Photoelectric Effect Equation

Equivalent forms

E_k = hc/\lambda - W

A linear equation that buried classical wave theory of light and launched quantum mechanics.

Symbol	Name	SI unit	Dimension	Range
$K_max$	Max kinetic energyoutput Maximum kinetic energy of ejected electron	J	M·L²·T⁻²	0 – 1e-17
$f$	Photon frequency Frequency of incident light	Hz	T⁻¹	10000000000000 – 3000000000000000
$h$	Planck constant Planck's quantum of action	J·s	M·L²·T⁻¹	6.62607015e-34 – 6.62607015e-34
$φ$	Work function Minimum energy to free an electron from the material	J	M·L²·T⁻²	1e-19 – 1e-18

Unit systems

Symbol	SI	CGS	Natural
$K_max$	J	erg	eV
$f$	Hz	Hz	Hz
$h$	J·s	erg·s	eV·s
$φ$	J	erg	eV

Where it holds

Valid for one-photon processes on metal surfaces in vacuum. Multi-photon processes (visible at very high intensities, e.g., lasers) violate the simple linear relation.

Dimensional analysis

K_\max \to [M\cdot L^{2}\cdot T^{-2}]

hf - \varphi \to [M\cdot L^{2}\cdot T^{-1}][T^{-1}] - [M\cdot L^{2}\cdot T^{-2}] = [M\cdot L^{2}\cdot T^{-2}]

Both sides are energy.

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Discovery

Albert Einstein · 1905

Einstein explained Lenard's photoelectric observations by postulating light quanta (photons) of energy hf. The discovery earned him the 1921 Nobel Prize — not relativity.

Try this

Why won't bright red light eject electrons but dim ultraviolet will?

UV light of wavelength 200 nm hits a sodium surface (work function 2.28 eV). What is the maximum kinetic energy of ejected electrons?

Research status: stable

Real-world applications

Photomultiplier tubes (PMTs)
X-ray photoelectron spectroscopy (XPS) for surface analysis
Solar cells (photovoltaic effect)
Image sensors (CCDs and CMOS)

Common misconceptions

K_max does NOT depend on light intensity — only frequency. Intensity affects the number of electrons, not their energy.
The work function is a property of the material surface, not of individual electrons.
Multi-photon photoemission (high intensity laser) allows electrons to be ejected below the classical threshold.

Experimental verification

Millikan's 1916 experiments verified the linear K_max vs f relation and measured h to 0.5% accuracy from the slope. Modern photoemission spectroscopy uses this to map band structures in solids.

Derivation

Photon energy

E = hf

is absorbed entirely by a single electron.

Energy conservation:

hf = \varphi + K

.

The maximum K is reached for the least-bound electron at the surface, giving

K_\max = hf - \varphi

.

Limiting cases

hf <

\varphi

⟶

K_\max = 0

No emission below threshold — even infinite intensity won't eject electrons.

hf = \varphi

⟶

K_\max = 0

(threshold)Marginal case — defines threshold frequency

f_0 = \varphi /h

.

hf ≫

\varphi

⟶

K_\max \approx hf

At very high frequencies, work function becomes negligible.

What if…

What if we doubled the light intensity?

Twice as many electrons emitted, but K_max stays the same.

What if we used a material with

\varphi = 0

?

Every photon would eject an electron — like an ideal photon detector.

1

UV on sodium

Given ·

f:: 1500000000000000
h:: 6.62607015e-34
φ:: 3.65e-19

Find · K_max

Steps

Step 1: Photon energy $hf = 6.626 \times 10^{-34} \times 1.5 \times 10^{15} = 9.94 \times 10^{-19} J = 6.21\,\mathrm{eV}$ .
Step 2: Subtract work function: $K_\max = 6.21 - 2.28 = 3.93\,\mathrm{eV}$ .
Step 3: Convert: $3.93 \times 1.602e-19 = 6.29 \times 10^{-19} J$ .

Result ·

K_\max = 6.626e-34 \times 1.5e15 - 3.65e-19 = 9.94e-19 - 3.65e-19 = 6.29 \times 10^{-19} J \approx 3.93\,\mathrm{eV}

Compton Scattering Wavelength Shift→Pair Production Threshold Energy→