Photoelectric effect — Einstein's explanation, work function

WHY does this topic even exist?

When light hits a metal, electrons can be ejected. Classical wave theory predicted things that experiments flatly contradicted:

Classical wave prediction	What actually happens
Brighter light ⇒ more energetic electrons	Brighter light ⇒ more electrons, same max energy
Any frequency works if you wait long enough	Below a threshold frequency, nothing happens, ever
There should be a time delay to "accumulate" energy	Emission is instantaneous (< $10^{-9}$ s)

The failure of waves forced Einstein (1905) to take Planck's quanta seriously.

WHAT are the key terms?

HOW Einstein derived the equation (from scratch)

Step 1 — Energy of one photon. Planck/Einstein: light of frequency $\nu$ comes in packets of energy $E_{\text{photon}} = h\nu.$ Why this step? We assume light is quantized; this is the bold postulate we test against experiment.

Step 2 — Energy bookkeeping for one electron. One photon is absorbed by one electron. That energy must pay two "bills":

The escape cost = work function $\phi$ .
Whatever's left becomes kinetic energy.

By conservation of energy: $h\nu = \phi + KE_{\max}$ Why $KE_{\max}$ and not just $KE$ ? Electrons deeper in the metal need extra energy to reach the surface, so they emerge slower. The electron right at the surface (binding $=\phi$ ) gets the maximum KE.

Step 3 — Connect to measurable stopping potential. We stop the fastest electron with potential $V_0$ , so the electrical work $eV_0$ equals its KE: $eV_0 = KE_{\max} = h\nu - \phi.$ Rearrange: $\boxed{V_0 = \frac{h}{e}\nu - \frac{\phi}{e}}$ Why this is gorgeous: plot $V_0$ vs $\nu$ ⇒ a straight line. Slope $=h/e$ (same for all metals!), x-intercept $=\nu_0$ , y-intercept $=-\phi/e$ . This let Millikan measure $h$ to confirm Einstein.

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

Reading the graphs (Dual Coding)

$V_0$ vs $\nu$ : slope independent of metal; metals shift only the intercept (bigger $\phi$ = line shifts right).
Photocurrent vs intensity (fixed $\nu>\nu_0$ ): straight line through origin — more photons, more electrons.
$KE_{\max}$ vs intensity: flat horizontal line — intensity does NOT change electron energy.

Worked examples

Worked example Example 1 — Find

$KE_{\max}$ and $V_0$ Sodium, $\phi = 2.3$ eV, light $\lambda = 400$ nm. Find $KE_{\max}$ and $V_0$ .

Step 1: Photon energy. Use $E=\dfrac{hc}{\lambda}=\dfrac{1240\,\text{eV·nm}}{400\,\text{nm}}=3.10$ eV. Why? The shortcut $hc=1240$ eV·nm gives energy directly in eV — fewer unit errors.

Step 2: $KE_{\max}=h\nu-\phi=3.10-2.3=0.80$ eV. Why? Einstein's equation — energy left after paying the work function.

Step 3: $V_0 = KE_{\max}/e = 0.80$ V. Why? In eV, KE in eV equals $V_0$ in volts numerically (the $e$ cancels).

Worked example Example 3 — Two frequencies, find

$h$ (Millikan style) $\nu_1=7\times10^{14}$ Hz gives $V_0=0.5$ V; $\nu_2=10\times10^{14}$ Hz gives $V_0=1.74$ V.

Step 1: Slope $=\dfrac{\Delta V_0}{\Delta\nu}=\dfrac{1.74-0.5}{(10-7)\times10^{14}}=4.13\times10^{-15}$ V·s. Why? From $V_0=\frac{h}{e}\nu-\frac{\phi}{e}$ , slope $=h/e$ .

Step 2: $h=\text{slope}\times e = 4.13\times10^{-15}\times1.6\times10^{-19}=6.6\times10^{-34}$ J·s. ✓ Why beautiful: we measured Planck's constant from a voltage graph.

Steel-manned mistakes

Recall Feynman: explain to a 12-year-old

Imagine a wall (the metal) and tiny balls (electrons) stuck to it with glue. Light is like a vending machine that only sells one-coin candies — each candy (photon) has a fixed energy set by its color (frequency), not by how many candies you buy. To pull a ball off you must pay the glue cost ( $\phi$ ). If one candy isn't worth more than the glue, the ball stays — no matter how MANY weak candies you throw. Use a richer candy (bluer light) and the ball flies off, with the leftover energy becoming its speed.

Flashcards

Why does increasing intensity not increase

KE_{\max}

Intensity = number of photons; each electron absorbs only one photon, so its energy is set by

h\nu

, not photon count.

State Einstein's photoelectric equation.

KE_{\max} = h\nu - \phi

Define work function

\phi

Minimum energy needed to free the least tightly bound electron from the metal surface.

What is the threshold frequency

\nu_0

The minimum frequency for emission;

\nu_0=\phi/h

(gives

KE_{\max}=0

Relation between stopping potential and KE?

eV_0 = KE_{\max} = h\nu - \phi

Slope of the

V_0

\nu

graph?

h/e

— the same for every metal.

Why is emission instantaneous classically impossible to explain?

Waves would need time to accumulate energy; photons deliver

h\nu

in a single absorption event.

Threshold wavelength formula?

\lambda_0 = hc/\phi

Convenient constant for eV–nm problems?

hc = 1240

eV·nm.

Why does a bigger

\phi

shift the

V_0

–

\nu

line?

It increases

\nu_0

(x-intercept) and lowers the y-intercept

-\phi/e

; slope stays

h/e

Connections

Planck's quantum hypothesis & blackbody radiation — origin of $E=h\nu$ .
Wave-particle duality of light — photon as particle aspect.
de Broglie wavelength — duality extended to matter.
Compton effect — photon momentum, more particle evidence.
Energy conservation — the bookkeeping backbone of the derivation.
Millikan oil drop & measurement of h — experimental confirmation.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, photoelectric effect ka asli twist yeh hai: light ko hum "wave" samajhte the jo continuous energy deti hai, par experiment ne bola "nahi bhai, light packets mein aati hai" — har packet ko photon kehte hain aur uski energy hoti hai $E=h\nu$ , yaani sirf frequency (color) pe depend karti hai, brightness pe nahi.

Ek electron ko metal se bahar nikalne ke liye thodi energy "toll" deni padti hai — usko work function $\phi$ kehte hain. Ek photon ko ek electron khaata hai. Toh energy ka hisaab simple hai: $h\nu = \phi + KE_{\max}$ . Matlab photon ki energy mein se pehle toll ( $\phi$ ) cut hoga, jo bachega woh electron ki speed (kinetic energy) ban jayega. Agar $h\nu < \phi$ , toh electron niklega hi nahi, chahe tum kitni bhi tej (bright) light maaro — kyunki har photon alag-alag akele hi kaam karta hai, energy jama nahi hoti.

Stopping potential $V_0$ woh ulta voltage hai jo sabse tej electron ko bhi rok de, toh $eV_0 = KE_{\max}$ . Yahin se famous straight-line graph aata hai: $V_0$ vs $\nu$ plot karo toh slope $h/e$ milta hai — aur kamaal yeh ki yeh slope har metal ke liye same hota hai! Sirf line ka position (intercept) badalta hai jab $\phi$ badalta hai. Isi graph se Millikan ne Planck's constant $h$ naap liya tha. Exam tip: eV-nm problems mein seedha $hc=1240$ eV·nm use karo, units ki tension khatam.

Photoelectric effect — Einstein's explanation, work function

WHY does this topic even exist?

WHAT are the key terms?

HOW Einstein derived the equation (from scratch)

Reading the graphs (Dual Coding)

Worked examples

Steel-manned mistakes

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Modern Physics

Connections