Question

Define the term “stopping potential” in relation to photoelectric effect.

Answer 1

Hint: Photoelectric effect is the phenomenon of ejection of electrons from the surface of a metal when electromagnetic radiation of suitably high enough frequency is irradiated on it. Stopping potential is the external potential difference that has to be applied to stop this ejection, even though electromagnetic radiation is still thrown on the object.

Formula used:
The maximum kinetic energy of a photoelectron ${{E}_{\max }}$ is given by
${{E}_{\max }}=h\nu -{{\phi }_{0}}$

where $\nu$ is the frequency of the electron, ${{\phi }_{0}}$ is the work function of the metal (minimum energy of radiation to eject an electron from the metal surface) and $h$ is the planck's constant equal to $6.636\times {{10}^{-34}}J.s$.

The energy $E$ gained or lost by an electron while going down or up a potential difference $V$ respectively is given by
$E=eV$
where $e=1.6\times {{10}^{-19}}C$ is the charge on an electron.

Complete step by step answer:
When electromagnetic radiation of suitably high frequency and energy is irradiated on the surface of the metal, some electrons on its surface gain enough energy to break the forces of attraction with the metal surface and become free electrons. Under a favorable potential bias, these ejected electrons (photoelectrons) can flow to constitute an electric current.
However, a negative potential difference (or reverse bias) can also be applied that opposes the ejection of the electrons so that they do not get ejected even if the electromagnetic radiation is continuously irradiated on the metal surface. This potential is known as the stopping potential.

Now, according to Einstein’s equation of photoelectric effect,

The maximum kinetic energy of a photoelectron ${{E}_{\max }}$ is given by

${{E}_{\max }}=h\nu -{{\phi }_{0}}$ --(1)

where $\nu$ is the frequency of the electron, ${{\phi }_{0}}$ is the work function of the metal (minimum energy of radiation to eject an electron from the metal surface) and $h$ is the planck's constant equal to $6.636\times {{10}^{-34}}J.s$.

Also, the energy $E$ gained or lost by an electron while going down or up a potential difference $V$ respectively is given by

$E=eV$ --(2)

where $e=1.6\times {{10}^{-19}}C$ is the charge on an electron.

Therefore, if a stopping potential has to stop the electrons, it must oppose the maximum energy that can be gained by a photoelectron. Therefore we have to equate (1) and (2). Doing so, we obtain,

$eV=h\nu -{{\phi }_{0}}$

$\therefore V=\dfrac{h}{e}\nu -{{\phi }_{0}}$

Hence, the stopping potential for a specific surface depends upon the frequency of the electromagnetic radiation that is used.

Note: Students must know that not all photoelectrons have the same amount of energy. Some have collisions with other particles and transfer their energy to these particles. However, the maximum theoretical energy that a photoelectron can have is given by equation (1). The stopping potential must be such that it must be able to stop even the electrons with the maximum kinetic energy.
On seeing the final equation for the stopping potential, students can also notice that the relation between stopping potential and frequency of electromagnetic radiation is a straight line with constant slope $\dfrac{h}{e}$ and y-intercept $-{{\phi }_{0}}$. Hence, different substances can be compared by reading the graphs and different characteristic properties can be found out about them.