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Einstein's equation and the potential well

If a quantum of radiation with an energy greater than the work function Eo, and therefore a frequency greater than fo, falls on a surface an electron will be emitted with some kinetic energy and escape from the surface.

The kinetic energy of the electron is then just the difference between the energy of the quantum and the work function of the metal.
This can be expressed in the following equation - called Einstein's photoelectric equation.

½mv2 = hf - hfo

This is shown in the diagram.

Example problems
Calculate the maximum kinetic energy of an electron that is emitted from a caesium surface when light of wavelength 210 nm falls on it. Work function for caesium = 1.88 eV.

Work function = 1.88x1.6x10-19 J
Frequency of incident radiation = 3 x 108/210x10-9 = 1.43 x 1015

Kinetic energy = hf - hfo = hf - W = 6.63x10-34x143x1015 - 1.88x1.6x10-19 = 6.46x 10-19 J = 4.04 eV

If we put a collecting electrode in front of the emitting surface in a vacuum we can detect the photoelectrons as a small current. Changing the frequency of the radiation will not have any effect on the current as long as it is bigger than fo but increasing the intensity will increase the current.
Now if we make the collecting electrode slightly negative compared with the emitting surface the electrons will find it difficult to get to it and electrons will only do that if their energy is greater than the "height" of the potential barrier.

They will be detected if ½ mv2 > eV

where V is the potential difference between the emitting surface and the collecting electrode.

Therefore Einstein's equation can be written as:

eV = ½mv2 = hf - hfo

If V is increased so that no more electrons can reach the detector this value for the potential is called the stopping potential for that surface and radiation.
© Keith Gibbs 2013