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Blackbody Radiation: Derivation of Planck’s Law Step-by-Step | CSIR-NET, GATE

Planck Blackbody Radiation Formula The spectral energy density of blackbody radiation as a function of frequency is given by Planck's radiation law: \[ u(\nu)d\nu = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/kT}-1} \, d\nu \] where \(h = 6.626 \times 10^{-34} \, \text{J·s}\) is Planck's constant \(u(\nu)\) is the energy density of radiation per unit frequency interval \(\nu\) is the frequency \(T\) is the temperature \(k\) is the Boltzmann constant In these notes we will derive the above formula in a step by step manner. Waves in a Box Consider an electromagnetic wave travelling with the speed of light in some arbitrary direction represented by the position coordinate \(x\). If the wavelength of the wave is \(\lambda\), the amplitude of the wave along the \(x\)-direction can be written as \[ A(x) = A_0 \sin \left(\frac{2\pi x}{\lambda}\right) \] This expression can also be written in terms of the wave number \(k\): \[ A(x) = A_0 \sin(kx) \] where the ...
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Quantum Harmonic Oscillator Coherent States – Complete Notes

Coherent State for the Harmonic Oscillator and its properties Discovered by R. J. Glauber in 1963. Glauber received the Nobel Prize in 2005 for the relevance of coherent states in quantum optics. The state describing a laser beam can be briefly characterized as: An indefinite number of photons. A precisely defined phase. Laser dynamics → coherent state. Normal light → unpolarized / incoherent. Uncertainty relation: \[ \Delta N \, \Delta \Phi \ge \frac{1}{2} \] Here \( \Delta N \) : fluctuation in occupation number \( \Delta \Phi \) : fluctuation in phase For a laser: \( \Delta \Phi \) → very small \( \Delta N \) → large For normal light: \( \Delta N \) → fixed / small \( \Delta \Phi \) → large (not in the same phase) Definition of Coherent States A coherent state \( |\alpha\rangle \) (also known as a Glauber state ) is defined as an eigenstate of the annihilation operator \( \hat{a} \) with eigenvalue \( \alp...

What is Carnot's Heat Engine ? Carnot's Cycle, Working and Efficiency|

 Heat Engine Any practical machine which converts heat into mechanical work is called a heat engine. Heat engines in their operation absorbs heat at Higher temperature, convert part of it into mechanical work, and reject the remaining heat at a low temperature. In this process, a workin substance is used. In steam engines, the working substance is water vapour, and in all gas engines the working substance is a combustible mixture of gases. The working substance goes through some processes that involes change of pressure(P), Volume(V) and temperature, and then returns to the initial state. This is called as one cycle of operation, since it is a cyclic process. Heat engines were from long time but were only  made useful at the time of industrial revolution in 18th century. Carnot's Ideal Heat Engine  French engineer  Sadi Carnot  conceived a theoretical engine in 1824. The engine is not practically possible. It has maximum efficiency. It is an ideal heat engine. ...

Friction in solids

Force is the most common term in the classical physics or classical mechanics. The type of force that only occurs when the two surfaces are in physical contact is called as a contact force . One of the example of contact force is friction. Friction is considered as an electromagnetic force. This is confusing for most of the people to imagine friction as an electromagnetic force, but it is. There are charged particles on both the surfaces. When there is a motion the forces that are exert by the charged particle on each other opposes the relative motion of both the bodies. This force is called as frictional force. One of the main characteristics of frictional force is that it opposes the relative motion between 2 bodies. For an example -    Let A and B be the 2 bodies that are in contact and have relative motion. Suppose A is moving more faster than B. So in this case where the friction will act on both the bodies? Be clear that friction wants to oppose relati...

Is light a particle or a wave? The great wave-particle struggle #1

Light in Seventeenth century In seventeenth century, there were very less facilities for experiments. But the nature was nature, as it is now. Many major behaviors of light was known at that time- Light cast shadows . Which shows that light travels in straight line. Light is reflected from smooth surfaces. The rules of reflection are- (i) The incident ray, the reflected ray and the normal are in the same plane (ii) The angle of incidence is equal to the angle of reflection. When light is travelling from one medium to another medium, it bends unless it falls on the second medium normally. The rules of this phenomenon, called refraction , are: (i) The incident ray, the refracted ray and the normal are in the same plane  Light comes in different colours such as red, yellow, green, blue, violet, etc. All the scientist were trying to understand that why light shows these characteristic behaviors of straight line motion, reflection, refraction, colours, etc Newton's corpuscle theory...

Nikola Tesla: Engineer and Inventor

The man who invented the 20th century is none other than Nikola Tesla. He is serbian-american inventor, electrical engineer, mechanical engineer and a futurist. Best known for Alternating current(AC) and electrical supply system. Nikola tesla  Born - 10 July, 1856. Died -  7 January, 1943 (83 years). Beginning with a quote of Nikola Tesla- " The scientists of today think deeply instead of clearly. One must be sane to think clearly, but one can think deeply and be quite insane. " Eidetic Memory  Tesla read many works. He can memorize complete books. He supposedly possessed a photographic memory. He was a polyglot, speaking eight languages: Serbo-Croatian, Czech, English, French, German, Hungarian, Italian, and Latin. Tesla related in his autobiography that he experienced detailed moments of inspiration. He often see blinding flashes of light before his eyes, also accompanied by visions. Many times the visions were related to the word or the idea that he was currently...

Understanding magnetic field in simple way

MAGNETIC FIELD Today we are going to talk about magnetic fields .   (Suggest more topics in the comment section) The fundamental problem electrodynamics hopes to solve is this(figure): We have some electric charges, q1,q2,q3,...........(call them source charges); what force do they exert on another charge, Q (call it test charge)? The position of the source charges are given (as functions of time); the trajectory of the test particle is to be calculated. Source charge and test charge According to the principle of superposition , it is sufficient to find the force of a single source  charge- the total is then the vector sum of all the individual forces. In the simplest case, electrostatics, the source charge is at rest(though the test charge need not be). We will consider the forces between charges in motion. To give you some sense of what is in store, imagine that I set up the following demonstration:  Two wires hang from the ceiling, a few centimetres apart; when I ...