Photoelectric effect — Einstein's photon model

WHAT is the photoelectric effect?

The 4 experimental facts classical wave theory could NOT explain:

There is a ==threshold frequency $\nu_0$ ==: below it, no electrons are emitted, even with intense light.
Above $\nu_0$ , emission is instantaneous (no time delay), even for dim light.
The kinetic energy of ejected electrons depends on frequency, not intensity.
Intensity only changes the number of electrons (the current), not their energy.

HOW Einstein fixed it — the photon model

Einstein (1905) borrowed Planck's quantum idea and made it physical: light itself is quantized into packets of energy called photons.

Deriving the photoelectric equation from first principles

Energy bookkeeping for ONE electron:

The photon delivers energy $h\nu$ . The electron must first "pay a toll" to escape the metal surface — this minimum escape energy is the work function $\phi = h\nu_0$ . Whatever is left over becomes kinetic energy:

$\underbrace{h\nu}_{\text{energy in}} = \underbrace{\phi}_{\text{cost to escape}} + \underbrace{KE_{\max}}_{\text{energy left over}}$

Rearranging:

Why $KE_{\max}$ and not just $KE$ ? Electrons deeper in the metal lose extra energy on the way out, so they emerge slower. The maximum KE belongs to a surface electron that pays only the minimum toll $\phi$ .

Stopping potential

To measure $KE_{\max}$ , apply a reverse voltage until the current just stops. The electron loses all its KE climbing that voltage:

$KE_{\max} = eV_0$

Combine with the photoelectric equation:

$eV_0 = h\nu - \phi \quad\Rightarrow\quad V_0 = \frac{h}{e}\nu - \frac{\phi}{e}$

This is a straight line $V_0$ vs $\nu$ with slope $h/e$ — a way to measure Planck's constant from a graph!

Figure — Photoelectric effect — Einstein's photon model

Worked Examples

Worked example Example 1 — Find

$KE_{\max}$ A metal has work function $\phi = 3.0\ \text{eV}$ . Light of wavelength $\lambda = 300\ \text{nm}$ hits it. Find $KE_{\max}$ of electrons.

Step 1 — Photon energy: $E = \dfrac{hc}{\lambda} = \dfrac{1240\ \text{eV·nm}}{300\ \text{nm}} = 4.13\ \text{eV}$ . Why this step? We first find how much energy each photon delivers; the handy shortcut $hc = 1240\ \text{eV·nm}$ avoids SI unit juggling.

Step 2 — Apply equation: $KE_{\max} = E - \phi = 4.13 - 3.0 = 1.13\ \text{eV}$ . Why this step? The electron keeps whatever is left after paying the work-function toll.

Answer: $KE_{\max} = 1.13\ \text{eV}$ ( $=1.81\times10^{-19}\ \text{J}$ ).

Forecast-then-Verify

Recall Forecast before reading

Q: You double the light intensity but keep the frequency fixed (above $\nu_0$ ). What happens to (a) number of electrons, (b) their $KE_{\max}$ , (c) stopping potential?

Verify: (a) doubles — more photons ⇒ more electrons. (b) unchanged — each photon still carries $h\nu$ . (c) unchanged — $V_0$ depends only on $KE_{\max}$ , hence on $\nu$ .

Common Mistakes (Steel-manned)

Active Recall

What are the two things a photon's energy is split into after ejection?

The work function

\phi

(escape cost) and the electron's

KE_{\max}

State Einstein's photoelectric equation.

KE_{\max} = h\nu - \phi = h(\nu-\nu_0)

What decides whether ANY electron is ejected?

The frequency: it must exceed the threshold frequency

\nu_0

(i.e.

h\nu>\phi

What does increasing intensity change?

The number of ejected electrons (photocurrent), NOT their kinetic energy.

Relation between work function and threshold frequency?

\phi = h\nu_0

How is stopping potential related to

KE_{\max}

eV_0 = KE_{\max}

, so

V_0 = KE_{\max}/e

Slope of the

V_0

\nu

graph?

h/e

(used to measure Planck's constant).

Handy shortcut for photon energy in eV·nm?

E(\text{eV}) = 1240 / \lambda(\text{nm})

Why does classical wave theory fail?

It predicts energy accumulates over time and depends on intensity, contradicting the instantaneous, frequency-dependent emission observed.

Formula for threshold wavelength?

\lambda_0 = hc/\phi

Recall Feynman: explain to a 12-year-old

Imagine a claw machine where each coin is a tiny packet of light. To win a toy (kick out an electron) you need a coin worth at least a certain amount. If your coin is too small (light too red), you can put in a million small coins and still win nothing — one coin, one try. If your coin is big enough (light blue enough), you win instantly, and whatever change is left over is how fast the toy shoots out. Putting in more coins of the same size just wins more toys, not faster toys.

Connections

Planck's Quantum Theory — where $E=h\nu$ was born (black-body radiation).
Wave-Particle Duality — photon as light's particle nature; de Broglie extends it to matter.
Work Function and Binding Energy — same idea appears in Photoelectron Spectroscopy.
Bohr Model of the Atom — quantized energy exchange with light (absorption/emission).
Compton Effect — further proof photons carry momentum, not just energy.
Electromagnetic Spectrum — why UV ejects electrons but red light often can't.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, photoelectric effect ka core idea simple hai: light ek continuous wave ki tarah energy nahi deti, balki chhote-chhote packets me deti hai jinhe photon kehte hain. Har photon ki energy $E=h\nu$ hoti hai — yaani sirf frequency (colour) pe depend karti hai, brightness pe nahi. Jab ek photon ek electron ko hit karta hai, to woh apni poori energy ussi ek electron ko de deta hai.

Ab metal se electron nikalne ke liye ek minimum "toll tax" bharna padta hai jise work function $\phi$ kehte hain. Agar photon ki energy $h\nu$ is toll se kam hai, to electron bilkul nahi niklega — chahe aap kitni bhi tez (bright) light maar do, kyunki brightness sirf zyada photons deti hai, bade photons nahi. Isiliye ek threshold frequency $\nu_0$ hoti hai jiske neeche kuch nahi hota. Yahi baat classical wave theory explain nahi kar paati thi, aur Einstein ne ise 1905 me solve karke Nobel prize jeeta.

Jo energy toll bharne ke baad bachti hai, woh electron ki kinetic energy ban jaati hai: $KE_{max}=h\nu-\phi$ . Isko measure karne ke liye reverse voltage lagate hain jab tak current ruk na jaye — us voltage ko stopping potential $V_0$ kehte hain, aur $eV_0=KE_{max}$ . Agar $V_0$ ko frequency $\nu$ ke against plot karo to straight line milti hai jiska slope $h/e$ hota hai — isse Planck's constant nikaal sakte ho!

Yaad rakhne ka mantra: "Frequency Frees, Intensity Increases" — frequency decide karti hai electron niklega ya nahi aur kitni speed se, jabki intensity sirf electrons ki ginti badhati hai. Exam me 90% questions bas $KE_{max}=h\nu-\phi$ aur $\lambda_0=hc/\phi$ se ban jaate hain, to yeh do formulae pakke kar lo.

Photoelectric effect — Einstein's photon model

WHAT is the photoelectric effect?

HOW Einstein fixed it — the photon model

Deriving the photoelectric equation from first principles

Stopping potential

Worked Examples

Forecast-then-Verify

Common Mistakes (Steel-manned)

Active Recall

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Quantum Atomic Structure

Connections