## Extra Credit 9

### The photoelectric effect

We can predict the positions of the maxima and minima in the diffraction and interference patters produced by single and multiple slits by assuming that light is an EM wave.  Where crests meet crests and troughs meet troughs we predict and observes maxima or bright regions, and where crests meet troughs we predict and observe minima or dark regions.

### So light is a wave, these experiments settle this?  Not so fast!

In this assignment you will simulate an experiment that suggests that light is a particle.  You will investigate the photoelectric effect.  To demonstrate this effect, light is shone on a metal surface and ejected electrons are detected.  To eject an electron from a metal surface a certain amount of energy Φ must be supplied to this electron.  Φ is called the work function of the metal.  (If no energy were required to free the electrons, they would just leave ordinary pieces of metal.)  In the wave picture the energy of the light beam does not depend on the frequency, but only on the intensity, which is proportional to the square of the amplitude.

### What is observed?

If the frequency of the light is higher than the cutoff frequency fc, then electrons are released.  No electrons are emitted if the frequency of the light falls below this cutoff frequency fc.  For many metal surfaces the frequency of blue light is greater than fc and the frequency of red light is less than fc.  If red light is shone on the surface, no electrons are emitted, no matter what the intensity of the light (within reasonable limits).  If blue light is shone on the surface, electrons are emitted.  The number of emitted electrons depends on the intensity of the light.  But even if the intensity is reduced to a very low value, electrons are still emitted, albeit at a very low rate.  These observations constitute the photoelectric effect.

Einstein explained the photoelectric effect by assuming that the energy and momentum transported by an electromagnetic wave are not continuously distributed over the wave front.  They are transported in discrete packages.  Photons are the particles of light.  Light is "quantized".  Photons always move with the speed of light.  The energy of each photon is E = hf = hc/λ.  The momentum of each photon is E/c = hf/c = h/λ.

h = 6.626*10-34J s = 4.136*10-15 eV s = Planck's constant.
unit of energy: 1 eV = 1.6*10-19 J.
useful product: hc = 1240 eV nm.

An electron can only receive the large amount of energy necessary to escape the metal from the EM wave by absorbing a single photon.  If this photon has enough energy, the electron is freed.  Excess energy appears as kinetic energy of the electron.  The maximum kinetic energy of the electron is given by E = hf - Φ.  If the photon does not have enough energy, then the electron cannot escape the metal.

You will direct light with different wavelength onto a metal surface and measure the kinetic energy of the photoelectrons ejected from the metal as a function of the frequency of the light used to eject the electrons.  You will measure the work function of the metal and also determine the value of Planck's constant h from your data.  This experiment reveals the "particle nature" of light.

Open a Microsoft Word document to keep a log of your experimental procedures, results and discussions.

### Experiment

You will use an on-line simulation from the University of Colorado PhET group.

Explore the interface. There are some non-obvious controls.

• You can select Show photons in the Options menu to show the light beam as composed of individual photons.
• You can select Control photon number instead of intensity in the Options menu to change the Intensity slider to a Number of photons slider.
• You can use the camera icon to take a snapshot of the graphs so that you can compare graphs for different settings.
• You can Pause the simulation and then use Step to incrementally analyze.

In the simulation photons strike a metal cathode and eject electrons.  Electrons are ejected with a range of energies, up to a maximum energy.  The electrons are collected on the anode and then flow back from the anode to the cathode through a wire.  The current in the wire is measured.  Electrons are negatively charged.  If the anode is at a positive voltage compared to the cathode, electrons are attracted and gain energy.  If the anode is at a negative voltage compared to the cathode, electrons are repelled and loose energy.

Let Vca = Vcathode - Vanode.  If this potential difference Vca (energy per charge) becomes large enough, electrons will no longer reach the anode and current will no longer flow in the wire.

It is convenient to measure electron energy in units of electron volt (eV).  In SI units 1 eV = 1.6*10-19 J.

If the potential difference between the cathode and anode is x volts (Vca = x V), then electrons ejected from the cathode need an energy of at least x eV to overcome this potential difference and to reach the anode.  By determining voltage Vca needed to reduce the current in the wire to zero, we can determine the maximum energy of the ejected electrons.  The maximum energy E of electrons that reach the anode in eV has the same numerical value as the voltage in Vca.

(a)  Exploration:

For a Sodium target discuss:

• For a fixed number of photons and zero battery voltage, how does the number of photoelectrons ejected depend on the wavelength?  Does every photon eject an electron?  Does the probability of ejection change with wavelength?   Discuss!
• For a fixed wavelength and zero battery voltage, how does the current depend on the light intensity?  Discuss!
• For a fixed wavelength and light intensity, how does the current depend on the battery voltage?
• For a fixed wavelength and light intensity, do all ejected electrons have the same energy?   How can you measure the maximum energy of the ejected electrons.

(b)  Measurement:

Use a Sodium target.  Set the intensity to 100%.  For the wavelengths listed in the table below, find the maximum energy of the ejected electrons by finding the battery voltage Vca that just prevents the most energetic electrons from reaching the anode.
Note:  The text box under the battery displays -Vca.

Wavelength (nm) Frequency (s) Maximum Electron Energy (eV)
150 2.00e15
200 1.50e15
300 1.00e15
400 0.75e15
500 0.60e15

Prepare a graph of the maximum electron energy versus the frequency of the light.  Choose a X-Y Scatter plot.  Plot maximum electron energy on the vertical axis and frequency on the horizontal axis.
Prediction: E = hf - Φ
This equation is of the form y = ax + b, with a being the slope and b being the y-intercept.  The slope of the plot of E versus f will yield Planck's constant and the y-intercept will yield the work function of the metal cathode.
Add a trendline and insert the equation for the trendline into your plot.  Use the slope of your trendline to find Planck's constant h and the intercept value to find the work function Φ of Sodium.  If there are data points that seem to deviate to far from the trendline, repeat those measurements until you are satisfied that you have done your best.
Since the electron energy is measured in eV and the frequency in seconds, Planck's constant h will have units of eV s and the work function will have units of eV.  Convert Planck's constant to SI units (J s) by multiplying the value you obtained from the slope by 1.6*10-19J/eV.