Difference between revisions of "Determination of Planck's Constant"

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=Description of the Experiment=
 
=Description of the Experiment=
In this experiment is possible study the photoelectric effect and calculate Planck's constant. Using 5 colored leds and a photoelectric cell.
+
The purpose of this experiment is to study the photoelectric effect and calculate Planck's constant using 5 different coloured leds and a photoelectric cell.
 +
 
  
 
=Experimental Apparatus=
 
=Experimental Apparatus=
[[File:Espectro_dos_leds.png|thumb|alt=Espectro dos Leds|Figure 1: Led's spectrum.]]
+
[[File:Espectro_dos_leds.png|thumb|alt=Led's spectrum|Figure 1: Led's spectrum.]]
The photoelectric cell is from the apparatus PASCO AP-9368. It works like a capacitor where an electric conductor emits photoelectrons.
+
The photoelectric cell is from the PASCO AP-9368 apparatus. It works like a capacitor where one of the plates emits photoelectrons when excited by light.
The potential of the photocell will increase with the emission of photoelectrons. And the stop potential will depend on the wavelength of the incident light, photoelectric effect.
+
The potential between the plates of the photocell will increase with the emitted photoelectron accumulation. After reaching a certain voltage, the stopping potential will be greater than the photoelectron's kinetic energy, and these will not have enough energy to reach the second plate. This voltage will depend on the wavelength of the incident light (photon energy).
  
The photocell is connected to ground to discharge, after each use.  
+
After each experiment the photocell is connected to ground to discharge.
  
The leds have different spectrum and intensities so the time to reach the stop potencial may vary.  
+
The leds have different efficiency, leading to different intensities for a chosen brightness. Therefore, the charging time will be different between colors.
 +
 
 +
The graph in figure 1 shows each led spectrum, table 1 shows more details.
 +
 
 +
Please note that the wavelength for each led depends on the junction temperature, and this will depend not only on room temperature, but also on the current passing through the junction. This effect (red shift) is more noticeable in warm-coloured led (green-red) than in the blue one, so the calculation of Planck's constant will suffer a shift.
  
 
{| border="1" style="text-align: center;"
 
{| border="1" style="text-align: center;"
|+ Table 1 – Led's spectrum peaks
+
|+ Table 1 – Led spectrum peaks; these values should be used considering the wavelength dispersion with 7 mA.
 
|-
 
|-
 
!Color
 
!Color
 
!Frequency (THz)
 
!Frequency (THz)
 
!Wavelegth (nm)
 
!Wavelegth (nm)
 
 
!Espectros dos leds
 
!Espectros dos leds
 
|-
 
|-
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=Protocol=
 
=Protocol=
The number of photoelectrons emited will increase with the intensity of light. (corpuscular behavior of light)
+
The number of photoelectrons emitted will increase with the intensity of light (corpuscular behaviour of light).
  
#Choose a led to light up on the photocell  
+
#Choose a led to light upon the photocell  
#Measure the stopping potential. Take note the time necessary to reach the maximum potential.
+
#Measure the stopping potential. Note the time necessary to reach the maximum potential.
#Repit step 2 for diferent intensities.
+
#Repeat step 2 for different intensities.
  
 
{| border="1" style="text-align: center;"
 
{| border="1" style="text-align: center;"
|+ Exemple of a table  
+
|+ Example of a table  
 
|-
 
|-
 
!Color #1 __________(name)
 
!Color #1 __________(name)
Line 88: Line 92:
  
  
[[File:Constante de Planck.png|thumb|alt=Constante de Planck|Figure 2: Potential vs. Peaks Frequency of the spectrum ]]
+
[[File:Constante de Planck.png|thumb|alt=Planck's constant|Figure 2: Potential vs. Peaks Frequency of the spectrum ]]
  
The photoelectron's kinetic energy depend only on the frequency of light. If the frequency of light increase the energy will increase.
+
The kinetic energy of the photoelectron depends only on the frequency of the incident light. If the frequency of light increases, the energy will increase.
  
#Obtain the stop potentials for different color leds.
+
#Obtain the stop potentials for different colour leds.
#Make a graphic of Stop Potential vs Frequency and obtain Planck's constant.
+
#Draw a graphic of Stop Potential vs Frequency. Fit it to \( V = \frac{h}{e} \nu - \frac{W_0}{e} \) and obtain Planck's constant.
  
  
 
{| border="1" style="text-align: center;"
 
{| border="1" style="text-align: center;"
|+ Exemple of a table  
+
|+ Example of a table  
 
|-
 
|-
!Color (name)
+
!Colour (name)
!Frequency (Hz)
+
!Frequency (THz)
!Stop Potencial (V)
+
!Stop Potential (V)
 
|-
 
|-
 
 
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|}
 
|}
  
=Protocolo Avançado=
+
=Advanced Protocol=
 +
# Study the photocell charging process for different intensities.
 +
# Find the wavelength expected value using the led spectra.
 +
# Use those values for a new graphical fitting and compare the results.
 +
# Redo the voltage vs. frequency graph, this time with error bars.
 +
 
 +
Note: This setup uses a 12bit ADC from 0V to 5V.
  
#Calcule a constante de tempo da montagem.
 
#Encontre os valores esperados do comprimento de onda apartir dos espectros dos leds.
 
#Refaça o gráfico tensão vs frequência com barras de erro.
 
  
Nota: Utilizou-se um ADC 12bit no intervalo de 0V a 5V.
+
=Theoretical Principles=
  
=Princípios Teóricos=
+
==Photoelectric effect==
 +
The photoelectric effect happens when a metal surface is illuminated by a light with a given frequency, causing electrons to be freed.
 +
When a photon with energy \( E \) collides with a metal, it transfers its energy to an electron in the crystal structure. The emission of electrons is deeply connected to the frequency of the light shone upon the surface.
 +
For each metal there is a critical frequency, \( \nu _0 \). If incident radiation has frequency bellow \( \nu _0 \), then there are no emitted electrons. If the former is above the latter, then the kinetic energy of the emitted electrons is proportional to the energy of the photons.
 +
The light intensity changes only the number of emitted photoelectrons, not its energy (this goes against what is expected in the classical theory of radiation).
  
==Efeito Fotoeléctrico==
+
Einstein proposed the following explanation: light is made of photons with a given energy, proportional to frequency (\( \nu \)):  
O efeito foto-eléctrico consiste na emissão de electrões da superfície de  um metal quando este é iluminado por luz de uma dada frequência (\( \nu \)). Um fotão de energia \( E = h \nu \) ao incidir no metal transfere a sua energia a um electrão pertencente a um dos átomos na rede cristalina do metal. A emissão de electrões do metal é muito dependente da frequência da luz incidente. Para cada metal, existe uma frequência critica, \( \nu _0 \), tal que para luz incidente com frequência inferior não há foto-electrões arrancados. Por outro lado, para frequências superiores, a energia dos foto-electrões emitidos aumenta linearmente com a energia dos fotões incidentes. A intensidade da luz incidente afecta somente o número de foto-electrões emitidos, mas não a sua energia, contrariamente ao que seria de esperar na teoria clássica da radiação.
 
Einstein propôs a seguinte explicação para o fenómeno: a luz é transportada por fotões com uma dada energia E associada à frequência da luz \( \nu \):
 
  
\[
+
<math>
E = h \times \nu  
+
E = h \nu  
\]
+
</math>
  
em que ''h'' é a constante de Planck. O efeito foto-eléctrico deve-se a uma colisão do fotão com o electrão, em que aquele transmite a este toda a sua energia. Tendo em conta que a energia de um electrão no vazio e dentro do metal é diferente (mais elevada no vazio), só se verifica efeito foto-eléctrico se a energia transmitida pelo fotão for superior à diferença entre estas  duas energias (ver Fig. 1). Assim, a energia com que o electrão abandona o metal é igual à energia do fotão menos a energia "gasta" para o electrão abandonar o metal:
+
Where \( h \) is Planck's constant. The photoelectric effect is, essentially, a collision between a photon (light) and an electron (in the metal) where the former gives all its energy to the latter. Since the electron's energy is higher when in vacuum (as opposed when it's bound to the crystal structure of the metal), the electron is only freed if the photon energy is higher than the difference between the energy of the electron in vacuum and in the metal (figure 1). Thus, the electron leaves the metal with energy equal to the energy of the photon minus the amount "spent" when the electron leaves the metal:
  
\[
+
<math>
E = h \times \nu - e \times \phi,
+
E = h \nu - e \phi
\]
+
</math>
  
em que ''e'' é a carga do electrão e \( \phi \) é a diferença de ''workfunction''.
+
where \( e \) is the electron charge and \( \phi \) is the diference in ''workfunction''. As the light frequency decreases, the photons have less energy. Bellow a certain critical frequency \( \nu _0 \), no more photoelectrons are emitted. In this case, \( E_{max} = 0 \) and we can write:
Tal como foi anteriormente enunciado, à medida que a frequência da luz incidente decresce, os fotões têm menos energia, e a partir de uma frequência critica \( \nu _0 \) não são emitidos mais foto-electrões. Neste caso, \( E _{max} = 0 \) e da Eq. l tiramos
 
  
\[
+
<math>
h \nu _0 = e \phi \quad ou \quad \nu _0 = \frac{e}{h} \phi
+
h \nu _0 = e \phi \quad or \quad \nu _0 = \frac{e}{h} \phi
\]
+
</math>
  
[[File:Plank-teo1.png|thumb]]
+
==In This Setup==
 +
A photoelectric cell is a device where light shines upon a metalic surface (cathode) and excites the electrons that will be collected by a concentric metalic surface (anode), as shown in the figure on the right (it is basically a semi-cilindric capacitor with very low capacity). If you connect the cathode and anode through an external circuit, you can measure the current created by the photoelectrons. In a more generic sense, the maximum kinetic energy of the electron is determined by applying a stopping potencial between the cathode and the anode to prevent electrons from reaching the anode. This way, when current no longer flows, we will know the value of the current.
  
 +
[[File:Plank-celula1.png|thumb]]
  
==Efeito Fotoeléctrico==
+
We begin by applying a voltage to the vell (approximatlly 9V, because the capacitor is discharged before starting the experiment by short-circuiting the terminals). With the cell in series with the capacitor, the latter will charge as electrons are generated in the cell. This creates an electric current through the cell. As the capacitor charges, the voltage between its terminals increases, reducing the volltage between the terminals of the cell(because \( V_{bat} = V_{Cap} + V_{cell} = constant \)). When the cell voltage reaches \( V_c = \frac{h  \nu - e \phi}{e} \), current no longer flows and the capacitor maintains a constant voltage.
Uma célula foto-eléctrica é um dispositivo onde a luz incidente sobre uma superfície metálica (cátodo) excita electrões que vão ser colectados numa superfície metálica concêntrica (ânodo), tal como exemplificado na figura, e que na prática é um condensador semi-cilíndrico de capacidade muito baixa. Ligando o ânodo e o cátodo por um circuito externo, podemos medir a corrente produzida pelos foto-electrões. No caso mais genérico, a energia cinética máxima dos foto-electrões emitidos é determinada aplicando um potencial de paragem, Vc entre o ânodo e o cátodo de modo a impedir que os foto-electrões emitidos pelo cátodo atinjam o ânodo. Deste modo, deixa de fluir corrente no circuito.
 
  
[[File:Plank-celula1.png|thumb]]
+
[[File:Constante de Planck.png|thumb|alt=Constante de Planck|Voltage (stopping power) vs. Maximum in the frequency of light]]
  
A célula inicialmente tem aplicada a tensão da fonte, aproximadamente 9V uma vez que o condensador é descarregado no início da experiência (é efectuado um curto-circuito aos seus terminais). Como a célula está em série com o condensador, este vai carregando à medida que são gerados foto-electrões, criando uma corrente eléctrica no circuito que atravessa a célula. À medida que o condensador carrega, aumenta a diferença de potencial aos seus terminais, o que obriga a diminuir a tensão aos terminais da célula (note que \( V_{bat} = V_{Cond} + V_{célula} = constante \)). Quando a diferença de potencial nas placas da célula for igual a \( V_c = \frac{h \times \nu - e \times \phi}{e} \), deixa de fluir corrente no circuito e o condensador passa a ter uma tensão constante aos seus terminais.  
+
Knowing two or more voltage values for certain frequencies, the user can do a linear regression and determine both \( \phi \) and Planck's constant. In the following graphic we can see the relation between the stopping potencial V and the light frequency for a given metal. This function is a linear function with slope = \( \frac{h}{\epsilon} \) and y-intercept = \( \phi \).  
  
Conhecendo dois ou mais valores dessa tensão para determinadas frequências [1] pode-se efectuar uma regressão linear e determinar não só \( \phi \) mas também a constante de Planck. No gráfico seguinte esquematizamos a dependência da tensão de paragem V em função da frequência da luz incidente para um dado metal. A função em causa corresponde a uma recta de declive \( \frac{h}{e} \) e coeficiente na origem \( \phi \).
 
  
[[File:Plank-celula2.png|thumb]]
+
=Historical Elements=
 +
In 1921, Albert Einstein won the Nobel Physics Prize for his work on the photoelectric effect.
  
  
=Elementos Históricos=
+
=Links=
Em 1921 foi atribuído a Albert Einstein o Prémio Nobel da Física pelas suas descobertas no efeito fotoeléctrico.
+
*[[Determinação da Constante de Planck | Portuguese version (Versão em Português)]]

Latest revision as of 18:47, 24 May 2015

Description of the Experiment

The purpose of this experiment is to study the photoelectric effect and calculate Planck's constant using 5 different coloured leds and a photoelectric cell.


Experimental Apparatus

Led's spectrum
Figure 1: Led's spectrum.

The photoelectric cell is from the PASCO AP-9368 apparatus. It works like a capacitor where one of the plates emits photoelectrons when excited by light. The potential between the plates of the photocell will increase with the emitted photoelectron accumulation. After reaching a certain voltage, the stopping potential will be greater than the photoelectron's kinetic energy, and these will not have enough energy to reach the second plate. This voltage will depend on the wavelength of the incident light (photon energy).

After each experiment the photocell is connected to ground to discharge.

The leds have different efficiency, leading to different intensities for a chosen brightness. Therefore, the charging time will be different between colors.

The graph in figure 1 shows each led spectrum, table 1 shows more details.

Please note that the wavelength for each led depends on the junction temperature, and this will depend not only on room temperature, but also on the current passing through the junction. This effect (red shift) is more noticeable in warm-coloured led (green-red) than in the blue one, so the calculation of Planck's constant will suffer a shift.

Table 1 – Led spectrum peaks; these values should be used considering the wavelength dispersion with 7 mA.
Color Frequency (THz) Wavelegth (nm) Espectros dos leds
Blue.ab 638.7 469.70 File:Espectro Azul.ab.txt
Blue 684.6 438.20 File:Espectro Azul.txt
Red 482.2 622.21 File:Espectro Vermelho.txt
Yellow 514.4 583.16 File:Example.txt
Green 530.8 565.22 File:Espectro Verde.txt

Protocol

The number of photoelectrons emitted will increase with the intensity of light (corpuscular behaviour of light).

  1. Choose a led to light upon the photocell
  2. Measure the stopping potential. Note the time necessary to reach the maximum potential.
  3. Repeat step 2 for different intensities.
Example of a table
Color #1 __________(name) Intensity (%) Stop Potential (V) Time (s)
  100    
  80    
  60    
  40  
  20  


Planck's constant
Figure 2: Potential vs. Peaks Frequency of the spectrum

The kinetic energy of the photoelectron depends only on the frequency of the incident light. If the frequency of light increases, the energy will increase.

  1. Obtain the stop potentials for different colour leds.
  2. Draw a graphic of Stop Potential vs Frequency. Fit it to \( V = \frac{h}{e} \nu - \frac{W_0}{e} \) and obtain Planck's constant.


Example of a table
Colour (name) Frequency (THz) Stop Potential (V)
 
 
 
 
 

Advanced Protocol

  1. Study the photocell charging process for different intensities.
  2. Find the wavelength expected value using the led spectra.
  3. Use those values for a new graphical fitting and compare the results.
  4. Redo the voltage vs. frequency graph, this time with error bars.

Note: This setup uses a 12bit ADC from 0V to 5V.


Theoretical Principles

Photoelectric effect

The photoelectric effect happens when a metal surface is illuminated by a light with a given frequency, causing electrons to be freed. When a photon with energy \( E \) collides with a metal, it transfers its energy to an electron in the crystal structure. The emission of electrons is deeply connected to the frequency of the light shone upon the surface. For each metal there is a critical frequency, \( \nu _0 \). If incident radiation has frequency bellow \( \nu _0 \), then there are no emitted electrons. If the former is above the latter, then the kinetic energy of the emitted electrons is proportional to the energy of the photons. The light intensity changes only the number of emitted photoelectrons, not its energy (this goes against what is expected in the classical theory of radiation).

Einstein proposed the following explanation: light is made of photons with a given energy, proportional to frequency (\( \nu \)):

[math] E = h \nu [/math]

Where \( h \) is Planck's constant. The photoelectric effect is, essentially, a collision between a photon (light) and an electron (in the metal) where the former gives all its energy to the latter. Since the electron's energy is higher when in vacuum (as opposed when it's bound to the crystal structure of the metal), the electron is only freed if the photon energy is higher than the difference between the energy of the electron in vacuum and in the metal (figure 1). Thus, the electron leaves the metal with energy equal to the energy of the photon minus the amount "spent" when the electron leaves the metal:

[math] E = h \nu - e \phi [/math]

where \( e \) is the electron charge and \( \phi \) is the diference in workfunction. As the light frequency decreases, the photons have less energy. Bellow a certain critical frequency \( \nu _0 \), no more photoelectrons are emitted. In this case, \( E_{max} = 0 \) and we can write:

[math] h \nu _0 = e \phi \quad or \quad \nu _0 = \frac{e}{h} \phi [/math]

In This Setup

A photoelectric cell is a device where light shines upon a metalic surface (cathode) and excites the electrons that will be collected by a concentric metalic surface (anode), as shown in the figure on the right (it is basically a semi-cilindric capacitor with very low capacity). If you connect the cathode and anode through an external circuit, you can measure the current created by the photoelectrons. In a more generic sense, the maximum kinetic energy of the electron is determined by applying a stopping potencial between the cathode and the anode to prevent electrons from reaching the anode. This way, when current no longer flows, we will know the value of the current.

Plank-celula1.png

We begin by applying a voltage to the vell (approximatlly 9V, because the capacitor is discharged before starting the experiment by short-circuiting the terminals). With the cell in series with the capacitor, the latter will charge as electrons are generated in the cell. This creates an electric current through the cell. As the capacitor charges, the voltage between its terminals increases, reducing the volltage between the terminals of the cell(because \( V_{bat} = V_{Cap} + V_{cell} = constant \)). When the cell voltage reaches \( V_c = \frac{h \nu - e \phi}{e} \), current no longer flows and the capacitor maintains a constant voltage.

Constante de Planck
Voltage (stopping power) vs. Maximum in the frequency of light

Knowing two or more voltage values for certain frequencies, the user can do a linear regression and determine both \( \phi \) and Planck's constant. In the following graphic we can see the relation between the stopping potencial V and the light frequency for a given metal. This function is a linear function with slope = \( \frac{h}{\epsilon} \) and y-intercept = \( \phi \).


Historical Elements

In 1921, Albert Einstein won the Nobel Physics Prize for his work on the photoelectric effect.


Links