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The Work Function (CIE A Level Physics)

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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:

E space equals space h f space equals space capital phi space plus thin space 1 half m v squared subscript m a x end subscript

  • 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:

y space equals space m x space plus thin space c

  • Comparing this to the photoelectric equation:

E subscript k m a x end subscript space equals space h f space minus space capital phi

  •  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

E thin space equals space h f space equals space capital phi thin space plus thin space E subscript k m a x end subscript

E subscript k m a x end subscript space equals space h f space minus space capital phi

y space equals space m x thin space plus thin space c

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

capital phi space equals space h f subscript 0 space equals space open parentheses 6.63 space cross times space 10 to the power of negative 34 end exponent close parentheses space cross times space open parentheses 4 space cross times space 10 to the power of 14 close parentheses space equals space 2.652 space cross times space 10 to the power of negative 19 end exponent space straight J

Step 4: Convert the work function into eV

  • 1 eV = 1.6 × 10-19 J    

E space equals space fraction numerator 2.652 space cross times space 10 to the power of negative 19 end exponent space over denominator 1.6 space cross times space 10 to the power of negative 19 end exponent end fraction space equals space 1.66 space eV            

Examiner Tip

You must recognise how to compare linear equations to the straight-line equation ymxc. This is very common in A-level Physics!

Intensity & photoelectric current

Kinetic energy & intensity

  • The kinetic energy of the photoelectrons is independent of the intensity of the incident radiation
  • This is because each electron can only absorb one photon
  • Kinetic energy is only dependent on the frequency of the incident radiation
  • Intensity is the rate of energy transferred per unit area and is related to the number of photons striking the metal plate
  • Increasing the number of photons striking the metal will not increase the kinetic energy of the electrons; it will increase the number of photoelectrons emitted

Why the kinetic energy is a maximum

  • Each electron in the metal acquires the same amount of energy from the photons in the incident radiation for any given frequency
  • However, the energy required to remove an electron from the metal varies because some electrons are on the surface whilst others are deeper in the metal
    • The photoelectrons with the maximum kinetic energy will be those on the surface of the metal since they do not require much energy to leave the metal
    • The photoelectrons with lower kinetic energy are those deeper within the metal since some of the energy absorbed from the photon is used to approach the metal surface (and overcome the work function)
    • There is less kinetic energy available for these photoelectrons once they have left the metal

Photoelectric current

  • Current is defined as the flow of charge, in this case, photoelectrons
  • The photoelectric current is a measure of the number of photoelectrons emitted per second
    • The value of the photoelectric current is calculated by the number of electrons emitted multiplied by the charge on one electron
  • Photoelectric current is proportional to the intensity of the radiation incident on the surface of the metal
  • This is because intensity is proportional to the number of photons striking the metal per second
  • Since each photoelectron absorbs a single photon, the photoelectric current must be proportional to the intensity of the incident radiation

2-4-3-ke-photocurrent-graphs

Sketch graphs showing the trends in the variation of electron KE with the frequency and intensity of the incident light and the variation of photocurrent with the intensity of the incident light

Examiner Tip

If you change the frequency of the incident light whilst keeping the number of photons emitted from the light source constant, then the photoelectric current will remain constant

This is because changing the frequency will change the energy of the emitted photons, but the number of photons will remain the same

If you change the frequency of the incident light whilst keeping the intensity constant, then the photoelectric current will change

This is because intensity is power per unit area which is equal to the rate of energy transfer per unit area

I space equals fraction numerator space P over denominator A end fraction space equals fraction numerator space E over denominator t A end fraction

The energy transferred comes from the photons, where the energy of a single photon is hf

I space equals fraction numerator space h f over denominator t A end fraction

So to account for n number of photons:

I space equals fraction numerator space n h f over denominator t A end fraction

If the frequency, f,  is increased and the intensity, I, remains constant, then the number of photons, n, must decrease

Planck's constant, h, and the area, A, of the metal plate do not change

This is because at higher frequencies, each photon has a higher energy, so fewer photons are required to maintain the intensity

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Ashika

Author: Ashika

Expertise: Physics Project Lead

Ashika graduated with a first-class Physics degree from Manchester University and, having worked as a software engineer, focused on Physics education, creating engaging content to help students across all levels. Now an experienced GCSE and A Level Physics and Maths tutor, Ashika helps to grow and improve our Physics resources.