The Planck Constant

The Planck constant (h) — the constant of proportionality between a photon's energy and its frequency:

h = 6.63 × 10⁻³⁴ J s

Its unit, the joule second, follows from E = hf: joules divided by hertz, and a hertz is a reciprocal second.

Why it is so small — h sets the scale at which quantum behaviour becomes noticeable. Because it is tiny, the energy steps between quantum states are negligible for everyday objects, which is why a swinging pendulum appears to take any energy at all. Quantisation is always present; it is only visible at atomic scales.

Measuring h with an LED — when the p.d. across an LED reaches its threshold, electrons have just enough energy to produce photons of the emitted wavelength:

eV = hc / λ

Rearranging gives

V = (hc / e) × (1 / λ)

The graphical method — measure the threshold voltage for several LEDs of known wavelength and plot V against 1/λ. This should be a straight line through the origin with gradient hc/e, so

h = gradient × e / c

Example: using an LED of wavelength 640 nm that lights at 1.9 V, h = eVλ/c = (1.60 × 10⁻¹⁹ × 1.9 × 640 × 10⁻⁹) ÷ (3.00 × 10⁸) = 6.5 × 10⁻³⁴ J s — close to the accepted value.

Why the graph beats a single LED — a single measurement carries the full uncertainty of one threshold voltage, which is hard to judge because an LED brightens gradually rather than switching on sharply. Several LEDs and a gradient average that judgement out, and a line that misses the origin reveals a systematic error in how the threshold was identified.

Determining the threshold consistently — increase the p.d. slowly in a darkened room and record the voltage at which emission is just detectable, using the same criterion for every LED. The absolute value matters less than applying the same judgement each time, because a consistent offset shifts the intercept rather than the gradient.