## Important topics for JEE mains : The photoelectric effect

Let us study about one of Important topics for JEE mains : The photoelectric effect .

## The photoelectric effect

In the first third of the twentieth century, some peculiarities of the electron began to be known in phenomena that settled the nascent theories of relativity and quantum mechanics. Among them, we can highlight the manifestations known as the photoelectric effect and the Compton effect, two forms of interaction between electrons and electromagnetic radiation.

In 1887, the German physicist Heinrich Hertz (1857-1894) accidentally discovered that ultraviolet light modified the voltage at which sparks were produced between metal electrodes. The German Philipp Lenard (1862-1947) described this phenomenon, called the photoelectric effect, as the emission of electrons by metallic surfaces when visible or ultraviolet light hits them, and he arrived at two basic conclusions:

1.The maximum kinetic energy that the emitted electrons can reach does not depend on the intensity of the incident radiation.

2.In the photoelectric effect, the emission of electrons is instantaneous.

## The Bohr-Sommerfeld Atomic Model

The atomic model devised by Ernest Rutherford at the beginning of the 20th century described the hydrogen atom as a system composed of a massive nucleus of positive electric charge and minimum dimensions around which a negative electron moved. When this model proved insufficient, Niels Bohr introduced a series of quantum postulates that established a new conceptual framework for the development of atomic theory.

## The hydrogen atom

If the hydrogen atom is considered as a set of nucleus and electron subjected to the laws of the dynamics of the central forces, the total energy and the angular momentum of the electron should be governed by the following expressions:

where r is the radius of the orbit of the electron, m its mass and Z the atomic number of hydrogen (expressed in symbolic form, although its value is 1).

##### Important topics for JEE Mains : Algebra

These expressions explain the mechanical behavior of the system, but not its electromagnetic properties. According to classical electromagnetism, if an electron emits radiation, it will inevitably fall under the influence of the atomic nucleus.

Model of Bohr-Sommerfeld

To understand the behavior of the hydrogen atom, the Danish Niels Bohr (1885-1962) incorporated to the previous model considerations of the quantum theory. Bohr supposed that the electron can only describe certain circular orbits around the nucleus, which he called stationary and which he identified with integers.

When an electron emits radiation, it moves from a stationary orbit n to another n¿, and the difference between its energies corresponds to the energy of the photon-emitter:

Since the number of possible orbits of the electron is discrete, so is the set of electromagnetic frequencies that it can emit. If an electron absorbs a photon, it acquires energy and moves to an orbit further away from the nucleus, and if it emits it, it loses energy and falls into an orbit closer to the nucleus.

##### Important topics for JEE Mains : Qualitative Analysis

He also proposed that the allowed orbits would be those whose angular momentum L was a multiple of the constant “h”, which means: L = h n , being n = 1, 2, 3 … In this way, the radii of the Bohr stationary orbits and the associated energy levels would be:

The German physicist Arnold Sommerfeld (1868-1951) completed this atomic model of Bohr considering that the described orbits were not circular, but elliptical, and developed the corresponding corrections.

Matter Waves by Louis De Broglie

Albert Einstein’s work on the photoelectric effect showed that electromagnetic waves are made up of elementary particles called photons. In reverse, the Frenchman Louis De Broglie predicted in 1924 that the material corpuscles of the exterior of the atoms, the electrons, should also show a wave behavior. The experimental verification of the particle and wave duality of electrons, which arrived a few years later, closed the circle of one of the most seductive proposals of quantum physics: everything that exists is, at the same time, wave and matter.

The controversies over the nature of light that had focused scientific debates for more than two centuries were resolved in 1905 when Albert Einsten, in his interpretation of the photoelectric effect, came to reconcile the two hypotheses handled and, until then, considered incompatible:

1.The wave, according to which the light radiation is simply a disturbance that moves in space.

2.The corpuscular, which held that light is made up of material corpuscles capable of interacting with matter.

Einstein concluded that light and, by extension electromagnetic waves, are both corpuscle and wave, since they are composed of massless and uncharged particles, called photons, which propagate in space as a wave motion, exchanging energy with the environment.

##### Important topics for JEE Mains : Mechanics

In a speculative study, which did not respond to any observed reality that had to be explained, the Frenchman Louis de Broglie (1892-1987) played with the possibility that, like photons, electrons also have the same wave duality and corpuscle.

## De Broglie waves

In a work published in 1924, De Broglie started with a comparison between the properties of the photon and the electron to assume that the latter particle could have energy-frequency and wave-momentum linear relationships analogous to the first, and expressed as:

being $\vec{n}$ a vector that shares direction with the wave vector $\vec{k}$ .

Starting from the relativistic hypothesis, an equivalence could be established between energy and the linear moment of the electron considered as a wave and as a material particle, from which it would be deduced that:

## De Broglie wavelength

From the comparison of the magnitudes of the behavior of the electron understood as a wave and as a particle, we obtain a value for the wavelength that the wave motion associated with the electron that is given by:

where v is the velocity of the particle and m is its mass. This magnitude, called De Broglie’s wavelength, increases with decreasing speed, and vice versa.

##### Important topics for JEE Mains : 2D, 3D & Vectors

If applied to the postulate of the Bohr atomic model (see t60), which holds that the orbits of electrons in atoms can only have certain radii quantified, it follows that:

According to this formula, the allowed (stationary) orbits in the Bohr model would be those whose radius was equal to an integer number of de Broglie wavelengths.

As for detecting wave behavior in light, it was necessary to handle dimensions of the order of their wavelength (for example, gratings that would cause diffraction patterns as light interferences), to observe the effects of waves associated with matter particles of very small mass and moving at low speed, for example, the electrons themselves, must be used. In these particles it would be possible to obtain de Broglie wavelength values ​​of the order of a few tenths of a nanometer.

I hope you know enough about Photoelectric effect. But if you still have some doubts you can go through this book.

Best of luck for your JEE Preparation. Do not forget to comment and share.

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