Threshold Frequency & Work Function (AQA A Level Physics): Revision Note

Threshold frequency

• The photoelectric effect is the phenomenon in which electrons are emitted from the surface of a metal upon the absorption of electromagnetic radiation

• Electrons removed from a metal in this manner are known as photoelectrons

• The photoelectric effect provides important evidence that light behaves as a particle

  • Light is quantised or carried in discrete packets

• This is shown by the fact that each electron can absorb only a single photon

• This means only the frequencies of light above a threshold frequency will emit a photoelectron

Threshold frequency and wavelength

• The threshold frequency is defined as:

The minimum frequency of incident electromagnetic radiation required to remove a photoelectron from the surface of a metal

• The threshold wavelength, related to threshold frequency by the wave equation, is defined as:

The longest wavelength of incident electromagnetic radiation that would remove a photoelectron from the surface of a metal

• Frequency and wavelength are related by the equation:

$v=f\lambda$

• Where:

  • $v$ = speed of the wave (m s^-1)

  • $f$ = frequency (Hz)

  • $\lambda$ = wavelength (m)

• Since photons are particles of light, $v=c$ (speed of light)

• Threshold frequency and wavelength are properties of a material, and vary from metal to metal

The work function

• The work function Φ, or threshold energy, of a material, is defined as:

The minimum energy required to release a photoelectron from the surface of a metal

• Consider the electrons in a metal as trapped inside an ‘energy well’ where the energy between the surface and the top of the well is equal to the work function Φ

• A single electron absorbs one photon

• Therefore, an electron can only escape from the surface of the metal if it absorbs a photon which has an energy equal to Φ or higher

• Different metals have different threshold frequencies and hence different work functions

• Using the well analogy:

  • A more tightly bound electron requires more energy to reach the top of the well

  • A less tightly bound electron requires less energy to reach the top of the well

• Alkali metals, such as sodium and potassium, have threshold frequencies in the visible light region

  • This is because the attractive forces between the surface electrons and positive metal ions are relatively weak

• Transition metals, such as zinc and iron, have threshold frequencies in the ultraviolet region

  • This is because the attractive forces between the surface electrons and positive metal ions are much stronger

Stopping potential

• Stopping potential, V_s, is defined as:

The potential difference required to stop photoelectron emission from occurring

• The photons arriving at the metal plate cause photoelectrons to be emitted

  • This is called the emitter plate

• The electrons that cross the gap are collected at the other metal plate

  • This is called the collector plate

• The flow of electrons across the gap results in an electromotive force (e.m.f.) between the plates that causes a current to flow around the rest of the circuit

  • Effectively, it becomes a photoelectric cell producing a photoelectric current

• If the e.m.f. of the variable power supply is initially zero, the circuit operates only on the photoelectric current

• As the supply is turned up, the emitter plate becomes more positive (because it is connected to the positive terminal of the supply)

• As a result, electrons leaving the emitter plate are attracted back towards it

  • This is because the potential difference (p.d.) across the tube opposes the motion of the electrons between the plates

• If any electrons escape with enough kinetic energy, they can overcome this attraction and cross to the collector plate

  • And if they don't have enough energy, they can't cross the gap

• By increasing the e.m.f. of the supply, eventually, a p.d. will be reached at which no electrons can cross the gap – this is the stopping potential, V_s

• At this point, the energy needed to cross the gap is equal to the maximum kinetic energy $E_{k\left(max\right)}$ of the electrons

$V_s=\frac{W}{Q}=\frac{E}{Q}=\frac{E_{k\left(max\right)}}{e}$

• Therefore, the maximum kinetic energy of the photoelectrons is: 

$E_{k\left(max\right)}=e{V}_s$

• Where:

  • $E_{k\left(max\right)}$ = maximum kinetic energy of photoelectrons (i.e. surface electrons) (J)

  • $e$ = charge on an electron (1.6 × 10^-19 C)

  • $V_s$ = stopping potential (V)

Intensity and stopping potential

• Increasing the intensity of the incident radiation on the plate increases:

  • the number of photons incident on the metal plate

  • the number of photoelectrons emitted from the plate, i.e. the photoelectric current

• For a given potential difference, increasing the intensity increases the photoelectric current but the stopping potential remains the same

  • This shows that the intensity does not affect the kinetic energy of the photoelectrons

• The maximum kinetic energy of the photons (and photoelectrons) depends only on:

  • the frequency (or wavelength) of the incident photons

  • the work function of the metal

• However, if the frequency or wavelength is changed whilst keeping the intensity constant, the photoelectric current will not be constant

• For example, increasing the frequency of the incident radiation whilst keeping the intensity constant will cause the photoelectric current to decrease. This is because: 

  • Increasing the frequency of a source means the energy of each photon increases

  • Keeping intensity the same means the energy transferred per unit area in a given time is constant

  • So, a higher frequency source must emit fewer photons per unit area in a given time than a lower frequency source (of the same intensity)

  • If there are fewer photons incident on a given area each second, the number of electrons emitted each second must decrease