Work Function & Stopping Potential

Equivalent forms

eV_0 = h f - \phi

K_{max}=h(f-f_0)

A straight line: plot stopping voltage vs frequency and its slope is h/e — Millikan measured Planck's constant this way while trying to prove Einstein wrong.

Symbol	Name	SI unit	Dimension	Range
$K_{max}$	Max kinetic energyoutput Kinetic energy of the fastest ejected electron	J	ML^2T^-2	—
$f$	Light frequency Frequency of incident light	Hz	T^-1	400000000000000 – 1500000000000000
$\phi$	Work function Minimum energy to free an electron from the surface	J	ML^2T^-2	3.2e-19 – 9.6e-19
$h$	Planck constant Quantum of action	J*s	ML^2T^-1	—
$V_0$	Stopping potential Voltage that just halts the fastest electrons	V	ML^2T^-3*I^-1	—
$e$	Elementary charge Electron charge magnitude	C	I*T	—

Where it holds

Single-photon absorption. Fails at very high intensities where multi-photon processes occur, and for photon energies below

\varphi

(no emission at any brightness).

Dimensional analysis

K_\max \to [M*L^2*T^-2]

(energy)

hf \to [M*L^2*T^-1][T^-1] = [M*L^2*T^-2]

Photon energy and kinetic energy share energy dimensions.

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Discovery

Albert Einstein · 1905

In his 'miracle year' Einstein proposed that light itself is quantized, not just its emission (as Planck had assumed). The photoelectric paper — not relativity — won him the 1921 Nobel Prize. Robert Millikan spent a decade trying to disprove the light-quantum idea, and instead confirmed the linear law and measured h to 0.5%.

Research status: stable

Real-world applications

Photomultiplier tubes and night-vision devices
Solar-cell surface engineering
X-ray photoelectron spectroscopy (XPS) for material analysis

Common misconceptions

Brighter light does NOT give faster electrons — only more of them
There is a sharp threshold frequency $f_{0} = \varphi /h$ below which nothing happens
Emission is essentially instantaneous, contradicting the classical energy-accumulation picture

Experimental verification

Millikan (1914–16) measured the

V_{0}-vs-f

line for sodium and lithium, confirming the slope equals h/e; modern photoelectron spectroscopy uses the same relation daily.

Derivation

Energy conservation for one photon absorbed by one electron: photon energy hf splits into the escape cost

\varphi

(binding to the metal) plus leftover kinetic energy K.

So

K_\max = hf - \varphi

.

To measure K_max, apply a reverse voltage until the current just stops; then the work done against the field

eV_{0}

equals K_max, giving

eV_{0} = hf - \varphi

.

Plotting

V_{0} vs f

gives a line of slope h/e and intercept

-\varphi /e

.

What if…

What if the light is below threshold frequency?

No electrons emerge no matter how intense the beam — the classic classical-physics failure.

What if you use a metal with lower work function?

The threshold frequency drops, so even red light can eject electrons — why cesium is used in photocathodes.

1

Stopping potential for sodium

Given ·

f:: 8.0e14 Hz
φ:: 2.28 eV

Find · stopping potential

V_{0}

Steps

$hf = 6.626e-34 \times 8.0e14 = 5.30e-19\,\mathrm{J} = 3.31\,\mathrm{eV}$
$K_\max = 3.31 - 2.28 = 1.03\,\mathrm{eV}$
$V_{0} = K_\max /e = 1.03\,\mathrm{V}$

Result ·

hf = 3.31\,\mathrm{eV}

;

V_{0} = (3.31 - 2.28) = 1.03\,\mathrm{V}

Photon Energy (Planck-Einstein Relation)→De Broglie Wavelength→Compton Scattering→