Showing posts with label de Broglie concept. Show all posts
Showing posts with label de Broglie concept. Show all posts

Problems and solutions on de Broglie's Hypothesis

Problem and solution

If 10,000 voltage is applied across an x-ray tube, what will be the ratio of the de  Broglie wavelengths of incident electron to that of the x-ray produced?

To solve this problem, we have to write the wavelength of the electron in terms of the de Broglie concept. Here in the place of the momentum we can write applied potential and the corresponding energy. Similarly to right the wavelength of the x-ray, we can use plank’s quantum concept as shown below.




Problems and solutions

Here we are going to solve two problems. In the first problem and alpha particle and the proton are passed through same magnetic field which is perpendicular to the velocity vectors. If both the charger particles are having the same radius, what is the ratio of their de Broglie wavelengths?

We can solve this problem by first of all understanding that the force due to the magnetic field on the charged particle provides the necessary centripetal force so that they takes a circular path. From that it can be concluded that the momentum of the particle is directly proportional to charge of the particle as well as the radius of the particle. Anyway in this problem being the radius is same, we can say momentum means directly proportional to the charge of the particle itself.

As per this concept, wavelength is inversely proportional to momentum that means wavelength is inversely proportional to the charge of the given particles.



The second problem is also solved basing on the same concept as shown above.

Problem and solution

Photons of energy 4.25 electron volt and 4.7 electron volt are allowed to incident on to metal surface S. If the maximum kinetic energies between them are having a difference of 1.5 electron volts and the they are wavelengths are in the ratio of 1:2, find the work functions of the two different metals?

We can express de Broglie wavelength in terms of the kinetic energy. Basing on this we can say that the ratio of the day Broglie wavelength is inversely proportional to Squire root of ratio of their respective kinetic energies. Taking this concept into consideration we can solve the problem as shown below.




Problem and solution

de Broglie wavelength of the proton accelerated  through a potential difference of 100 V is given. If a alpha particle is accelerated through same potential difference, what will be its wavelength?

According to de Broglie hypothesis, we know that the wavelength of a particle is the ratio of Planck’s constant to the momentum of the particle. We can express the momentum in terms of kinetic energy. Further we can express the kinetic energy in terms of the applied potential. By substituting the given data in that equation, we can solve the problem as shown below.




Problem and solution

A light particle of a certain mass at rest explodes into two particles having masses in the ratio of 2:3 . What is the corresponding ratio of the de Broglie wavelengths?

As the particle is in the state of rest, its initial momentum is equal to 0. The explosion is happened due to internal forces and hence law of conservation of momentum is very much valid here. According to this law the initial momentum of the system is equal to the final momentum when no external forces are acting on the system. As the momentum is as well as the Planck’s constant are same, the wavelengths are also going to be the same.



Problem and solution

What will be the wavelength of electron having energy of hundred electron volts?

We can solve this problem basing on dual nature of the particle. Here the particles shall be having a wave nature and hence it shall have certain wavelength which is equal to the ratio of Planck’s constant to momentum of the particle. We can express the momentum in terms of kinetic energy and the kinetic energy can be further expressed in terms of the applied voltage.

By substituting the data in the given problem we can solve the problem as shown below.



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De Broglie Hypothesis and Wave Nature of Particle

Particle nature of light

Photoelectric effect of light can be explained only when we assume that light these interacting with the matter like a particle. These wave packets are called quanta and they have fixed amount of energy. The energy particles are also called photons and they do not have the rest mass. It can be confirmed that when the light is interacting with the matter, it behaves like a particle.

With the increase of intensity, there will be more number of photons are available in the given light. These photons are electrically neutral and they are not deflected in electric and magnetic fields. When photon collides with a particle, its total energy and momentum are always conservative.

De Broglie Hypothesis

To explain the properties of the light like interference, diffraction and polarization, we shall depend on wave nature of the light. Simultaneously to explain the properties like photoelectric effect and Compton effect, light shall have particle nature.

As the same light is exhibiting both the set of properties, it shall also have dual nature.
In the universe the energy is broadly in the format of matter and radiation. There is a basic concept in the nature which tells that nature loves symmetry. We see so many things in the nature which are symmetrical.

We can explain the concept of symmetry easily. We know that the solar system is a macroscopic system where the sun is at the Center and the planets revolving around it. We also know that the item is a microscopic system where the nucleus is at the Center and the electrons are revolving around it. So we can see a broader symmetry in the design of the nature.
It is already proved that light is having dual nature. It travels like a wave and it interacts like a particle. We know that in the nature both light and the particles are forms of energy’s. As one form of energy is having dual nature and as nature loves symmetry, it is understood that the matter is also having dual nature.

It means to say that all the particles that we see are also having the wave nature. It means the electrons, the protons are not only having the behavior of the particles, and they can also behave like the waves under suitable conditions. This concept is called dual nature of matter. When the particles are travelling like waves they are called as matter waves.

As per the day Broglie’s concept the wavelength of the particle is equal to the ratio of Planck’s constant to the momentum of the particle. Experimentally it is proved that the particle can have wave nature.



So it can be concluded that electrons can behave like waves and can undergo diffraction. We can express the momentum in terms of kinetic energy as well as the applied potentiality as shown below.



We can also express the wavelength of the particle in terms of absolute temperature. It is based down the total energy particle can have for a particular temperature. This equation is derived basing on kinetic theory of gases.




In the above diagram we are also having a problem where the applied potential is given for an electron and we are supposed to calculate the corresponding wavelength of the electron. By converting the momentum into the kinetic energy and further in terms of the voltage with can solve the problem as shown above.


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