## torsdag 3 april 2014

### Water Dam Analog of Photoelectric Effect

Open sluice gates in the Three Gorges Dam in the Yangtze River.

Einstein was awarded the 1921 Nobel Prize in Physics for his "discovery of the law of the photoelectric effect", connecting frequency $\nu$ of light shining on a metallic surface with measured potential $U$:
• $h\nu = h\nu_0 + e\, U$ or $h(\nu -\nu_0) = e\, U$,
where $h$ is Planck's constant with dimension $eVs = electronvolt\,\times second$,  $\nu_0$ is the smallest frequency releasing electrons and $U$ in Volts $V$ is the stopping potential bringing the current to zero for $\nu >\nu_0$ and $e$ is the charge of an electron. Observing $U$ for different $\nu$ in a macroscopic experiment shows a linear relationship between $\nu -\nu_0$ and $U$ with $h$ as scale factor with reference value
• $h = 4.135667516(91)\times 10^{-15}\, eVs$,
with Millikan's value from 1916 within $0.5\%$.

Determining $h$ this way makes Einstein's law of photoelectricity into an energy conversion standard attributing $h\nu$ electronvolts to the frequency $\nu$, without any implication concerning the microscopic nature of the photoelectric effect.

The award motivation "discovery of the law of the photoelectric effect" reflected that Einstein's derivation did not convince the committee as expressed by member Gullstrand:
• When it was formulated it was only a tentatively poorly developed hunch, based on qualitative and partially correct observations. It would look peculiar if a prize was awarded to this particular work.
To give perspective let us as an analog of the law of the photoelectric effect consider a water dam with sluice gates which automatically open when the level of water is $\nu_0$.  The sluice gates will then remain locked as long as the water level $\nu <\nu_0$.  Lock the sluice gates and let the dam fill to some water level $\nu >\nu_0$ and then unlock the sluices. The sluices will then open and water will flow through under transformation of potential energy into kinetic energy. Assuming the work to open the sluices corresponds to a level loss of $\nu_0$, a net level of $\nu -\nu_0$ potential energy will then be transformed into kinetic energy by the water flow through the sluices.

The dam can be seen as an illustration of the photoelectric effect with the water level corresponding to frequency $\nu$ and the gravitational constant corresponding to $h$ and the width of the dam corresponding to the amplitude of incoming light. If $\nu <\nu_0$ then nothing will happen, if $\nu >\nu_0$ then the kinetic energy will scale with $h\nu$ and the total flow will scale with the width of the dam.

Notice that noting in this model requires the water to flow in discrete lumps or quanta. The only discrete effect is the threshold $\nu_0$ for opening the sluices.