The photoelectric effect equation

​​In a nutshell

When light is shone on a metal plate, each photon interacts with a single electron, transferring its energy to it. Whether electrons are emitted from a metal or not is not related to the intensity of the radiation. After the energy has been used to overcome the work function, the remaining energy is transferred to kinetic energy. This kinetic energy is a maximum value as some electrons are closer to the nucleus so require more energy to be freed. 

Equations

Constants

Variables

Conservation of energy

Einstein used the conservation of energy for photons and electrons in order to derive his photoelectric equation. The energy of each individual photon must be conserved. Each photon frees a single electron from the surface in a one-to-one interaction and the remainder is transferred to kinetic energy of the photon. 

From this, the photoelectric equation can be deduced:

​$$hf = \Phi + E_{k,\,max}$$

Example

When radiation of unknown frequency is shone on a metal plate with a work function of $$2.2 \,eV$$, photoelectrons are emitted with a maximum kinetic energy of $$7.25 \times 10^{-19}\, J$$.  Calculate the unknown frequency.

State variables:

​$$\Phi= 2.2\, eV \, \, \\ E_{k,\,max} = 7.25 \times10^{-19}\,J$$

State equation:

​$$hf = \Phi + E_{k,\,max}$$

Convert $$2.2 \,eV$$ into $$J$$​:

$$2.2\,eV = (2.2 \times 1.6 \times 10^{-19}) \,J\\[0.1in]=3.52 \times 10^{-19}\,J$$​​

Rearrange and substitute in:

$$f=\dfrac{\Phi + E_{k,\,max}}{h} \newline \\[0.15in] f=\dfrac{3.52 \times 10^{-19} + 7.25 \times 10^{-19}}{6.63 \times 10^{-34}} \\[0.15in] f=1.6 \times 10^{15}\, Hz$$

The frequency of radiation is $$\underline{1.6 \times 10^{15}\, Hz}$$

Maximum kinetic energy

The maximum kinetic energy, $$E_{k,\,max}$$, is a maximum value as some electrons may be closer to the nucleus, requiring more energy than the work function to be released. It is only the energy left over after the work function has been used to overcome the work function that is transferred to kinetic energy. 

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When the frequency of the source equals the threshold frequency, the maximum kinetic energy is zero.

Electron emission

Whether electrons are emitted from a metal or not is not related to the intensity of the radiation. No matter how high the intensity of radiation is, if the frequency is not above the threshold frequency, electrons will not be released. 

If the frequency is above the threshold frequency, an increase in intensity of radiation will increase the rate of electron emission. This is because increasing intensity increases the amount of photons available to interact with electrons, therefore freeing them from the metal.

Obtaining Planck's constant

The value of Planck's constant can be obtained experimentally using LEDs, and the graph below is produced.

From this graph, you can calculate:

• Planck's constant $$h$$​ which is the gradient
  
• the threshold frequency $$f_t$$ which is the $$x$$-intercept
• the work function $$\Phi$$ which is the threshold frequency multiplied by Planck's constant​
  

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