Question

Learning Goal: To understand the experiment that led to the discovery of the photoelectric effect.
In 1887, Heinrich Hertz investigated the phenomenon of light striking a metal surface, causing the ejection of electrons from the metal. The classical theory of electromagnetism predicted that the energy of the electrons ejected should have been proportional to the intensity of the light. However, Hertz observed that the energy of the electrons was independent of the intensity of the light. Furthermore, for low enough frequencies, no electrons were ejected, no matter how great the intensity of the light became. The following problem outlines the methods used to investigate this new finding in physics: the photoelectric effect.
Suppose there is a potential difference between the metal that ejects the electrons and the detection device, such that the detector is at a lower potential than the metal. The electrons slow down as they go from higher to lower electric potential; since they must overcome this potential difference to reach the detector, this potential is known as the stopping potential. To reach the detector, the initial kinetic energy of an ejected electron must be greater than or equal to the amount of energy it will lose by moving through the potential difference.

If there is a potential difference V between the metal and the detector, what is the minimum energy Emin that an electron must have so that it will reach the detector?

Express your answer in terms of V and the magnitude of the charge on the electron,e .

Answer 1

Concepts and reason

The main concepts required to solve this problem are the work done and potential difference.

Initially, write the concept of the work done and the potential difference. Use these concepts and find the minimum energy that an electron must have so that it will reach the detector.

Fundamentals

The work is said to be done if an external force is applied on an object to move it.

The electric potential difference is defined as the work done by external force on a charged particle to move from infinitive to a point in the electric field.

The equation for the potential difference can be expressed as follows,

$V = \frac{W}{q}$

Here, V is the potential difference, W is the word done, and q is the charge of the charged particle.

Assume that there is a potential difference between the metal surface and the detection device. Here, the electrons are ejected from the metal surface. Here, some electric work is done on the electrons to move them from the higher potential to lower potential. The electrons should overcome this potential difference to hit the detector and hence, this potential is known as stopping potential. The work is done on the electrons to move from higher potential to lower potential, this work done is equal to the energy.

For the incidence of light to eject the electrons, the light must impart some amount of energy to the electrons to overcome the force that constrain the electrons within the metal. The minimum amount of energy that required to overcome these forces is called as work function. The work function is denoted by $\phi$ . There will be different values of work functions for the difference metals. The light should impart enough energy to overcome the electrons for both the work function and stopping potential to reach the detector.

Let the charge of the electron be q, and potential difference be V.

The work done on the electron to move from the higher potential to the lower potential is,

$W = qV$ …… (1)

Here, q is the charge and V is the potential difference.

The minimum energy required by the electron to reach the detector is equal to the work done on the electron, that is,

$W = {E_{{\rm{min}}}}$

Substitute ${E_{{\rm{min}}}}$ for W in above equation (1).

${E_{{\rm{min}}}} = qV$

The charge of the electron is denoted by e.

Substitute e for q in above equation.

${E_{{\rm{min}}}} = eV$

Ans:

The minimum energy required for an electron to reach the detector is eV.