki energy exactly

ϕ

ke barabar ho, woh electron ko high speed par eject karta hai.

False.

h ν = ϕ

par

K E_{m a x} = 0

hota hai: electron bahut mushkil se essentially zero speed ke saath escape karta hai. Yahi precisely threshold case hai.

T/F: Bahut high intensity wali red light us metal se electrons eject kar sakti hai jisse low intensity wali green light aasaani se eject karti hai.

False (generally). Agar red light

ν_{0}

se neeche hai, toh koi bhi intensity kaam nahi karegi, jabki low-intensity green (agar

ν_{0}

se upar hai) har baar electrons eject karegi. Kya emission hogi yeh frequency decide karti hai, brightness nahi.

T/F: Kisi metal ki work function us metal ke free atom ki ionization energy ke barabar hoti hai.

False.

ϕ

bulk solid se least-bound electron ke liye surface escape energy hai; yeh usually isolated atom se electron strip karne ki energy se chhoti hoti hai.

T/F: Fixed intensity par light ki frequency badhane se (threshold ke upar) ejected electrons ki sankhya per second badh jaati hai.

False (aur thoda subtle bhi). Fixed intensity (fixed energy per second) par, zyada frequency wale photons mein se har ek zyada energy carry karta hai, isliye per second kam photons hote hain — isliye kam electrons nikalte hain, haalaanki har ek tez hoga. Electrons ki sankhya photon count ke saath chalti hai, frequency ke saath nahi.

Spot the error

Neeche diye har statement mein ek galat step hai. Use identify karein aur theek karein.

"Kyunki

K E_{m a x} = h ν - ϕ

hai,

ϕ = 0

wala metal kisi bhi light ke liye electrons emit karega, isliye

ϕ = 0

metals sabse achhe photodetectors hain."

Error: koi bhi real metal

ϕ = 0

nahi hota; electrons hamesha kuch na kuch bound rehte hain, isliye

ϕ > 0

hamesha hota hai. Equation sahi hai, premise physically impossible hai.

"Photoelectric equation

h ν = ϕ + K E_{m a x}

dikhata hai ki photon electron ko free karne par energy create hoti hai."

Error: kuch bhi create nahi hota — yeh energy conservation hai. Photon ka

h ν

puri tarah kharach hota hai:

ϕ

escape ke liye aur

K E_{m a x}

bacha hua motion ke roop mein. Dekho Energy conservation.

"Kyunki emission instantaneous hai, electron ko energy light ki travel speed se tez absorb karni padegi, jo relativity violate karta hai."

Error: instantaneous ka matlab ek single absorption event hai, superluminal energy transfer nahi. Ek photon ek interaction mein

h ν

deta hai; koi accumulation nahi, koi speed limit break nahi hoti.

"Stopping potential woh voltage hai jo sabse tez electron ko collector tak pahunchne mein speed up karta hai."

Error: yeh ek reverse (retarding) voltage hai jo sabse tez electron ko collector se ठीक pehle rok deta hai. Jab

e V_{0} = K E_{m a x}

hota hai, tab sabse tez electron bhi wapas ho jaata hai.

K E_{m a x} = h ν - ϕ

, isliye

K E_{m a x}

ko intensity ke against plot karne par slope

h

wali straight line milti hai."

Error:

K E_{m a x}

intensity par bilkul depend hi nahi karta — intensity ke against iska graph ek flat horizontal line hoga. Slope

h / e

wali straight line

V_{0}

vs frequency ka plot hai, yeh alag plot hai.

"Ek hi frequency ki light se eject hone wale saare electrons same speed se nikalte hain, yaani

K E_{m a x}

se."

Error: electrons

0

K E_{m a x}

tak speeds ki ek range ke saath nikalte hain. Sirf least-bound (surface) electron ko poora leftover milta hai; deeper electrons surface tak pahunchne mein extra energy kharach karte hain aur slower nikalte hain.

Why questions

Why

V_{0}

-versus-

ν

graph ka slope har metal ke liye same kyun hota hai, jabki metals alag-alag hote hain?

Kyunki slope

h / e

hai — sirf Planck's constant aur electron charge se bana hai, dono universal hain. Metal sirf

ϕ

ke zariye aata hai, jo intercept shift karta hai, slope nahi. Yahi universality hai jo Millikan ko $h$ measure karne deti hai.

Why classical wave theory emission mein time delay predict karta hai, aur woh prediction galat kyun hai?

Ek spread-out wave gradually energy deposit karegi, isliye ek dim wave ko ek electron par enough energy accumulate karne mein time lagega. Reality: ek photon ek instant event mein

h ν

deliver karta hai, isliye emission brightness chahe kuch bhi ho

\sim 1 0^{- 9}

s ke andar shuru ho jaati hai.

Why equation mein

K E_{m a x}

hota hai, sirf

K E

kyun nahi?

Sirf surface (least-bound) electron exactly

ϕ

pay karta hai; deeper electrons surface tak pahunchne ke liye zyada pay karte hain aur slower nikalte hain. Maximum leftover energy surface electron ki hoti hai, jo ceiling

K E_{m a x} = h ν - ϕ

deti hai.

Why bada work function threshold ko higher frequency (shorter wavelength) par push karta hai?

ν_{0} = ϕ / h

se, bada toll

ϕ

use just cover karne ke liye zyada energetic photon maangta hai, isliye

ν_{0}

badhta hai. Kyunki

λ_{0} = h c / ϕ

hai, threshold wavelength ghatti hai — aksar UV mein chali jaati hai.

Why intensity photocurrent control karti hai lekin stopping potential nahi?

Intensity = photons per second = electrons per second = current. Lekin har electron ki energy ek photon ke

h ν

se fixed hai, isliye stopping voltage (jo sabse tez electron ki energy counter karta hai) kitne bhi aayein, untouched rehta hai.

Why photoelectric effect ne physicists ko light ki particle nature accept karne par majboor kiya?

Threshold, instant emission, aur

K E_{m a x}

ki intensity-independence — yeh sab inexplicable hain agar light continuous wave ho, lekin turant explain ho jaate hain agar light energy packets

h ν

mein aaye. Data ne quanta maange.

Edge cases

Edge: Jab

h ν = ϕ

exactly ho toh

K E_{m a x}

kya hoga?

Exactly zero — electron liberate toh ho jaata hai lekin uske paas move karne ke liye koi spare energy nahi hoti. Yahi threshold define karta hai:

ν = ν_{0}

λ = λ_{0}

Edge: Jab

h ν < ϕ

ho (photon below threshold) toh kya hota hai?

Koi bhi electron emit nahi hota — single packet toll pay nahi kar sakta, aur packets add up nahi hote. Photocurrent exactly zero hota hai.

Edge: Agar do low-energy photons almost saath ek hi electron par aayein (below-threshold light) toh kya hoga?

Ordinary photoelectric regime mein yeh electron ko liberate nahi karta — absorption one-photon-at-a-time hoti hai aur electron dusra aane se pehle de-excite ho jaata hai. (Multi-photon emission ke liye enormous laser intensities chahiye, jo is topic se bahar hai.)

Edge:

V_{0}

-versus-

ν

line

ν = 0

par kya predict karti hai?

Y-intercept

- ϕ / e

hai, ek negative stopping potential — purely mathematical extrapolation, kyunki koi real light

ν = 0

nahi hoti aur

ν_{0}

se neeche koi emission hoti hi nahi.

Edge: Jab frequency

ν \to \infty

ho,

K E_{m a x}

ka kya hoga?

Yeh

K E_{m a x} = h ν - ϕ \to \infty

ke saath unbounded badhti jaati hai (non-relativistic idealization mein). Bahut high

ν

(X-rays) waakai bahut tez electrons eject karte hain — aur eventually Compton effect regime, jahan photon ek momentum-carrying particle ki tarah behave karta hai, takeover kar leta hai.

Edge: Kya threshold ke upar light ke liye stopping potential kabhi negative ho sakta hai?

Nahi. Threshold ke upar

K E_{m a x} = h ν - ϕ > 0

hota hai, isliye

V_{0} = K E_{m a x} / e > 0

hoga. Genuinely emitted electron ko rokne ke liye hamesha ek positive retarding voltage chahiye hota hai.

Recall One-line self-test

Agar koi dost kahe "bas light ko aur bright karo aur dim red light finally electrons eject kar degi," toh aap kya jawab denge? ::: Threshold se neeche, brightness irrelevant hai — har red photon individually $ϕ$ pay karne mein fail karta hai, aur photons apni energy pool nahi karte; sirf ek bluer (higher- $ν$ ) photon hi yeh kar sakta hai.

Connections

Photoelectric effect — Einstein's explanation, work function — woh parent note jinhein ye traps stress-test karti hain.
Energy conservation — yahan har "spot the error" honest energy bookkeeping par hi aata hai.
Wave-particle duality of light — kyun quanta, waves nahi, data explain karte hain.
Compton effect — high-frequency limiting behaviour yahan extend hota hai.
Millikan oil drop & measurement of h — universal slope $h / e$ jise kai items probe karte hain.