Objectives:

• To use LEDs and the equation  $eV = \frac{hc}{\lambda}$  to estimate the value of Planck constant h.
• To be able to determine the Planck constant using different coloured LEDs.

The Planck constant was introduced by Max Planck back in 1900. The significance of Planck’s constant is that a photons energy (a ‘quanta’ of energy) can be determined by frequency of radiation and Planck’s constant.

The Planck constant can be determined with the use of an LED and power source (as well as an Ammeter and Voltmeter for measurements to be taken). The theory is that with the potential difference up high enough to emit radiation from the LED, the energy the electrons are giving the photons is enough to emit them from the device. Due to the nature of how LED’s work, the potential difference across the LED needs to only just be enough to emit photons. If the e.m.f. supplied to the circuit increases beyond this, the voltage goes on to increasing the current around the circuit and so increases the number of electrons, rather than giving energy to the electrons and therefore photons.

The minimum voltage required to emit a photon from an LED is known as the threshold voltage, this value is partly dependent on the colour of light that the LED is emitting.

A circuit required to determine Planck’s constant could look like one of the following;

The power source is supplying a potential difference of $V = \frac{W}{Q}$ to the LED. If this LED is only just emitting light, the energy supplied from the power source, $W$  is going into the energy of the photons that are being emitted $E = hf$ , we can therefore write;

$V = \frac{hf}{Q}$

Since it is the electrons that are emitting the photons, $Q = e$ ;

$V = \frac{hf}{e}$

Rearranging for  $h$ gives;

$h = \frac{eV}{f}$

So providing the frequency is known, Planck’s constant can be determine. The frequency can be determined using a number of methods; sometimes the supplier will give the frequency of the light, otherwise a data sensor could be used or the methods used in diffraction could be utilised.

Doing this practical with a red LED of wavelength 485 nm gave a p.d. of  1.80 V, substituting these values into an appropriate equation gives;

$h = \frac{eV}{f} = \frac{eV\lambda}{c}$

$h = \frac{1.6 \times 10^{-19} \times 1.80 \ times 685 \times 10^{-9}}{3.00 \times 10^{8}}$

$h = 6.58 \times 10^{-34}$

For one value of voltage and wavelength this is not a bad first result – just 0.81 % off the true value.

The experiment can be carried out with numerous LED lights (and hence several different wavelengths) and a number of different values for the threshold voltage, $V_{threshold}. With multiple pieces of data, a graph can be drawn to compare the variation of wavelengths to$latex V_{threshold} $’s. $eV = hf = \frac{hc}{\lambda}$ Rearranging for $V$ gives; $V = \frac{hc}{\lambda e}$ Since $c$ and $e$ are constants, providing $h$ is a constant, this arrangement of the equation tells us that a graph of $V$ against $\frac{1}{\lambda}$ should prove to be a straight line; $V = \frac{hc}{e} \times \frac{1}{\lambda}$ $y = m \times x$ As this equation has no y-intercept value, we should expect the line of best fit to pass through the origin. The gradient of a graph of $V$ against $\frac{1}{\lambda}$ will be equal to $\frac{hc}{e}$ . A Planck’s Constant Apparatus kit could also be used to gather the relevant information (an example is in the image below): Using the following data, you can complete the table, draw a graph, determine the gradient and hence identify what Planck’s constant is; The following graph can be drawn from this data; The equation of the line of best fit shown here is $y= 1.247 \times 10^{-6}x - 0.0064$ Based on the equation shown earlier, the line should pass through the origin (which it almost does); $V = \frac{hc}{e} \times \frac{1}{\lambda}$ $y = m \times x$ Using the gradient of the line, $1.247 \times 10^{-6}, Planck's constant can be found. Since$latex \frac{hc}{e} = 1.247 \times 10^{-6}$, this can be rearranged for h;

$h = \frac{1.247 \times 10^{-6} \times e}{c}$

$h = 6.65 \times 10^{-34} Js$

The actual value is $h = 6.63 \times 10^{-34} Js$  so the value calculated here is just 0.3 % off the actual value.

In order to increase accuracy the experiment should be carried out in a darkened room OR the LED should be viewed through a narrow tube so that the observer can identify more precisely when the LED first begins emitting light (with an increasing voltage), this way a more accurate threshold voltage can be determined.

CAUTION – In doing this, the voltage can be turned up to quickly and the LED may illuminate very brightly and directly into the users eye.