The photoelectric equation
- Since energy is always conserved, the energy of an incident photon is equal to:
The threshold energy + the kinetic energy of the photoelectron
- The energy within a photon is equal to hf
- This energy is transferred to the electron to release it from a material (the work function) and gives the emitted photoelectron the remaining amount as kinetic energy
- This equation is known as the photoelectric equation:
- Where:
- h = Planck's constant (J s)
- f = the frequency of the incident radiation (Hz)
- hf = energy of a single photon (J)
- Φ = the work function of the material (J)
- ½mv2max= the maximum kinetic energy of the photoelectrons (J)
- This is always written as Ek max
- This equation demonstrates:
- If the incident photons do not have a high enough frequency (f) and energy to overcome the work function (Φ), then no electrons will be emitted
- Above energy hf0 = Φ, where f0 = threshold frequency, photoelectric emission can occur
- Ekmax depends only on the frequency of the incident photon, and not the intensity of the radiation
- The majority of photoelectrons will have kinetic energies less than Ekmax
Graphical representation of work function
- The photoelectric equation can be rearranged into the straight-line equation:
- Comparing this to the photoelectric equation:
- A graph of maximum kinetic energy Ekmax against frequency f can be obtained
Graph of kinetic energy against frequency
The graph of maximum kinetic energy of photoelectrons against photon frequency is straight line with an x-intercept
- The key elements of the graph are:
- The work function Φ is the y-intercept
- The threshold frequency f0 is the x-intercept
- The gradient is equal to Planck's constant h
- There are no electrons emitted below the threshold frequency f0
Worked example
The graph below shows how the maximum kinetic energy Ek of electrons emitted from the surface of sodium metal varies with the frequency f of the incident radiation.
Calculate the work function of sodium in eV.
Answer:
Step 1: Write out the photoelectric equation and rearrange it to fit the equation of a straight line
Step 2: Identify the threshold frequency from the x-axis of the graph
- When Ek = 0, f = f0
- Therefore, the threshold frequency is f0 = 4 × 1014 Hz
Step 3: Calculate the work function
- From the graph at f0, Ek = 0
Step 4: Convert the work function into eV
- 1 eV = 1.6 × 10-19 J
Examiner Tip
You must recognise how to compare linear equations to the straight-line equation y = mx + c. This is very common in A-level Physics!