This essay focuses on using LEDs (Light Emitting Diodes) to explore the photoelectric effect, a physical phenomenon by which if the surface of an electrode is struck by light of sufficient frequency, an electron will absorb a photon and be ejected from that surface.
The scope of this experiment is to investigate the relationship between activation voltage, which is the minimum voltage required for the LED to light up, and the wavelength of various LED radiations, and, using this information, to get an experimental measurement of Planck’s constant (h). The experiment involves two methods: for the first, the activation voltage of the LEDs will be determined by performing the experiment in a dark room and looking at when the LEDs cease to emit light; the second entails setting up a circuit with a voltmeter connected in parallel and an ammeter connected in series to a variable resistor and a LED. The current will then be decreased until it is equal to 0 by increasing the resistance of the circuit, and the voltage across the LED when this happens (known as the activation voltage) will be recorded. This will then be extended to include a range of wavelengths spanning from Infrared to blue radiation.
I was quite sure that the topic of my essay would be related to Quantum Physics, a branch of Physics which I find interesting and peculiar because it concerns the extremely small. In the end, I opted for an experimental essay on the photoelectric effect because I am fascinated by its implications, which truly revolutionised the whole of classical Physics, and I would like to see it myself. Specifically, I chose to focus on a method using LEDs rather than a classical experiment using a cathode and an anode because I find that this method sheds more light on the photoelectric effect by looking at it from the perspective of the activation voltage rather than looking at well-known results involving the frequency of light. I believe this essay is relevant to me because it allows for an exploration of a field of Physics which is still unknown to me and deepens my knowledge of Physics by delving into a cutting-edge topic such as Quantum Physics.
How does activation voltage depend on the wavelength of LED radiation measured in the range 400 to 800 nm and how can this information be used to measure Planck’s constant?
Between 1886 and 1887 Heinrich Hertz discovered that an electric discharge between two electrodes has a higher chance of occurring if ultraviolet light falls on one of the electrodes. Lenard discovered that this is because it facilitates the discharge by causing electrons to be ejected from the cathode’s surface. The emission of electrons from a surface, due to light, is called the photoelectric effect.
A value V_{0} , the stopping voltage or potential, is reached when the photoelectric current drops to 0. This is because when this happens, electrons have the minimum energy needed to be ejected from the cathode, barely reach the collecting plate and then accelerate backwards towards the cathode. Hence, there is no current. The stopping voltage multiplied by the charge on an electron gives a measurement of the kinetic energy (K_{max}) of the fastest electron ejected from the cathode-
K_{max} = eV_{0} (1)
Albert Einstein found that the photoelectric effect revolutionised physics, because there are three main aspects of it which cannot be explained in terms of the classical wave theory of light-
In 1905, Einstein proposed that electromagnetic or radiant energy, once radiated, was quantised into “concentrated bundles”, which we now call photons. Einstein assumed that this bundle of energy is initially located in a small volume of space, and that, as it moves away from the source with velocity c, it stays localised. Another assumption made by Einstein was that during the photoelectric effect one photon is completely absorbed by one electron in the cathode.
Planck, however, was the first to assume that the energy, E, of the photon (bundle) is related to its frequency, f, by the equation-
E = hf (2)
When one electron is ejected from the surface of a metal, its kinetic energy K is given by-
K = hf - w (3)
where hf gives the energy of the absorbed photon and w is the work needed to force the electron to be ejected from the metal. Work is required to overcome the attraction felt by the electron to the other atoms on the surface of the metal. Moreover, there are losses of kinetic energy to be considered. If there are no internal losses of energy involved in the ejection of the electron, that is, the electrons do not collide with other particles, losing some energy as thermal energy, then the kinetic energy is given by-
K_{max} = hf - w_{0} (4)
where w_{0} is the work function of the metal, which is the minimum energy required for an electron to pass through the metal’s surface and escape the attractive forces binding it to the metal.
Equation (4) shows that if K_{max} = 0, the electrons have just enough energy to be ejected from the metal, and we have-
hf_{0} = w_{0} (5)
This shows that a photon, having frequency f_{0}, would have the minimum energy required for the electrons to be ejected. This is further evidence that if the frequency is reduced below f_{0}, regardless of the illumination, the photons will not eject any electron.
Equation (4) can be rewritten using a substitution K_{max} = eV_{0}, which comes from equation (1). This gives-
eV_{0} = hf - w_{0} (6)
Einstein’s equation can be rearranged further, using the wave equation \(f=\frac{c}{\lambda}\) , into-
\(eV_0=\frac{hc}{\lambda}-\frac{hc}{\lambda_0}\) (7)
where λ_{0} is the minimum wavelength needed for a photon to eject an electron.
This is related to the threshold frequency, which is the minimum frequency of light for an electron to be ejected from a metal. After dividing by e, we get-
\(V_0=\frac{hc}{e}\frac{1}{\lambda}-\frac{hc}{e\lambda_0}\) (8)
Which is in the form y = mx + c. A graph of V_{0} against \(\frac{1}{\lambda}\) will therefore have a gradient equal to \(\frac{hc}{e}\) and a y-intercept of \(-\frac{hc}{e\lambda_0}\).
There are, however, other methods for conducting the photoelectric experiment, using semiconducting devices known as LEDs. LEDs are made from a thin layer of doped semiconductor material (in this experiment GaP) and when forward biased emit light at a specific wavelength, converting electrical energy into light energy. Lossev experimented on electro-luminescence effects and concluded that they represented the inverse of the photoelectric effect – a current producing light rather than light producing a current.
In 1951, Welker combined equiatomic quantities of an element from the third group and one element from the fifth group of the Periodic Table, creating compounds which showed a wide range of band gaps (E_{g}). The frequency f of these compounds is given by
\(f=\frac{E_g}{h}\) (9)
Most of these frequencies correspond to wavelengths in the visible region of light. Today, most of these compounds are used as LEDs.
LEDs consist of a p-n junction diode. If no external voltage is applied, a voltage V_{D} is generated in the area between the n- and p- type materials, known as a depletion layer. This layer prevents electrons from leaving their region and entering the other region.
If an external voltage V is applied, the layer between the n- and p- regions reduces to e(V_{D} − V).
If V ≈ V_{D} the barrier’s thickness is almost 0 and electrons can flow from one side to the other. When electrons are injected into the depletion region, they eject a photon of energy
hf = E_{g }(10)
For doped semiconductors, such as the materials used in my LEDs,
eV_{D} = E_{g}_{ }(11)
Assuming that all the energy of the electrons in the depletion region is converted into light, the frequency of this light is given by equation (10). Therefore, substituting eV_{D}_{ }for Eg in equation (10)-
hf = eV_{D} (12)
Or, equivalently, the photoelectric effect can be demonstrated with LEDs because the energy of the ejected photons is related to the activation voltage by the equation
eV_{threshold} = E_{light} (13)
where V_{threshold} is the activation voltage and E_{light} = hf
This shows that V_{D} ≡ V_{threshold}, which is the activation voltage. Planck’s constant can then be measured, since the wavelength of light emitted is known.
I think that as the independent variable, the wavelength of LED radiation, increases, the dependent variable, the activation voltage, will decrease, hence I predict the two are strongly negatively correlated. Their inverse proportionality can be inferred from the equation E = hf, where E is the radiant energy, f is the frequency of radiation and h is the constant of proportionality between the two which is nowadays known as Planck’s constant, which sets the scale for these kinds of experiments, concerned with low energies and phenomena at a microscopic scale. The radiant energy is equivalent to the kinetic energy; from equation (1), K_{max} = eV_{0}. Then, \(eV_0=\frac{hc}{\lambda}\) which shows that activation voltage and wavelength have an inverse relationship.
However, these variables will not be inversely proportional. This is because the best-fit line will not pass through the origin since the work function of the LEDs will determine the y–intercept of the graph of V against \(\frac{1}{\lambda}\) . I predict that the value for the y–intercept will be small and negative because looking at the y–intercept (the constants \(-\frac{hc}{e\lambda_0}\)), I can see that their orders of magnitude approximately cancel out, hence the y–intercept should be close to 1, indicating a very strong negative correlation between the two variables and excluding the possibility of an indirect proportionality.
I also predict that the activation voltage for the blue LED will be significantly higher because it has quite a small wavelength compared with the other three. Similarly, I also think that the infrared LED will have quite a small activation voltage because it has a substantially higher wavelength than the other LEDs.
Independent variable- the wavelength of LED radiation, which changes for every different LED. The values for the wavelength of the LED radiation are obtained from the data sheets of Arduino, Shenzhen Jingyuanxing Photoelectric Co., Ltd and Kingbright, the companies from which the LEDs used in this experiment come from. The value for the LEDs used is “dominant wavelength”; this is because it is the wavelength which most probably is most similar to the wavelength of the light emitted by the LEDs in the experiment. The absolute uncertainties on the wavelength vary from LED to LED, and are as follows-
\(\Delta\lambda_{blue}=±5nm\)
\(\Delta\lambda_{red}=±5nm\)
\(\Delta_{green}=±10 nm\)
\(\Delta\lambda_{yellow}=±5nm\)
Dependent variable- the activation voltage, measured by eye and by using a voltmeter, in two different methods. The first method involves seeing when the LED stops emitting light and is enhanced by the use of a dark room. The second method involves measuring the activation voltage by looking at when the current through the LED reaches 0 (at this point the LED will have stopped emitting light). The absolute uncertainty on the voltmeter is ∆V = ±0.001 V; the absolute uncertainty in the current measured by the ammeter is \(\Delta I=±0.05A\).
Control variables
Method 1– obtaining the activation voltage in the dark
Method 2– obtaining the activation voltage using an ammeter and voltmeter in the light
Hazard | Risk | Control |
---|---|---|
Electricity | Getting a shock | Handle apparatus carefully. Turn off the power supply in between trials. |
Heat from the wires and power supply | Getting burnt. | Turn off the power supply during trials so as to not let it heat up. Do not touch the wires throughout the experiment. |
Light from the LED | Damage to the eye | Especially when in the dark room, do not stare at the LED closely nor directly. Stare at the LED only when the intensity of the light being emitted has been reduced. |
Water, especially if in contact with electricity | Electrocution | Ensure that any sources of water are far away from the electrical equipment. If water is on the floor, dry the floor using paper towels. |
Wavelength of radiation (nm) [λ] | Activation voltage (V) [V_{0}] ∆V_{0} = ±0.001 V |
---|---|
630 | 0.938 |
630 | 0.969 |
630 | 0.894 |
630 | 0.914 |
630 | 0.918 |
Average activation voltage= 0.93±0.04V | |
588 | 0.952 |
588 | 0.944 |
588 | 0.987 |
588 | 0.956 |
588 | 0.933 |
Average activation voltage= 0.95±0.03V | |
570 | 1.020 |
570 | 0.975 |
570 | 0.984 |
570 | 0.949 |
570 | 0.940 |
Average activation voltage= 0.97±0.04V | |
460 | 1.332 |
460 | 1.311 |
460 | 1.317 |
460 | 1.280 |
460 | 1.303 |
Average activation voltage= 1.31±0.03V |
Wavelength of radiation (nm) [λ] | Activation voltage (V) [V0] ∆V0 = ±0.001V |
---|---|
630 | 1.026 |
630 | 1.029 |
630 | 1.021 |
630 | 1.026 |
630 | 1.025 |
Average activation voltage= 1.025±0.004V | |
588 | 1.097 |
588 | 1.099 |
588 | 1.096 |
588 | 1.099 |
588 | 1.105 |
Average activation voltage= 1.099±0.004V | |
570 | 1.148 |
570 | 1.150 |
570 | 1.142 |
570 | 1.149 |
570 | 1.143 |
Average activation voltage= 1.146±0.003V | |
460 | 1.433 |
460 | 1.425 |
460 | 1.405 |
460 | 1.436 |
460 | 1.422 |
Average activation voltage= 1.42±0.02V | |
790 | 0.620 |
790 | 0.617 |
790 | 0.612 |
790 | 0.610 |
790 | 0.617 |
Average activation voltage= 0.615±0.005V |
The absolute uncertainty on the voltage was found by considering the number of decimal places with which the voltmeter delivered recordings for the voltage across the LED, which was three. The uncertainty lies on the last digit, therefore ∆V = ±0.001 V.
The absolute uncertainty on the wavelength of each LED is variable, and was obtained by considering the range of dominant wavelength in the data sheet of the LEDs and dividing it by two, so for example for the red LED: \(\Delta\lambda=\frac{635-625}{2}=±5nm\)
The uncertainty in the current is ∆I = ±0.05 A, obtained by taking the smallest measurement of the analogue ammeter (0.10 A) and dividing it by two.
The uncertainty on the gradient for experiment a was calculated using the formula-
\(\Delta m=\frac{m_{max}-m_{min}}{2}\)
where m ≡ \(\frac{hc}{e}\)
Substituting the values for method 1-
\(\Delta m=±100.10^{-9}m kgs^{-3}A^{-1}\)
Therefore-
m = (700 ± 100).10^{-9} m kgs^{-3} A^{-1}
Similarly, for method 2-
\(\Delta m =±90.10^{-9}mkgs^{-3}A^{-1}\)
Thus-
m = (900 ± 90).10^{-9}m kgs-^{3}A^{-1}
The uncertainty on the y–intercept is obtained in a similar way-
\(\Delta c=\frac{c_{max}-c_{min}}{2}\)
For method 1-
\(\Delta\frac{hc}{e\lambda _0}=±0.2kgm^2s^{-3}A^{-1}\)
Therefore-
\(\frac{hc}{e\lambda_0}=(-0.2±0.2)kgm^2s^{-3}A^{-1}\)
For method 2-
h = 4.10^{-34}kgm^{2}s^{-1}
The uncertainty on Planck’s constant is calculated by considering the absolute uncertainties and absolute values for the speed of light in a vacuum, the electronic charge and for the gradient of the graphs, which corresponds to \(\frac{hc}{e}\)
The formula used for this is: \(\Delta h=h\sqrt{(\frac{\Delta e}{e})^2+(\frac{\Delta m}{m})^2}\)
The term \((\frac{\Delta c}{c})^2\) is not included because the value for the speed of light in a vacuum is exact, as it is defined in terms of the second (s) and has no uncertainty.
For method 1-
\(\Delta h =\pm0.9.10^{-34}kgm^2s^{-1}\)
For method 2-
\(\Delta h=\pm0.6.10^{-34}kgm^2s^{-6}\)
Hence, the experimental values obtained for Planck’s constant are-
h = (4 ± 0.9).10^{-34}kgm^{2}s^{-1}
for the method in the dark, and
h = (5 ± 0.6).10^{-34}kgm^{2}s^{-1}
for the method in the light.
The first relationship which can be established from both the data and the graphs is that as the wavelength of the radiation emitted by the LED increases the activation voltage decreases. This is equivalent to saying that the activation voltage increases as the reciprocal of the wavelength increases. It can also be noticed that values for the activation voltage for Infrared light and blue light are quite smaller and bigger respectively compared with the values for red, yellow and green light. This is due to the fact that the wavelengths of these two LEDs were significantly higher and lower respectively when compared to the wavelengths of the other LEDs. Taking a closer look at the graphs shows that the activation voltage and the reciprocal of the wavelength are not directly proportional. This is because it is not possible to fit a straight line through all error bars and through the origin.
From the data analysis above, it can be seen that the values obtained for Planck’s constant for both methods are at more than two experimental uncertainties away from the accepted value. These are quite big errors, which show that although it is possible to combine the gradient of the graph of V_{0} against \(\frac{1}{\lambda}\) with the constants e and c, it is not clear whether this method ensures great accuracy when determining Planck’s constant; this is further reinforced by the fact that the uncertainties on the gradients of both graphs are quite large, therefore the experimental value of Planck’s constant could be much closer or much farther from the accepted value.
I believe that the background information I collected has been very useful in guiding me and in enhancing my understanding of the experiment. This is because not only was I able to fully derive relevant equations for my experiment, but I was also helped by them– for example, in predicting the relationship between my variables, and in knowing the physical meaning of the quantities in the y − intercept and slope of the graphs I used. I think that the high quality of the sources I used comes from their purpose, which is to provide guidance to students on the photoelectric effect, and their origin, since they come from people such as scientists who I consider to be authorities in their field, because they have published or have links to a lot of research on this topic.
The above analysis strongly suggests that my hypothesis was correct; I was able to correctly predict the relationship and correlation between my variables. However, I feel like to make an even stronger hypothesis I could have used a more convincing mathematical argument in support of the strong negative correlation between activation voltage and wavelength.
The procedure I followed also has its merits. First of all, having two different methods of determining activation voltage was an advantage because it allowed for comparison of the two methods and shed more light on the errors involved in this kind of experiment. Moreover, I believe that taking five trials for each LED improved my method because it allowed for a lot of data to be used to produce graphs and it reduced random errors. Other strong points of the method I used include the fact that it was very safe, and I didn’t find any risks for my safety throughout the experiment.
Nevertheless, the methods used suffered from a lot of weaknesses. First of all, something which must be considered is that there is some ambiguity in this experiment. It was chosen to take the activation voltage as the point at which the LED starts emitting light. However, the activation voltage can also be defined in terms of the I–V curve of a LED as when the current starts increasing exponentially.
It can be argued that both methods were unsuccessful, for various reasons; first of all, the ammeter and voltmeter were subject to systematic error, because sometimes even though there was no voltage applied to the circuit they were still measuring a current/voltage– this means my results are slightly skewed towards higher values. This error was compensated for by first recording the measurements of the voltage, and then adding on an amount equal to the absolute uncertainty on the voltmeter, which has eliminated this error from the experiment.
Other errors come from my own inability to fully concentrate: for example, in the dark room, the values for the activation voltage are certainly affected by some degree of random uncertainty, since it is not possible for my eye to always see the LED fade away at the same exact voltage. Unfortunately, it is very difficult to compensate for this type of random error. Furthermore, a limitation in this method is that, even in the dark room, some light was very faintly passing through the jackets used to cover the bottom of the doors, which impacts on the activation voltage in such a way that when there is light, I perceive the LED turning off faster than I would in complete darkness, hence this skewed my values to higher values.
For the method in the light, although great care was taken in trying to avoid parallax errors as far as possible, it is still subject to random uncertainty because estimating when the current is 0 using an analogue device entailed that the ‘hand’ of the ammeter would be in different positions; I believe that a digital ammeter would have probably been better for this experiment, because the measurement taken using an analogue ammeter is subject to both its inherent uncertainty as well as the human errors arising from my inability to see exactly when the ‘hand’ was indicating 0 current.
A further experiment to extend this method could involve two steps: firstly, the I–V characteristics of the LEDs used would have to be determined. This involves a graph which initially roughly approximates a straight line and then, for higher values of voltage, looks like an exponential graph. Then, a plot of ln(I) against V could be used to turn it into a straight-line graph. The x– intercept would then be used as the value for the activation voltage.
Something which could also be interesting to explore would be the work function of the LEDs. Several LEDs made out of different materials could be used in order to compare a range of work functions, for example to explore which materials are best for cathode ray tubes.
Activation voltage
Anode
Band gap
c
Cathode
Doped semiconductor
e
Electrode
Forward biased
h
Intensity
Kinetic energy
LED
n-type semiconductor
p-n junction
Photoelectron
Potential difference
Quantised energy
Radiation
Semiconductor
Threshold frequency
Wavelength
Work function
the minimum amount of energy that must be supplied for an electron to escape the surface of a metal.
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