Measuring the Planck's Constant at Home Using Multicolored LEDs

(This experiment is based on the article https://www.scienceinschool.org/article/2014/planck/ )

The Planck constant, or Planck's constant, denoted by ℎ , is a fundamental physical constant of foundational importance in quantum mechanics: a photon's energy is equal to its frequency multiplied by the Planck constant, and the wavelength of a matter wave equals the Planck constant divided by the associated particle momentum.

The constant was postulated by Max Planck in 1900 as a proportionality constant needed to explain experimental black-body radiation. Planck later referred to the constant as the "quantum of action". In 1905, Albert Einstein associated the "quantum" or minimal element of the energy to the electromagnetic wave itself. Max Planck received the 1918 Nobel Prize in Physics "in recognition of the services he rendered to the advancement of Physics by his discovery of energy quanta".

In metrology, the Planck constant is used, together with other constants, to define the kilogram, the SI unit of mass. The SI units are defined in such a way that, when the Planck constant is expressed in SI units, it has the exact value ℎ= 6.62607015×10−34 J/Hz. It is often used with units of electronvolt (eV), which corresponds to the SI unit per elementary charge. (Wikipedia https://en.wikipedia.org/wiki/Planck_constant ).

Named after German physicist Max Karl Planck (1858–1947), the Planck constant tells us how the energy of individual photons relates to the wavelength of their radiation, as this key equation shows:

Ep = hc/λ

Where Ep is the energy of a single photon (in joules), h is the Planck constant, c is the speed of light in a vacuum, and λ is the radiation’s wavelength.

A light-emitting diode (LED) is a semiconductor device that emits light when current flows through it. Electrons in the semiconductor recombine with electron holes, releasing energy in the form of photons. The color of the light (corresponding to the energy of the photons) is determined by the energy required for electrons to cross the band gap of the semiconductor. The turn on voltage is typically close to the bandgap of the semiconductor used in fabrication of the LED. The turn on voltage is the voltage that the LED begins to emit light. When a resistor is connected in series with an LED, the voltage across the LED is close to the turn on voltage and can be used roughly as a measure of the bandgap of the LED.

The wavelength of the light emitted by the LED is also determined by the bandgap. By measuring the turn on voltage of different LEDs and their emission wavelengths, we can estimate the Planck's constant h. This is what we are doing in this demonstration.

Supplies

Helper Hands to hold LED and fiber from spectrometer

Arduino UNO as a voltage source

Jumper cables

Voltmeter

Multicolored LED kit

430 Ohm (or about 1 KOhm) resistors

Spectrometer (e.g. Spectryx Blue)

Step 1: Connect the LEDs to Power Through Resistor

Use jumper cables to connect the Arduino to the LED, inserting a 430 Ohm resistor in between as shown in image. Connect the jumper cables to LED on one end. Connect the other jumper cables to Arduino UNO 5 V and GND.

Note that LEDs must be connected with correct polarity or won't work.

Step 2: Measure the Voltage Across the LED and the Wavelength

While the LED is on, measure the voltage across the LED using the voltmeter. Note this value. Also measure the LED spectrum with the help of crocodiles on the helping hand. Make sure you don't short the LED when clamping. Record the spectral data or simply note the peak wavelength for the color you are measuring. Make a table of the peak wavelengths and LED voltages for the colored LEDs in the kit. I recommend using UV, Blue, Green, Yellow, and Red LEDs. You can make a plot of the measurement results as shown in figure using your software of choice (Octave recommended, because it can also be used to do a linear line fit).

Step 3: Convert the Wavelength Into Frequency and Replot

Frequency is f=c/λ , where c is the speed of light (299,792,458 m/s). Energy in Joules is voltage of LED times e=1.60217663E-19 coulombs (electronic charge). Replot the graph as frequency vs energy as shown in figure.

Step 4: Linear Line Fit to Plot and Extract Slope

Do a line fit and extract slope. Octave has a built in function for this purpose, polyfit , where an order of n=1 can be specified to fit a line to a dataset.

Please take a look at the attached Octave script that includes all the data and methods to generate the plots in this demo.

In the plot f vs E, h=1/slope, which is found as 6.6921e-34 J/Hz . This is very close to the tabulated value of 6.62607015×10−34 J/Hz.

Note that, room temperature has an effect on the bandgap of the LEDs, affecting the emission wavelength and turn on voltages. There seems to be slight day to night variations in the measurement which are probably due to the temperature fluctuations of the room.