The Photoelectric Effect

The photoelectric effect was first discovered in 1887 by Heinrich Hertz, who most people know for the unit of frequency called hertz, or Hz. He noticed that electrons were emitted when metals were exposed to light, but the properties of the ejected electrons didn't match the predictions of classical electromagnetism.

For example, high-intensity light was expected to eject more electrons than low-intensity light, regardless of the frequency of the light. However, electrons would only be emitted if the frequency of the incident light was above a certain threshold, regardless of the intensity. This minimum frequency was different for each metal.

In 1905, Albert Einstein came to the rescue. He proposed (based on the results of work done by Max Planck on blackbody radiation) that light, on small scales, doesn't actually behave the way classical electromagnetism predicted.

In classical electromagnetism, light is a wave. Einstein proposed that light is actually made up of particles, which we now call photons. In other words, light is quantized. (In fact, photons and all other particles have both particle-like and wave-like behavior. This includes big particles like you. But that's a topic for another cache.)

In addition to light being made of particles called photons, Einstein proposed that the energy of each photon was given by E = hf, where h is Planck's constant and f is the frequency of the light in Hz.

This was the key to the puzzle. When a photon with sufficient energy hit an electron in a metal, the electron would gain enough energy to escape. The energy required to remove an electron from a particular metal is called the metal's work function, ϕ.

This also explained the observed kinetic energy of the ejected electrons. If a photon imparts E = hf energy to an electron, but the electron required an energy of ϕ to escape, the kinetic energy of the freed electron would be K = hf - ϕ. If a photon with an energy smaller than ϕ struck the metal, the electron would not be freed.

The Puzzle

The cache is hidden at 39° 58.ABC' N, 104° 57.XYZ' W, where each letter corresponds to a single digit. To find the digits, you must answer the following questions.

Values

• Planck's constant, h = 4.14x10^-15 eV*s
• The work function for silver, ϕ = 4.7 eV
• Speed of light, c = 3.0x10⁸ m*s^-1
• To convert a wavelength to a frequency: f = c/λ

I have given you all the values and equations you need, either here, above, or in the problems themselves. If you use values other than the ones provided, your answers may differ from mine.

A Note on Units

The electronvolt, or eV, is a measure of energy used in particle physics. All of your answers will be in units of eV.

You will need to convert nm (nanometer) and pm (picometer) to meters and MHz (megahertz) to Hz to get the correct answers. Use the following conversions:

• 1 nm = 1x10^-9 m
• 1 pm = 1x10^-12 m
• 1 MHz = 1x10⁶ Hz

Problems

While doing these problems, note the relative magnitude of the energies. See if you can determine the relationship between the wavelength (or frequency) of the photon and its energy.

Problem 1: What is the energy of a single photon of red light with a wavelength of λ = 700 nm?

Problem 2: What is the energy of a single photon of yellow light with a wavelength of λ = 550 nm?

Problem 3: What is the energy of a single photon of purple light with a wavelength of λ = 400 nm?

Problem 4: A local radio station broadcasts at 90.1 MHz. What is the energy of a single radio photon from this station?

Problem 5: A gamma-ray burst (GRB) was detected emitting photons at a wavelength of λ = 1 pm. What is the energy of a single photon from this GRB?

Problem 6: A photon with a wavelength of λ = 200 nm strikes a silver target. What is the kinetic energy of the ejected electron?

Mapping Answers to Letters

To get A, take the first non-zero digit from the answer to problem 1 and subtract 1.

To get B, take the first non-zero digit from the answer to problem 2 and add 6.

To get C, take the first non-zero digit from the answer to problem 3 and subtract 3.

To get X, take the first non-zero digit from the answer to problem 4 and add 6.

To get Y, take the first non-zero digit from the answer to problem 5 and subtract 1.

To get Z, take the first non-zero digit from the answer to problem 6 and add 7.

A+B+C = 8.

X+Y+Z = 17.

You can check your answers for this puzzle on GeoChecker.com.

Appendix

This cache has been placed with the permission of the Quail Valley HOA.

I used "Modern Physics, For Scientists and Engineers" by Taylor, Zafiratos, and Dubson (all of whom are or were once professors at CU Boulder, by the way. Go Buffs!) as a reference for this puzzle.