KNOW PHYSICS

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

2026-03-14T18:05:00.000+05:30

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 wave number is defined as

\[ k = \frac{2\pi}{\lambda} \]

Time Dependence of the Wave

After a time $t$, the wave propagates in space and the position becomes

\[ x' = x - ct \]

Therefore the wave function becomes

\[ A(x) = A_0 \sin[k(x-ct)] \] or equivalently \[ A(x) = A_0 \sin(kx - kct) \]

Relation Between Frequency and Wave Number

\[ kc = \frac{2\pi c}{\lambda} = 2\pi\nu = \omega \]

Hence the wave can be written in the standard form

\[ A(x) = A_0 \sin(kx - \omega t) \]

and therefore the speed of the wave is

\[ c = \frac{\omega}{k} \]

Electromagnetic Modes in a Box

Consider a cubical box of side $L$ and imagine electromagnetic waves bouncing back and forth inside it. The walls of the box are assumed to be perfectly conducting and rigid.

Because the walls are conducting, the electric field of the electromagnetic wave must vanish at the boundaries of the box. Therefore, only standing waves that satisfy the boundary conditions can exist inside the box.

This means that the waves that fit inside the box must be integer multiples of half a wavelength.

Allowed Wavelengths

\[ L = \frac{\ell \lambda_x}{2}, \qquad \ell = 1,2,3,\dots \] \[ L = \frac{m \lambda_y}{2}, \qquad m = 1,2,3,\dots \] \[ L = \frac{n \lambda_z}{2}, \qquad n = 1,2,3,\dots \]

Thus the allowed wavelengths along each direction are quantized.

Standing Wave Solution

The spatial dependence of the wave along the $x$-direction can be written as

\[ A(x) = A_0 \sin(k_x x) \]

The wave number is related to the wavelength by

\[ k_x = \frac{2\pi}{\lambda_x} \]

Using the boundary condition $L = \ell \lambda_x /2$, we obtain

\[ \lambda_x = \frac{2L}{\ell} \]

Substituting into the expression for the wave number gives

\[ k_x = \frac{\pi \ell}{L} \]

where

\[ \ell = 1,2,3,\dots \]

Wave Numbers in Three Dimensions

Similarly, the allowed wave numbers in the $y$ and $z$ directions are

\[ k_y = \frac{\pi m}{L} \] \[ k_z = \frac{\pi n}{L} \]

where $m,n = 1,2,3,\dots$

Electromagnetic Modes in Three Dimensions

The standing wave solution inside the cubical box can be written as

\[ A(x,y,z) = A_0 \sin(k_x x)\sin(k_y y)\sin(k_z z) \]

The electromagnetic wave satisfies the wave equation

\[ \frac{\partial^2 A}{\partial x^2} + \frac{\partial^2 A}{\partial y^2} + \frac{\partial^2 A}{\partial z^2} = \frac{1}{c^2} \frac{\partial^2 A}{\partial t^2} \]

For the standing wave solution we obtain

\[ |k|^2 = k_x^2 + k_y^2 + k_z^2 \]

Using the relation between frequency and wave number

\[ |k|^2 = \frac{\omega^2}{c^2} \]

Substituting the quantized wave numbers

\[ k_x = \frac{\pi \ell}{L}, \qquad k_y = \frac{\pi m}{L}, \qquad k_z = \frac{\pi n}{L} \] we obtain \[ \frac{\pi^2}{L^2}(\ell^2+m^2+n^2) = \frac{\omega^2}{c^2} \] Let \[ p^2 = \ell^2 + m^2 + n^2 \] Thus \[ \frac{\omega^2}{c^2} = \frac{\pi^2 p^2}{L^2} \]

Counting the Modes

To count the number of electromagnetic modes, we consider the points $(\ell,m,n)$ in the positive octant of a sphere in mode space.

The volume of a spherical shell of radius $p$ and thickness $dp$ is

\[ 4\pi p^2 dp \]

Since only the positive octant is allowed,

\[ dN(p) = \frac{1}{8} 4\pi p^2 dp \]

Relation Between $p$ and $k$

\[ k = \frac{\pi p}{L} \] \[ dk = \frac{\pi}{L} dp \]

The wave number is related to frequency by

\[ k = \frac{\omega}{c} = \frac{2\pi \nu}{c} \]

The volume of the box is

\[ V = L^3 \]

Number of Modes

Substituting these relations we obtain \[ dN(p) = \frac{L^3}{2\pi^2} k^2 dk \] or \[ dN = \frac{V}{2\pi^2} k^2 dk \] Using \[ k = \frac{2\pi \nu}{c} \] we obtain \[ dN = \frac{V}{2\pi^2} \left(\frac{2\pi \nu}{c}\right)^2 \left(\frac{2\pi}{c} d\nu\right) \] Finally, \[ dN = \frac{4\pi \nu^2 V}{c^3} d\nu \]

Mode Density per Unit Volume

From the previous result, the number of modes in the frequency interval $ \nu $ to $ \nu + d\nu $ inside a volume $V$ is

\[ dN = \frac{4\pi \nu^2 V}{c^3}\, d\nu \]

Since each electromagnetic mode has two possible polarization states, we multiply by 2. Hence

\[ dN = \frac{8\pi \nu^2 V}{c^3}\, d\nu \]

For unit volume, $V=1$, so the number of modes per unit volume is

\[ dN = \frac{8\pi \nu^2}{c^3}\, d\nu \]

Average Energy per Mode and Ultraviolet Catastrophe

The energy density of radiation per unit frequency interval per unit volume is

\[ du = u(\nu)\, d\nu = \frac{8\pi \nu^2}{c^3}\,\bar{E}\, d\nu \]

where $ \bar{E} $ is the average energy of each mode. In classical physics, by the equipartition theorem,

\[ \bar{E} = kT \]

Substituting this into the above expression, we obtain

\[ u(\nu)\, d\nu = \frac{8\pi \nu^2}{c^3} kT\, d\nu \]

Therefore,

\[ u(\nu) = \frac{8\pi \nu^2}{c^3} kT \]

This is the Rayleigh–Jeans law.

Ultraviolet Catastrophe

The total energy density is obtained by integrating over all frequencies:

\[ \int_0^\infty u(\nu)\, d\nu = \int_0^\infty \frac{8\pi \nu^2}{c^3} kT \, d\nu \to \infty \]

Thus, according to classical theory, the energy density of blackbody radiation diverges at high frequencies. This failure of classical physics is known as the ultraviolet catastrophe.

Planck's Law

Planck proposed that the energy of each electromagnetic mode is quantized. The allowed energies are

\[ E_n = n h \nu, \qquad n = 0,1,2,3,\dots \]

where

$h$ is Planck's constant
$\nu$ is the frequency

Boltzmann Distribution

According to the Boltzmann distribution, the probability of a state with energy $E_i$ is proportional to

\[ P_i \propto e^{-E_i/kT} \]

Including the normalization factor, the probability becomes

\[ P_i = \frac{1}{Z} e^{-E_i/kT} \] where $Z$ is the partition function. Thus, \[ P(n) = \frac{e^{-E_n/kT}} {\sum_{n=0}^{\infty} e^{-E_n/kT}} \]

Average Energy of a Mode

The average energy of a mode is given by

\[ \bar{E}_\nu = \sum_{n=0}^{\infty} E_n P(n) \] Substituting the Boltzmann probability, \[ \bar{E}_\nu = \frac{\sum_{n=0}^{\infty} E_n e^{-E_n/kT}} {\sum_{n=0}^{\infty} e^{-E_n/kT}} \] Since $E_n = nh\nu$, \[ \bar{E}_\nu = \frac{\sum_{n=0}^{\infty} nh\nu \, e^{-nh\nu/kT}} {\sum_{n=0}^{\infty} e^{-nh\nu/kT}} \]

Simplifying the Expression

Let \[ x = e^{-h\nu/kT} \] Then \[ \bar{E}_\nu = h\nu \frac{\sum_{n=0}^{\infty} n x^n} {\sum_{n=0}^{\infty} x^n} \] Expanding the series, \[ \bar{E}_\nu = h\nu \frac{x + 2x^2 + 3x^3 + \dots} {1 + x + x^2 + \dots} \]

Evaluation of the Average Energy

Using the series expansions

\[ 1 + x + x^2 + x^3 + \dots = \frac{1}{1-x} \] and \[ 1 + 2x + 3x^2 + 4x^3 + \dots = \frac{1}{(1-x)^2} \] we can simplify the expression for the average energy.

From the previous result

\[ \bar{E}_\nu = h\nu \frac{\sum_{n=0}^{\infty} n x^n} {\sum_{n=0}^{\infty} x^n} \] Substituting the series identities, \[ \bar{E}_\nu = h\nu \frac{\frac{x}{(1-x)^2}} {\frac{1}{1-x}} \] \[ \bar{E}_\nu = \frac{h\nu x}{1-x} \]

Since $x = e^{-h\nu/kT}$,

\[ \bar{E}_\nu = \frac{h\nu}{e^{h\nu/kT}-1} \]

Classical Limit

In the classical limit where

\[ \frac{h\nu}{kT} \ll 1 \] the exponential can be approximated as \[ e^{h\nu/kT} \approx 1 + \frac{h\nu}{kT} \] Thus, \[ \bar{E}_\nu = \frac{h\nu}{\frac{h\nu}{kT}} = kT \]

This reproduces the classical equipartition result.

Planck's Radiation Law

Substituting the average energy into the expression for the energy density,

\[ u(\nu)d\nu = \frac{8\pi\nu^2}{c^3}\,\bar{E}_\nu\, d\nu \] we obtain \[ u(\nu)d\nu = \frac{8\pi\nu^2}{c^3} \frac{h\nu}{e^{h\nu/kT}-1} \, d\nu \]

Therefore the final result is

\[ u(\nu)d\nu = \frac{8\pi h\nu^3}{c^3} \frac{1}{e^{h\nu/kT}-1} \, d\nu \]

This is Planck's Blackbody Radiation Law that we wanted to derive.

Quantum Harmonic Oscillator Coherent States – Complete Notes

2026-03-14T01:25:00.001+05:30

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 $ \alpha \in \mathbb{C} $.

\[ \hat{a} |\alpha\rangle = \alpha |\alpha\rangle \]

Since $ \hat{a} $ is a non-Hermitian operator, $ \alpha $ must be a complex number, i.e.

\[ \alpha = |\alpha| e^{i\phi} \]

Coherent State for Harmonic Oscillator

We know that for a one-dimensional harmonic oscillator

\[ \hat{H} = \frac{\hat{p}^2}{2m} + \frac{1}{2}m\omega^2 \hat{x}^2 \]

The time-independent Schrödinger equation is

\[ \hat{H} |\psi\rangle = E |\psi\rangle \]

Here $ |\psi\rangle $ is an energy eigenstate and $ E $ is the corresponding energy eigenvalue (a real number).

\[ E_n = \hbar\omega \left( n + \frac{1}{2} \right) \]

In the energy basis,

\[ \langle \hat{x} \rangle = 0, \qquad \langle \hat{p} \rangle = 0 \] \[ \langle \hat{x}^2 \rangle = \frac{(2n+1)\hbar}{2m\omega} = \sigma_x^2 \] \[ \langle \hat{p}^2 \rangle = \frac{(2n+1)m\hbar\omega}{2} = \sigma_p^2 \] \[ \sigma_x \sigma_p = (2n+1)\frac{\hbar}{2} \]

For $ n=0 $,

\[ \sigma_x \sigma_p = \frac{\hbar}{2} \]

which is the minimum value and corresponds to the Gaussian wavefunction of the ground state.

The Ladder Operator Method

The ladder operators are defined as

\[ \hat{a} = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} + \frac{i}{m\omega}\hat{p} \right), \qquad \hat{a}^\dagger = \sqrt{\frac{m\omega}{2\hbar}} \left( \hat{x} - \frac{i}{m\omega}\hat{p} \right) \]

and the following are the representations of $ \hat{x} $ and $ \hat{p} $:

\[ \hat{x} = \sqrt{\frac{\hbar}{2m\omega}} \left( \hat{a}^\dagger + \hat{a} \right) \] \[ \hat{p} = i\sqrt{\frac{\hbar m\omega}{2}} \left( \hat{a}^\dagger - \hat{a} \right) \]

Properties of Ladder Operators

The operator $ \hat{a} $ is not Hermitian, since

\[ \hat{a} \ne \hat{a}^\dagger \]

Consider an energy eigenstate $ |n\rangle $. Then

\[ \hat{a}^\dagger |n\rangle = \sqrt{n+1}\,|n+1\rangle \] \[ \hat{a} |n\rangle = \sqrt{n}\,|n-1\rangle \]

The commutation relations are

\[ [\hat{x},\hat{p}] = i\hbar \] \[ [\hat{a},\hat{a}^\dagger] = 1 \]

Number Operator

Let us define the number operator $ \hat{N} $ with the following properties:

\[ \hat{N} = \hat{a}^\dagger \hat{a} \] \[ \hat{N}|n\rangle = n|n\rangle \]

The commutation relations with ladder operators are

\[ [\hat{N},\hat{a}^\dagger] = \hat{a}^\dagger \] \[ [\hat{N},\hat{a}] = -\hat{a} \]

The Hamiltonian can be written in terms of the number operator as

\[ \hat{H} = \hbar\omega \left(\hat{N} + \frac{1}{2}\right) \]

Hence the eigenstates of $ \hat{N} $ are also eigenstates of the Hamiltonian.

\[ \hat{H}|n\rangle = \hbar\omega\left(\hat{N}+\frac{1}{2}\right)|n\rangle = \hbar\omega\left(n+\frac{1}{2}\right)|n\rangle \]

Thus the energy eigenvalues are

\[ E_n = \hbar\omega \left(n+\frac{1}{2}\right) \]

Action of the Number Operator

And,

\[ \hat{N}(\hat{a}^\dagger |n\rangle) = (n+1)\hat{a}^\dagger |n\rangle \] \[ \hat{N}(\hat{a}|n\rangle) = (n-1)\hat{a}|n\rangle \]

Note: The smallest eigenvalue of $ \hat{N} $ is $0$, therefore

\[ \hat{a}|0\rangle = 0 \]

Hence,

\[ n = \langle n|\hat{N}|n\rangle = \langle n|\hat{a}^\dagger \hat{a}|n\rangle = (\hat{a}|n\rangle)^\dagger (\hat{a}|n\rangle) \ge 0 \]

Arbitrary Eigenstate

Since

\[ \hat{a}^\dagger |n\rangle = \sqrt{n+1}\,|n+1\rangle \]

Now,

\[ |n\rangle = \frac{\hat{a}^\dagger}{\sqrt{n}}|n-1\rangle \] \[ = \frac{(\hat{a}^\dagger)^2}{\sqrt{n(n-1)}}|n-2\rangle \]

\[ |n\rangle = \frac{(\hat{a}^\dagger)^n}{\sqrt{n!}}|0\rangle \]

Solving the Eigenvalue Equation

We have the eigenvalue equation for the coherent state $ |\alpha\rangle $:

\[ \hat{a}|\alpha\rangle = \alpha|\alpha\rangle \]

Let us assume the following expansion:

\[ |\alpha\rangle = \sum_{n=0}^{\infty} C_n |n\rangle \]

Determination of the Coefficients

Starting from

\[ \hat{a}|\alpha\rangle = \hat{a}\sum_{n=0}^{\infty} C_n |n\rangle = \alpha |\alpha\rangle \]

we get

\[ \sum_{n=1}^{\infty} C_n \sqrt{n}\,|n-1\rangle = \alpha \sum_{n=0}^{\infty} C_n |n\rangle \]

Now change the index by setting $ n-1 = m $. Then

\[ \sum_{m=0}^{\infty} C_{m+1}\sqrt{m+1}\,|m\rangle = \alpha \sum_{m=0}^{\infty} C_m |m\rangle \]

Comparing the coefficients of $ |m\rangle $, we obtain

\[ C_{m+1}\sqrt{m+1} = \alpha C_m \]

Therefore,

\[ C_{m+1} = \frac{\alpha}{\sqrt{m+1}}\,C_m \]

Iteratively,

\[ C_1 = \alpha C_0 \] \[ C_2 = \frac{\alpha}{\sqrt{2}} C_1 = \frac{\alpha^2}{\sqrt{2}} C_0 \] \[ C_3 = \frac{\alpha^3}{\sqrt{3!}} C_0 \]

Hence, in general,

\[ C_n = \frac{\alpha^n}{\sqrt{n!}}\,C_0 \]

Therefore,

\[ |\alpha\rangle = \sum_{n=0}^{\infty} C_n |n\rangle = \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} C_0 |n\rangle \]

Normalization

We have obtained

\[ |\alpha\rangle = \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} C_0 |n\rangle \]

Now compute the normalization:

\[ \langle \alpha | \alpha \rangle = \sum_{n=0}^{\infty} \frac{|\alpha|^{2n}}{n!} |C_0|^2 \langle n|n\rangle \]

Since $ \langle n|n\rangle = 1 $,

\[ 1 = |C_0|^2 \sum_{n=0}^{\infty} \frac{|\alpha|^{2n}}{n!} \]

Using the exponential series

\[ \sum_{n=0}^{\infty} \frac{|\alpha|^{2n}}{n!} = e^{|\alpha|^2} \]

we obtain

\[ 1 = |C_0|^2 e^{|\alpha|^2} \] \[ C_0 = e^{-|\alpha|^2/2} \]

Normalized Coherent State

Hence the normalized coherent state is

\[ |\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle \]

Thus a coherent state is a superposition of number states (Fock states).

Checking the Eigenvalue Equation

Now that we have derived the coherent state $ |\alpha\rangle $, let us verify that it satisfies the eigenvalue equation.

\[ |\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} |n\rangle \]

Let us check that the state satisfies the eigenvalue equation.

\[ \hat{a}|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=0}^{\infty} \frac{\alpha^n}{\sqrt{n!}} \hat{a}|n\rangle \]

Using $ \hat{a}|n\rangle = \sqrt{n}|n-1\rangle $, we obtain

\[ \hat{a}|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{n=1}^{\infty} \frac{\alpha^n \sqrt{n}}{\sqrt{n!}} |n-1\rangle \]

Changing the index $ n-1=m $,

\[ \hat{a}|\alpha\rangle = e^{-|\alpha|^2/2} \sum_{m=0}^{\infty} \frac{\alpha^{m+1}}{\sqrt{m!}} |m\rangle \]

Factoring out $ \alpha $,

\[ \hat{a}|\alpha\rangle = \alpha e^{-|\alpha|^2/2} \sum_{m=0}^{\infty} \frac{\alpha^m}{\sqrt{m!}} |m\rangle \] \[ \hat{a}|\alpha\rangle = \alpha |\alpha\rangle \]

Probability Distribution

\[ \langle \hat{N} \rangle = \langle \alpha | \hat{a}^\dagger \hat{a} |\alpha\rangle = (\hat{a}|\alpha\rangle)^\dagger(\hat{a}|\alpha\rangle) \] \[ \bar{n} = (\alpha^* \alpha) \langle \alpha | \alpha \rangle \] \[ \bar{n} = |\alpha|^2 \]

Thus the photon number is proportional to the square of the amplitude.

Classical Electromagnetic Wave

Average energy density

\[ u = \frac{\epsilon_0}{2} |E_0|^2 \]

Total energy

\[ U = \frac{\epsilon_0 V}{2} |E_0|^2 \]

Quantum Mechanical Analog

\[ \bar{n} = |\alpha|^2 = \frac{U}{\hbar\omega} = \frac{\epsilon_0 V}{2\hbar\omega} |E_0|^2 \]

Probability Distribution

The probability distribution is

\[ P(n) = |\langle n|\alpha\rangle|^2 \]

Using the coherent state expansion,

\[ P(n) = \left| \left\langle n \left| e^{-|\alpha|^2/2} \sum_{m=0}^{\infty} \frac{\alpha^m}{\sqrt{m!}} |m\rangle \right.\right\rangle \right|^2 \] \[ = \left| e^{-|\alpha|^2/2} \frac{\alpha^m}{\sqrt{m!}} \delta_{mn} \right|^2 \] \[ = \left| e^{-|\alpha|^2/2} \frac{\alpha^n}{\sqrt{n!}} \right|^2 \] \[ = \frac{e^{-|\alpha|^2}|\alpha|^{2n}}{n!} \]

Since $ \bar{n} = |\alpha|^2 $, this becomes

\[ P(n) = \frac{e^{-\bar{n}}\bar{n}^{\,n}}{n!} \]

Thus the photon number distribution for a coherent state is a Poisson distribution:

\[ P(X=n)=\frac{\mu^n e^{-\mu}}{n!} \]

with mean $ \mu = \bar{n} $.

Hence, the probability of finding the Fock state $ |n\rangle $ in a coherent state is

\[ P_n(\alpha) = |\langle n|\alpha\rangle|^2 = \frac{e^{-|\alpha|^2}|\alpha|^{2n}}{n!} \]

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

2022-05-10T20:02:00.000+05:30

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.

It has following parts:

A cylinder which have perfectly non-conducting walls, a perfectly conducting base, with a perfectly non-conducting piston which moves without friction in the cylinder.
The cylinder contains a perfect gas.
Source : A reservoir maintained at a constant temperature T1. It has infinite thermal capacity. Any amount of heat can be drawn from it , and still the temperature will not change.
Insulating stand : A perfectly non-conducting platform acts as a stand for adiabatic processes.
Sink : A reservoir maintained at a constant lower temperature T2(T2<T1). Engine can reject any amount of heat to it, the temperature of sink would still remain constant. It also has infinte thermal capacity.

Some important terms-

1) Quasi-Static Process-

A Quasi-Static process is a process carried out very very slowly so that the system under consideration remains at all times arbitrarily close to the equilibrium state. The thermodynamic processes should be Quasi-Static for the P-V diagram to be meaningful.

2) Cyclic Process-

A cyclic process is a thermodynamic process in which the system returns to its initial state after undergoing a series of changes.

It may be a closed cycle or an open cycle.

3) Reversible Process-

A reversible process is that which can be retraced in the opposite direction, so that the working substance passes through exactly the same conditions as it does in the direct process.

In other words it means that in a step where heat is absorbed in the direct process, it is given out in the reverse process and vice versa.

4) Isothermal Process-

The process in which the temperature of the system remains constant is called a isothermal process.

Also by Boyle's law at constant temperature P(pressure) is inversely proportional to V(volume).

5) Adiabatic process-

The process in which no heat transfer takes place. The heat neither transfer inside nor outside the system.

Carnot's Cycle

It is a reversible cyclic process. This cycle consists of isothermal and adiabatic curves on a P-V diagram and is known as Carnot's cycle.

1) Isothermal expansion-

Suppose the cylinder contains one mole of perfect gas. Let the initial temperature of the gas in the cylinder be T1 K. Let the temperature of heat source is also T1 K.

The cylinder is placed on the source, so that the gas absorbs heat at a constant temperature T1. Since the gas absorbs heat from source at constant temperature it undergoes isothermal expansion. It is an quasi-static expansion.

Let the initial state of the working substance be A(P1,V1,T1). And the final state of the working substance during this isothermal expansion be B(P2,V2,T1). This can be expressed in a P-V graph as the curve AB at constant temperature T1.

if the gas absorbs Q1 heat from the source at T1 and does work W1. Since it is an isothermal process

ΔU = 0

Therefore, by first law of thermodynamics

Q1 = W1

Hence,

2) Adiabatic expansion-

The cylinder is now removed from the source and is placed on the insulating stand. Since the stand is insulating, the processes that will happen , will happen without the exchanging of heat.

The gas is allowed to expand slowly. The expansion is called slow adiabatic expansion. The expansion or the work done by the gas is done at the expense of its internal energy, which will decrease the temperature from T1 to T2. Where T2 is the temperature of the sink.

The initial state of the substance is B(P2,V2,T1) and let the final state at the end of this process be C(P3,V3,T2). If the work in this process be W2

3) Isothermal Compression-

Now the cylinder is removed from the insulating stand and is placed on the sink which is at the temperature T2. In this process the gas is compressed. The work is done on the gas. The gas rejects the heat to the sink, hence the temperature of the gas remains constant. Thus the gas undergoes isothermal compression at constant temperature T2 and rejects some heat to the sink.

This process can be represented by the isothermal curve CD. Where C(P3,V3,T2) to the state D(P4,V4,T2). Let the heat rejected by the gas be Q2 to the sink at T2 and work W3 is done on the gas is given by

4) Adiabatic Compression-

The cylinder is now removed from sink and again kept on the insulating stand. Again because the stand is insulating, the process will be adiabatic. The gas will compress, and the temperature will rise. Adiabatic Compression continues till the original condition arrives.

Thus, this adiabatic process can be represented by a adiabatic curve DA. Where D(P4,V4,T2) is the initial state and A(P1,V1,T1) be the final state in this process.

In this process if the work done on the gas is W4 and is given by

Efficiency of heat engine

The efficiency of the heat engine is the ratio of quantity of heat converted into work(useful output) per cycle to the total amount of heat absorbed per cycle.

It T1=T2 i.e. the temperature of the source is equal to the temperature of sink, then efficiency is zero i.e. the engine does not work.

Friction in solids

2021-09-19T13:16:00.000+05:30

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 relative motion. So frictional force will act in such a way that it will make A to move slowly and B to move fast, so that the relative motion could become zero. In other words friction will deaccelerate A and accelerate B.

Now frictional force can operate between a given pair of solids , a solid and fluid or in given pair of fluids. Frictional force exerted by fluids is called as viscous force. Here we are only talking about solids.

For understanding friction we have to understand few basic terms or definitions, that are as follows-

Contact Surface- Contact surface is that surface of plane on which both the bodies are in contact.
Contact Force- Resultant of all the forces exerted by a body on another body due to their contact is called a contact force. Each body exerts it on one another.
Normal force- The component of the contact force perpendicular to the contact surface or the component of contact force normal to the contact surface is called normal force.

Now suppose that a mass M is kept on ground. The object is at rest. Then an external force F is applied on the obect in the direction shown in the figure. Let f be the frictional force acting on the mass in the direction opposite to the external force.

Let F=f.

Then we can say that the object will be at rest. In this case the forces that operate are- weight, normal reaction, external force, frictional force.

But the forces that are applied by the ground to the mass because of their contact are - frictional force and normal force. Hence for calculating contact force we have to only consider friction and normal reaction.

From the above figure, let theta be the angle between normal and contact force. Then tan(theta) will be equal to (f/N). Contact force can also be calculated using the above formula.

Friction coefficient - Basically it defines the roughness of the surface, i.e. the value of frictional force. The coefficient is dimensionless and unit less.
We can also say that the ratio of frictional force to the normal reaction is called friction coefficient, which in all case is a constant. This constant ratio is called the coefficient of friction and is usually symbolized by the Greek letter mu (μ). Since it is only a ratio of forces , the coefficient is dimensionless.
The value of friction coefficient can also be calculated using tan(theta), because tan(theta) is also the same ratio.

Types of friction

There are two types of friction. When there is no relative motion between two surfaces or when there is no slipping of one body on another then the friction which operates is called as static friction.

Static frictional force can vary from zero to a limiting value F(L). It depends upon the external force that is being operated on the body. If the external force (F) will be less than or equal to the limiting static frictional force then the body will be at rest. But if the external force is more than the limiting friction then the body will start motion. Friction coefficient of static friction is denoted by mu_s (μs).

When the body starts slipping on another body or when the relative motion starts, then the friction acting between surfaces is called as kinetic friction. Friction coefficient of kinetic friction is denoted by mu_k (μk). It has a constant value. It does not vary. It is less than static friction. So it becomes easier to move an body after it starts moving.

Friction is an important factor to consider in experiments of classical mechanics. To reduce the friction, lubricants are used. Which reduces the value of friction coefficient of the surface. Only in ideal situations coefficient of friction is considered as zero.

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

2020-09-05T18:56:00.003+05:30

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

Newton, who was respected by all who had interest in science, came out with what can be called "particle model of light". According to this model when you put on a bulb, a candle or any other source of light, the source emits special kinds of particles-- the particles of light.

Source- https://en.m.wikipedia.org/wiki/Isaac_Newton

These particles were commonly called corpuscles and the description of light given by Newton, corpuscle model of light. The model was very simple in nature and was able to explain the observations available in those periods.

Newton's first law of motion tells that every particle moves in a straight line with a constant speed if no force acts on it. So the particles of light should also move in straight lines in free space‐— a simple explanation for rectilinear motion of light.

Reflection is also simple to understand using the corpuscle model. A rubber ball hitting a smooth, hard surfaces rebounds. For a perfectly elastic oblique collision from a hard plane surface, the angle of incidence is equal to the angle of reflection, and the two velocity vectors and the normal to the surface are in the same plane. Similarly, a particle of light when strikes a smooth surface surface, say a mirror, reflects obeying the known laws of reflection.

If light goes from air to another medium such as glass, the particles of the second medium attract the particles of light, resulting in deflection in their paths. Once the particle is well within the second medium, force is exerted on it from all sides. This makes the resultant force zero and the particle moves along a straight line. The deflection at the surface causes bending of the light ray. The force of attraction on the particles of light by the medium will also increase their speed and according to the Newton's model, speed of light in water or glass should be larger than that in air.

However, in Newton's times there was no way to measure the speed of light in a medium and this prediction could not be verified. To explain colours, Newton assumed that there are different types of particles of light, each corresponding to a particular colour.

The corpuscular model of light

Many believe that the corpuscular model of light was known much before Newton. According to Joyce and Joyce, Rene Descartes, in 1637, derived Snell's law using a corpuscular model of light. Newton was only about eight years old when Descartes (1596-1650) died and therefore Descartes did not get the corpuscular model from Newton!

The front covers of the first edition(1704) of OPTICKS: OR, A Treatise of the Reflexions, Refractions, Inflexions and Colours of LIGHT by Newton.

According to Wikipedia,

the corpuscular theory of light, arguably set forward by Pierre Gassendi and Thomas Hobbes states that light is made up of small discrete particles called "corpuscles" (little particles) which travel in a straight line with a finite velocity...... About a half-century after Gassendi, Isaac Newton used existing corpuscularian theories to develop his particle theory of the Physics of light. And it became very famous.

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Huygens wave model

Source-
https://en.m.wikipedia.org/wiki/Christiaan_Huygens

Newton's theory, though very successful in understanding the known behaviour of light, did not go unchallenged.

The Dutch physicist Christian Huygens, a contemporary of Newton, had a very different perspective on this problem. He suggested that light travels as a wave motion in a medium, more like a sound wave. When you put a source of light on, a disturbance is created in the equilibrium state of the medium. This disturbance travels in the medium. But then how does light cast shadow? A sound wave does not appear to cast shadow. If you put a cardboard between yourself and a source of sound, you still 'hear' the sound quite well.

If you keep the same cardboard between yourself and a source of light, you do not see the source. Huygens argued that the wavelength of light waves may be much smaller than the dimensions of the cardboard and the other usual obstacles. In such cases the bending at the edges(diffraction) will be negligible and light will cast a sharp shadow.

Huygens believed that ether vibrated in the same direction as light, and formed a waveitself as it carried the light waves. In a later volume, Huygens' Principle, he ingeniously described how each point on a wave could produce its own wavelets, which then add together to form a wavefront.

The struggle: Newton or Huygens

The reflection or refraction of light can also be explained on the basis of wave theory. All waves, including sound waves, show these phenomena. Huygens proposed a way to construct the wavefronts and showed that the laws of reflection and refraction are the same for the waves as were known for light.

However, if light were waves, the speed of light in water or glass should be less than that in air. This was in contrast to the prediction of corpuscle model. But there was no way to measure the speed of light. Hence it was not the deciding factor between wave and corpuscular models.

Wave theory was having an additional advantage. Different wavelengths of light can be associated with different colours. It was much simpler than Newton's corpuscle model.

At this stage all experimental observations about light could be explained by the corpuscle model and also by the wave model of Huygens. Newton, enjoying a very high respect because of the great success achieved in the field of dynamics, had an advantage and particle model of light by and large remained the popular choice for about 150 years.

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Nikola Tesla: Engineer and Inventor

2020-08-19T18:31:00.005+05:30

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 working. Often if was a solution to a problem that he had encountered during his work.

He could visualise the item by just hearing it with realistic details. He could visualise his invention in mind with great precision, in all dimensions also in working mode before actually moving to the construction stage of that machine.

So he typically thinks about his invention greatly in his mind. Using techniques like Picture thinking.

Work Habits

It is popular belief that Tesla worked from 3:00 am to 11:00am. He slept very less. Only for 3-4 hours. And additional naps. He never married. So he was having all his time to spend on work. He once reportedly worked for 84 hours straight in his lab without any rest.

When Tesla was in college first year. His principal sent a letter to Tesla's father that his son might die due to overwork. But during his 2nd and 3rd years in college he got heavily addicted to gambling. Due to this he was not able to study for his final exmas. He asked for some time but he didn't got. So he never received the degree.

Tesla walked between 8 and 10 miles per day. He curled his toes one hundred times for each foot every night, saying that it stimulated his brain cells.

Tesla vs Edison

In 1881, Tesla moved to Budapest, Hungary, to work under Tivadar Puskás at a telegraph company, the Budapest Telephone Exchange.

In 1882, Tivadar Puskás got Tesla another job in Paris with the Continental Edison Company.

Moving to US

In june 1884, tesla moved to US, where he worked for Edison in his company. Tesla was troubleshooting all the problems with the machines, making them even more effective.

Tesla worked there for 6 months before he quit. In his autobiography, Tesla stated the manager of the Edison Machine Works offered a $50,000 bonus to design "twenty-four different types of standard machines" "but it turned out to be a practical joke".

Some versions of this story have Thomas Edison himself offering and then reneging on the deal, quipping "Tesla, you don't understand our American humor".

War of currents

In 1887, Tesla developed an induction motor that ran on alternating current (AC), a power system format that was rapidly expanding in Europe and the United States. Tesla

Engineers working for the Westinghouse Electric & Manufacturing Companyreported to George Westinghouse that Tesla had a viable AC motor and related power system—something Westinghouse needed for the alternating current system he was already marketing.

After purchasing patents, Westinghouse soon hired tesla for consulting in his company.

Tesla was having a lot of money, because of his patents.

But Edison was marketing large-scale low-voltage direct current (DC) indoor incandescent lighting.

Soon a war of current started. Edison started spreading misinformation about Alternating current(AC), that it is harmful, and more dangerous than Direct current(DC). This propaganda was demonstrated by Edison publicly using DC current, in the name of AC.

But Nikola Tesla also showed everyone by doing public demonstrations that AC is safe. Soon people started Believing in Tesla. Edison has lost the war. He misinterpreted the potential of AC current.

In 1897 Edison sold his remaining stock in Edison Electric Illuminating of New York to finance his iron ore refining prototype plant.

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Later years

At the 1932 party( birthday conference), Tesla claimed he had invented a motor that would run on cosmic rays.

In 1933 at age 77, Tesla told reporters at the event that, after 35 years of work, he was on the verge of producing proof of a new form of energy. He claimed it was a theory of energy that was "violently opposed" to Einsteinian physics.

At the 1934 occasion, Tesla told reporters he had designed a superweapon he claimed would end all war.[216][217] He called it "teleforce", but was usually referred to as his death ray.

On 7 January 1943, at the age of 83, Tesla died alone in Room 3327 of the New Yorker Hotel. His body was later found by maid Alice Monaghan after she had entered Tesla's room, ignoring the "do not disturb" sign that Tesla had placed on his door two days earlier.

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Understanding magnetic field in simple way

2020-08-03T21:12:00.000+05:30

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 turn on a current, so that it passes up one wire and back down the other, the wires jump apart- they evidently repel one another.

How do we explain this? You might suppose that the battery (or whatever drives the current); is actually charging up the wire, and that force is simply due to the electrical repulsion of like charges. But this is incorrect.

I could hold up a test charge near these wires, and there would be no force(not precisely) it, for the wires are in fact electrically neutral.( It's true that electrons are flowing down the line- that's what a current is- but there are just as many stationary plus charges as moving minus charges on any given segment.) Moreover, if I hook up my demonstration so as to make the current flow up both wires, they are found to attract!

Whatever force accounts for the attraction of parallel currents and the repulsion of antiparallel ones is not electrostatic in nature. It is our first encounter with a magnetic force. Whereas a stationary charge produces only an electric field in the space around it, a moving charge generates, in addition, a magnetic field.

In fact, magnetic fields are a lot easier to detect, in practice---- all you nedd is a compass. How these devices work is irrelevant at the moment; it is enough to know that the needle points in the direction of the local magnetic field.

Ordinarily, this means north, in response to the earth's magnetic field, but in the laboratory, where typical fields may be hundreds of times stronger that that, the compass indicates the direction of whatever magnetic field is present.

Now if you hold up a tiny compass in the vicinity of a current-carrying wire, you quickly discover a very peculiar thing: The field does not point toward the wire, nor away from it, but rather it circles around the wire. In fact, if you grab the wire with your right hand-thumb in the direction of the current--your fingers curl around in the direction of the magnetic field.

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What Happens: When an object is freely falling

2020-07-19T20:49:00.000+05:30

Let us first understand what gravity is. Gavity also known as gravitation, is a phenomenon (not a force) of attraction of all the things that have mass-energy like planets, galaxies and light towards one another.

What happens: when an object is freely
Falling?

Don't get confuse between gravity and Gravitational Force. They are separate terms. Gravity is a natural phenomenon. Whereas, Gravitational force is a force that is a measure of attraction.

Gravity is the phenomena or cause of, why we are able to stand. Whereas gravitational force is responsible for our weight. If you go to moon then your weight becomes 1/6th of the weight on earth.

Gravity is the reason, why planets revolve around sun., and moons revolve around planets. It is the phenomena, which plays an important role in galaxy formation. Most of the galaxies have a black hole as their centre. Black hole have very large gravitational effect.

Now from Newton's laws of motion, we know that acceleration is the ratio of force applied(F) to mass of the object(m). i.e., a= F/m.

Now if F is the gravitational force and m is the mass of the object that is being influenced by gravity, then a will be the acceleration of the object that is caused by gravity. The acceleration due to gravity(of earth) is represented by 'g'. Its value is 9.8m/s². At heights, the value may be different.

Free Fall

Now an interesting point is that the acceleration due to gravity does not depend on the mass of the object.
If in vacuum, you drop a stone and a feather at the same time , from same height, both will reach the ground at the same time. That means for every object the acceleration is same.

So does that means that mass of the object doesn't matter at all? No. The object with more mass, will experience more force of gravity. And the object with less mass will experience less force of gravity.

But why this happens? Why 'g' does not depends upon the object. For finding acceleration due to gravity we divide the force by mass of the object. Hence the resultant is independent of the mass of the object.

Gravitational Force

The 'g' only depends on the mass and radius of the earth. For any other planet, it will depend on mass and acceleration of that planet. If mass or radius of our planet will be changing, then the value of 'g' will also be changing accordingly. But that's not the case.

Now imagine you are standing on a mountain. And you are on its edge. You are holding a stone in your hand. And suddenly you release the stone from your hands.(Remember that you have to only release it, not throw it). Now tell, what will be the acceleration of the stone? Is it 'g', above 'g' or below 'g'.

Well when you release an object from this much height, it will suffer some air resistance. Wind speeds are also very high at heights.

So, in practical life we can say that different objects will hit the ground at different time, even if they are released from same height, due to atmosphere. In practical life there are many forces acting on an object other than gravity, like air resistance, buoyant force, force due to wind speed, climate change etc.

When there is only Gravitational force acting on a body, then it is considered as free fall in Newtonian physics. There are misconceptions that in free fall something must be falling. If an object is going upwards, but the only force acting on it is gravity, then also the object is under free fall.

On earth the value of 'g' is 9.8 m/s². So an object under free fall on earth should have an acceleration of 9.8m/s².

Whereas, on moon the value of acceleration due to gravity is 1/6th of the value on earth.

Astronaut David scott demonstrated free fall on moon in August 2, 1971. He released a hanmer and a feather simultaneously from the same height. Both fell and hit the ground at the same time.

In free fall, since the only force to consider is gravity. Hence it becomes easy to calculate values of different variables such as acceleration, final velocity, displacement, time taken etc. at any part of the motion using Newton's equations of motion.

If some initial parameters are given then we can easily calculate rest of the variables.

Objects in free fall includes:

A spacecraft in space with propulsions off. (May be in an orbit.)
An object dropped at the top of drop tube.
An object thrown upwards or person jumping off.( as long as air resistance is negligible)

Objects not in free fall includes:

An aircraft flying (additional forces such as lift)
Standing on ground. (Gravity is balanced by normal force form ground).
Jumping from height with parachute.( it balances gravity)

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Isaac Newton was an alchemist ? Was Isaac Newton searching for The Philosophers Stone ?

2020-07-17T19:52:00.002+05:30

Many of you have an image of Sir Isaac Newton. You probably think Issac Newton as a person who contributed immensely in Physics through his laws of motion, gravitation which were the foundation of classical mechanics which we study today.

Was Isaac Newton an alchemist?

Or many of you think Isaac Newton as a mathematician who invented calculus(debatable).

Or the person who proved the society wrong by proving that the light is made up of different colours, with his research on optics.

But there is also an other side of his personality. He was heavily interested in alchemy and other occult practices. The world came to know this side of him by the unpublished papers written by him on these topics.

Sir Isaac Newton was heavily involved in Alchemy.
It is considered as an occult science. Alchemy is the medieval forerunner of chemistry, concerned with the transmutation of matter, in particular with attempts to convert base metals into gold or find a universal elixir.

Some of the alchemy practices were banned in England during Newton's Lifetime. As many people used to fool others. Fake gold was a very big problem. The one who was found to do such practices was given severe punishments.

Newton would perform experiments with different types of chemicals and then observe their reactions.
Since at that time chemistry was still in infancy. He would spends hours near the furnace, for making the correct chemical combinations.

He hated to meet people, socialize and wanted to work in isolation. In his college days he used to stay in his room, doing his work. He also hates criticism. That is why he published many paper lately.

Most of the Newton's work on alchemy was lost in a fire occurred in his lab. It was believed that the fire was caused due to his dog. So the work which we are able to observe is very less. So, we would never know about the true extent of his work.

Newton copied the recipe by hand from a text by American-born alchemist George Starkey, then scribbled his own lab notes on the back.
Souce-https://api.nationalgeographic.com/distribution/public/amp/news/2016/04/160404-isaac-newton-alchemy-mercury-recipe-chemistry-science
PHOTOGRAPH BY CHEMICAL HERITAGE FOUNDATION

One of the aim of Newton's alchemy was to discover Philosophers Stone. It is a stone(or chemical) which was believed to turn base metals into gold. He aimed to find a recipe to make Philosophers Stone.

He was also trying to find the "Elixir of life". Elixir of life was believed to provide immortality to the one who drinks it. It can also cure all the diseases.

For all this, he heavily studied all the researches, recipes, experiments that were done before. And try to perfom all of them as it is. We would find an equation of optics and an experiment of alchemy on the same page of his notebooks. He believed that science and alchemy and also other occult practices go hand in hand.

Newton has been associated with many of the secret societies. But due to lack of publicized materials and doubtful motives, it is difficult to establish Newton's memberships.

Newton was too much involved in alchemy. He would work continuously for many hours. Which also affected his health. He was also suffering from a nervous breakdown during this period.

At the time of his death, Newton was having 169 book on alchemy in his personal library. Although he was having more books in his alchemy period.

At the time of his death, the papers on alchemy by Newton were considered "unfit" to publish by Newton's estate.

After purchasing and studying Newton's alchemical works in 1936, Keynes, opined in 1942 at the tercentenary of his birth that "Newton was not the first of the age of reason, he was the last of the magicians."

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What is Dark Matter? Dark Matter in a Nutshell

2020-07-13T18:08:00.001+05:30

Dark Matter in a nutshell

Introduction

The matter in which the modern scientists are higly interested in is the Dark Matter.

So, what is Dark Matter? How it was formed? Who suggested the idea of Dark matter? How much amount of it is in the universe? Does it interact with other particles? What is Baryonic Matter? Can we detect dark matter or has anyone detected it till now?
Let find out.
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What is Dark Matter?

Dark matter is a type of matter. Dark matter is called Dark because it does not appear to interact with electromagnetic field, which means it doesn't absorbs, reflect or emit electromagnetic radiations, and is therefore difficult to detect.

Primary Evidences

The primary evidence comes from the calculations done after studying about different galaxies in the universe.It was observed that may of the galaxies would fly apart, not move as they move or even do not form until and unless they contain a large amount of unseen matter inside them.

There are also other evidences which includes observations in Gravitational lensing, Cosmic microwave background and also in fields like observations in formation and evolution of galaxies.

Discovery of Dark matter

The first to suggest the existence of Dark Matter using stellar velocities was Dutch astronomer Jacobus Kapteyn in 1922. Fellow Dutchman and radio astronomy pioneer Jan Oort also hypothesized the existence of dark matter in 1932.

Originally known as the “missing mass,” dark matter’s existence was first inferred by Swiss American astronomer Fritz Zwicky, who in 1933 discovered that the mass of all the stars in the Coma cluster of galaxies provided only about 1 percent of the mass needed to keep the galaxies from escaping the cluster’s gravitational pull.

Amount Present in Universe

The total mass–energy of the universe contains 5% ordinary matter and energy, 27% dark matter and 68% of a form of energy known as dark energy.

Thus, dark matter constitutes 85% of total mass, while dark energy plus dark matter constitute 95% of total mass–energy content.

Detection of Dark Matter

If dark matter is made up of sub-atomic particles, then millions or even billions, of such particles must pass through every square centimeter of the Earth in each second.

Based on these there are experiments performed in order to detect dark matter. These experiments can be classified into two types: 1) direct and 2) indirect.

Direct detection experiments aims to detect the dark matter particles directly with the help of an detector. Till now there is no well established claim of direct detection of dark matter.

Indirect detection experiments search for the products of the self-annihilation or decay of dark matter particles in outer space. For example, in regions of high dark matter density (e.g., the centre of our galaxy) two dark matter particles could annihilateto produce gamma rays or Standard Model particle–antiparticle pairs.

Conclusion and future

Since the dark matter is not been observed directly it is still a mystery. It it exists, then it must barley interact with the ordinary Baryonic matter and radiation. Most of the dark matter is thought to be non-baryonic.

There can be a new type of elementary particle which has not been discovered yet.

Although the concept of dark matter is widely accepted, but since it is not directly observed , the arguments exists.

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What is Brownian Motion?

2020-07-09T17:51:00.000+05:30

BROWNIAN MOTION

Introduction

One of the most asked question is- what is Brownian Motion? Brownian motion was first observed in 1827 by Brown, a botanist. The phenomenon of Brownian movement can be easily observed in a laboratory if a colloidal solution is examined under and ultramicroscope. As the direction of illumination is perpendicular to the axis of the microscope, the suspended particles in the solution look like bright illuminated spots.

Is eliminated particles continuously move to and fro in a random haphazard way. The particles spin, rise, sink and rise again. The movement of the particles is continuous and spontaneous. This non-stop random and haphazard motion is perpetual and spontaneous.

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So, what is Brownian Motion? This non-stop, random and haphazard motion is called Brownian Motion.

According to Albert Einstein's theory of transitional Brownian Motion, the particles tend to diffuse into the medium in coyrse of time. Consequently, the diffusion coefficient must be related to the Brownian movement.

The diffusion can be calculated in two different ways:
1. From the irregular motion of the suspended particles.
2. From the difference in osmotic pressure caused by the difference in concentrations of the suspended particles.

Now let's see that what are the properties of Brownian Motion or what are the essential features of Brownian Motion.

Essential Features

1. In Brownian Motion, the motion of each particle is completely irregular and random. No two particles are found to execute the same motion.

2. In Brownian Motion, the motion is continuous and takes place for ever.

3. In Brownian Motion, the smaller the particles appear to be more agitated than the larger ones.

4. In Brownian Motion, the motion is independent of the nature of the suspended particles. Two particles of the same weight and size move equally fast at the same temperature.

5. In Brownian Motion, the motion becomes more vigorous when the temperature is increased.

6. In Brownian Motion, the motion is more vigorous conspicuous in a liquid of lower viscosity.

7. In Brownian Motion, the motion is not modified due to the shaking of the containing vessel.

Basis of Brownian Motion

Brownian Motion is due to the bombardment of the dispersed particles by molecules of the medium which are themselves in a state of continuous haphazard motion. When thr particles are sufficiently large, the forces due to molecular impact almost completely balance.

This is why Brownian Motion cannot be observed with particles of large size. But when the size of the particles is very small, each particle will be acted upon by a resultant unbalanced force and will consequently move in a haphazard manner in response to the magnitude and direction of this force.

In other words, the particles move in response to the molecules of the liquid in a completely haphazard manner. The motion of the molecules of a gas is similar to Brownian Motion of suspended particles in colloidal solution. Thus the laws of kinetic theory of gases are applicable to Brownian Motion.

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Quantum Mechanics #3- Classical Mechanics failed to explain Black Body Radiation, led Planck to the discovery

2020-07-05T19:34:00.001+05:30

Quantum Mechanics #3

Black Body

A perfectly black body is one which absorbs totally all the radiation of any wavelength which fall on it. As it neither reflects nor transmits any radiation, it appears black; whatever be the colour of incident radiation. The main characteristic of such a body is that when heated to a suitable high temperature, it emits full or total radiation.

As it is a perfect absorber, it is also a perfect radiator, its emission being the greatest possible for every wavelength at any given temperature.

BLACK BODY

Black Body in Practice

In practice, a perfectly black body is not available. Lamp-black and Platinum black are the nearest approach to a black body. However, a body showing close approximation to a perfectly black body can be constructed.

Black Body absorber

Black body emitter

A closed chamber, say a hollow sphere (known as hollow spherical cavity) whose inner surface is coated with platinum black or lamp black having a small opening O in the surface and a projection P opposite to the opening, acts as a black body.

It should be noted that perfectly black body is an ideal concept similar to a perfect gas which does not exist, i.e. it can not be 100% realized in actual practice. The platinum black surface absorbs 98% and lamp black 96% the incident radiation.

A body emits radiation when it is heated. The emitted radiation from the hot body is called thermal radiation. When a black body is heated, it emits more radiations per unit area than any other body at a particular temperature.

Since a perfectly black body absorbs all the radiation falling upon it , so it has absorptivity equal to 1. A perfectly black body has emissivity also equal to 1.

Fery's Black Body

A double walled hollow copper sphere is taken and coated with lamp black on its inner surface. A fine hole O acts as a very narrow opening. When the radiation enter through the narrow opening they suffer multiple reflections after falling on a pointed projection P which is made just in front of the hole.

After suffering multiple reflections, the radiation are completely absorbed. This body acts a a black body absorber.

Now , when this body is placed in a bath at a fixed temperature, the hear radiation come out of the hole, as shown in the figure below.

Fery's black body emitter

The hole acts as a black body radiator. It should be remembered that only the hole and not the walls of the body, acts as the black body radiator.

Wien's Black Body

Wien also constructed a black body in the form of cylinder. This black body is commonly used nowadays, as it is more convenient to use.

Wien's Black Body

It consists of a hollow cylindrical chamber C of brass or platinum having heating coil of thin platinjm wire HH wound over it. The cylinder is protected by co-axial porcelain tubes PP. When a suitable current is passed through the coil, the cylinder is heated to a desired temperature.

The inner surface of the chamber C is painted black, so that it reaches the equilibrium position in a small time. The radiation from the cylinder is limited by blackened concave duaphragms D and emerges out of the hole O. The temperature of the chamber is measured by the thermocouple T. The hole O will act as a black body radiator.

Distribution Of Energy in Black Body Spectrum

Lummer and Pringesheim investigated the distribution of energy among the radiation emitted by a black body at different temperatures.

The experimental arrangement of Lummer and Pringesheim is shown in above figure. The radiation from a black body pass through the slit S1 and falls on thr reflector M1, which passes through rock-salt or fluorspar prism ABC kept on a lotating(turn) table of spectrometer. The emergent light is focused by the concave reflector M2 on a line bolometer placed behind the slit S2.

The bolometer is connected to a sensitive galvanometer. The turn table is slowly rotated so that the different parts of the radiation spectrum successively falls on thr bolometer and the corresponding deflection in galvanometer are recorded.

The intensity of each line is proportional to the deflection in the galvanometer. Then the curve is drawn between intensity and the wavelength(λ).

Then the body is heated to different temperatures and the curves are drawn for various temperatures. The curves obtained are -

Results. From the above curves, we can conclude that

1). At a given temperature the energy is not uniformly distributed in the radiation spectrum of a black body.

2). At a given temperature, the intensity of heat radiation increase in wavelength and at a particular wavelength its value is maximum. With further increase in wavelength, the intensity of heat radiation decreases.

3). An increase in temperature causes an increase in the emission for all wavelengths.

4) An increase in temperature causes a decrease in

λ(m), where λ(m) is the wavelength fot which the energy is maximum, such that

λ(m)T = constant

Explanation: By Classical Laws

There were two main explanations given. One is Wien's Law and the other one is Rayleigh-Jean's Law.

Wien's Law. The success of Wien's law is that it hold good only in the region of shorter wavelengths at lower temperatures. It does not hold good at longer wavelengths and higher temperatures.

Rayleigh-Jean's Law. The problem in this explanation is that the energy density will continuously increase with increase in frequency and approaches infinity (as shown in above graph).This is contradiction to observed results.

It is clear that, the energy distribution of a black-body

could not be explained on the basis of classical concept.

Quantum Explanation: Planck's Radiation Law

Max Planck

The failure of classical mechanics led Planck(in1900) to the discovery that radiation is emitted in quanta.

For Planck's Quantum Theory visit- Planck's Quantum Theory

Planck's modified Rayleigh-Jean's Law with his assumption that a black body is associated with large mumber of harmonic oscillators.

Based on this he derived an equation, as shown below:

Above equation is called Planck's Radiation Law. This law can explain all the experimental results for black body radiation.

The failure of classical mechanics was a key in the discovery of quantum mechanics by Max Planck.

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Quantum Mechanics #2- Planck's Quantum Theory

2020-07-01T15:47:00.002+05:30

INTRODUCTION

The failure of classical mechanics led Planck(in 1900) to the discovery that radiation is emitted in quanta whose energy is E=hν. In 1901, Planck was able to derive an empirical formula to explain the experimentally observed distribution of energy in the spectrum of a black body, on the basis of his revolutionary hypothesis known as quantum theory of heat radiation. This was the Origin of quantum Mechanics(click here)

Max Karl Ernst Ludwig Planck

According to this theory, the energy distribution is given by

This realtion agrees (and hence completely fit) with the experimental curves obtained. This formula of distribution of energy with wavelengths, on the basis of his quantum concept, was deduced using following assumptions, which may be called as Planck's quantum postulates.

PLANCK'S QUANTUM POSTULATES

1) A black body contains the atomic oscillators capable of vibrating with all possible frequencies. An oscillator of frequency ν can not have any energy but only discrete values of energy given by

E(n) = nhν ----(1)

where n= 0,1,2,3,4,.....

ν= oscillator frequency

h= Planck's constant

2) The atomic oscillator can not emit or absorb energy continuously. But it can emit or absorbs the energy in discrete units of energy, a tiny packet called 'quanta'. The quantum energy is due to jump of the oscillator from one energy level to the other and is given by

ΔE(n) = Δn(hν).
where Δn = difference of levels. For example, if the transition energy levels are n=8 and n=5, then
ΔE= E(8) - E(5) = 8hν - 5hν = 3hν

EXPLANATION

From the first assumption, given by equation (1) it is clear that each quantum of energy depends upon the frequency of radiation(ν) and is not divisible. Moreover, the exchange of energy between matter and radiation is in terms of integral multiple of hν. Therefore, the total energy exchanged may be 0, hν, 2hν, 3hν,......., nhν in discrete manner.

However, according to classical theory, the atom can possess any amount of energy from zero to infinity, hence exchange of energies are continuous i.e. from 0 to infinity which is inadequate to quantum concept.

According to Planck's theory, the atom is associated with the allowed energy states called quantum states and its energies are quantized.

The matter can emit or absorb the radiation i.e. quantum of energy by jumping from one energy state to other.

The oscillator emits energy only when it passes from a higher energy state to a lower energy state and absorbs energy when it goes from a lower energy state to a higer energy state. No emission or absorption of energy takes place when the oscillator is in a given state.

The smallest amount of energy which can be emitted or absorbed by the oscillator is hν.

In other words a radiation of frequency ν is emitted as quantum of energy E= hν where h is Planck's universal constant having a value equal to
6.62607015 × 10−34 joule second. The quantum is the basic unit of energy and cannot be subdivided. It is the quantum of energy and Einstein named it as a photon.

What is a photon? Photon is a particle of zero rest mass. Its dynamical variables are energy(hν) and momentum(h/λ). A photon has zero charge and spin equal to one quantum unit. Two charges or two magnetic poles exert a force on each other due to exchange of photons. A photon interacts with all the charged particles and can interact with some neutral particles.

A photon behaves as a small elastic sphere for all purposes. When a photon collides with an electron both momentum and energy are conserved as in the case of collision between two elastic spheres. This fact has been verified by Compton in his experiments on X-Rays(click here to know more)

For derivation of Planck's energy distribution formula visit-Planck radiation law.

This theory has been able to explain a number of phenomena which could not be explained according to the classical conception of radiation. For example, it gives a very satisfactory explanation of variation of specific heat of solids with temperature, Photo-electric Effect, Compton Effect etc. and has been successfully applied by Bohr in the theory of the hydrogen spectrum.

NEXT BLOG - Black body radiation.

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Quantum Mechanics #1- The Origin

2020-06-28T20:13:00.001+05:30

INTRODUCTION

What is Quantum mechanics?| Quantum Mechanics is the theory which attempts to explain the behaviour of matter and interaction with energy on the scale of atoms and atomic particles i.e., particles of the size of the order of 1/10¹⁰ metres.

Phenomena such as motion of mechanical objects involving distances larger than about 1/10⁶ metres can be explained satisfactorily by laws of classical theoretical physics which is based on the following basic laws :

1) Newton's Laws of motion,

2) Newton's inverse square law of gravitational attraction between two bodies,

3) Coulomb's inverse square law of attraction or repulsion between two electrically charged bodies,

4) The law of force on a moving charge in a magnetic field, i.e., the Lorentz force.

However, certain phenomena such as spectral distribution of energy in blackbody radiation, Photoelectric effect etc; and phenomena involving distances of the order of 1/10¹⁰ metres could not be explained by classical mechanics.

These limitations or failure of classical mechanics led to the development of quantum mechanics, the major revolution in Physics. The above limitation i.e., the energy distribution in the spectrum of a black body was successfully explained by Max planck in 1900 who proposed the revolutionary concept of quantum hypothesis. This was origin of the Quantum theory.

FAILURE OF CLASSICAL MECHANICS

There are many important experimental results which could not be explained on the basis of classical mechanics. Some of them are:

1) Stability of atoms and molecules. According to classical mechanics, the atom is stable. When electron revolves around the nucleus in a circular orbit, it has acceleration. We know that any accelerating charge emits the radiation.

The electron during its motion around the nucleus should have limited energy continuously and as a result the radius of revolution would have reduced continuously and ultimately the electron would have much into the nucleus. But this is against the stability of the atom.

In other words, the classical theory could not explain the stability of the atoms and molecules. The classical theory, has no explanation to the behaviour in the region of atomic dimensions.

2) The atomic spectra. Characteristic frequency emission or absorption by the elements.

3) The regularities among the spectral lines.

4) The specific heat of solids at low temperatures.

5) The Black Body radiation. The quantity and quality of the emitted radiation depends only on the temperature of the body and not on the nature of the material of the body. Moreover, the radiation emitted at a particular temperature consists of wide range of frequencies with different intensities.

6) The photoelectric-effect(click here) . The photo-electric emission and scattering of free electrons.

7) The emission of X-Rays(click here for Details), radioactivity and other nuclear activities.

CONCLUSION

The faliure of classical mechanics (which includes Newton's laws of motion, inverse law of gravitational attraction between two bodies, coulomb's inverse law of attraction or repulsion between two charged bodies, law of Lorentz force etc.) led to the development of Quantum Mechanics. To overcome, this discrepancy, German scientists Max Plank in 1900, proposed his new theory, now known as old quantum theory. This theory was applied to many crucial problems of microscopic dimensions and got the satisfactory solution.

Many distinguished physicists Albet Einstein, Neils Bohr, Louis de Broglie, Max Born, Paul Dirac, Werner Heisenberg, Wolfgang Pauli Erwin Schrödinger, Richard Feynman, Compton contributed to the development of Quantum mechanics.

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Production , Properties and Uses Of X-Rays

2020-06-25T21:47:00.002+05:30

X-Rays

Introduction

X rays discovered by William Roentgen in 1895 honoured by Nobel Prize in 1901. Basically are high energy photons (1- 100Kev) or electro-magnetic radiation, having a very short wavelength of the order of 1A⁰. Typical x-ray wavelength lies between the range of

1 A⁰ and 100A⁰. Accordingly they are classified into hard X-rays and soft X-rays.

Uses of X rays are multifold. Usually they are used in surgery, radiotherapy, engineering, industry, detective department, scientific research and so many other areas. Hence, the name X-rays is popularly known even to a common man.

Production Of X-RAYS

The modern X-ray tube was designed by Dr.Coolidge and is known after his name as coolidge tube. It is widely used for commercial and medical purposes.

Construction and working of Coolidge Tube

The construction of coolidge tube to produce X-rays is schematically shown below.

The cathode C consists of a tungsten filament heated by passing a current through it. It then emits a large number of electrons called thermions which travel towards the anti-cathode A, also called the target with a velocity depending upon the potential difference applied between A and C.

By surrounding the cathode with a molybdenum shield M, maintained at a negative potential with respect to the filament circuit, the rays can be focused to a fine spot on A, as without the shield the electrons would fly away in all directions. The tube is completely exhausted to the no-discharge stage, so that no ionisatikn can take place.

A current of 30 mA at a voltage of 100,000 volts is used; the high voltage is obtained from a step-up transformer. The tube acts as its own rectifire, unless the anode attains such a high temperature that it begins to emit electrons. In large plants, the current from the transformers is rectified by thermoionic rectifires and high voltage capacitors.

The anti-cathode in a Coolidge tube must have the following characteristics:

High atomic weight, to produce hard X rays.
High melting point to withstand the high temperature developed, as most of the energy of the impinging electron is converted into heat.
High thermal conductivity, to get rid of the heat produced.
Low vapour pressure at high temperatures.

Tantalum, platinum and tungsten fulfil these requirements, but tungsten is the best and is widely used. In actual practice, a heavy block of this metal is mounted on a thick copper rod and the face is sloped at about 45⁰ to the electron beam. The heat generated by the impact of electrons is removed by special devices. In some cases, the anode is water-cooled.

Control Of Intensity and Quality

The higher the potential difference between the cathode and the anti-cathode, the greater is the velocity of electrons emitted. The electrons possessing greater energy produce more penetrating Express. The quality of X-rays, therefore, where is with the difference of potential between the cathode and the anti-cathode.
The number of electrons given out by the filament is proportional to its temperature, which can be adjusted by wearing the current in the filament circuit. The intensity of X-rays, therefore, where is with the filament current. Thus separate control of quality and intensity can be achieved.

Properties of X rays

They are not deflected by electric and magnetic fields. This property distinguishes them from the cathode rays and indicates that the X-rays do not consist of charged particles.
They are highly penetrating and can pass through many solids, which are Opaque to ordinary light, example, wood, flesh etc. The transparency depends upon the the density of the material. The density of substance the less transparent it is to X-rays. A sheet of lead 1 millimetre thick, absorbs these rays, where as the rays can pass through the same thickness of aluminium.
The penetrating power of X-Rays depends upon: (a) The applied potential difference (b) The atomic weight of the material of the anti-cathode. The greater the potential difference and higher the atomic weight, the most penetrating are the X-rays produced. The X-rays having high penetrating power are known as hard X rays and those with low penetrating power are known as soft X-rays.
They cause fluorescence in many substances, example, barium platino-cyanide, cadmium tungstate, zinc sulphide, etc.
They produce of photochemical reaction and effect of photographic plate and they are even more effective than light.
They ionise a gas and also eject electrons from certain metals on which they fall(For Photo electric effect click here)
They are propagated in straight lines with the velocity of light.
Like light, X-Rays consist of Electromagnetic waves of very short wavelength and show reflection, refraction, interference, diffraction and polarisation in a similar way. These effects can be demonstrated by suitable arrangements.
When X-Rays are incident on matter, they gave rise to a complex phenomenon known as secondary radiations, which consist of three different types of rays.
They have a destructive effect on living tissues. Exposure of human body to X-Rays causes reddening of the skin and surface sores.

Types of Rays after the incidence of X-rays on matter

SCATTERED X-Rays: They are practically of the same nature and wavelength as the original or the primary rays. Their properties do not depend upon the nature of the scattering substance.
Corpuscular Rays: They consist of fast moving electrons produced by the photoelectric process. Their properties are independent of nature of the scattering substance, but depends upon the quality of primary X-rays.
Characteristic X-Rays: The wavelength of the characteristic X-Rays is equal to or less than the primary X-Rays, but it is characteristic of the substance upon which the primary X-Rays are incident and is independent of the wavelength of the primary X-Rays.

Uses of X-Rays

1) Surgery - X-Rays can pass through flesh and not through bones. They are of extensive use in Surgery to detect fractures, foreign bodies diseased organs etc. The patient is made to stand between an X-Ray source and a fluorescent screen.

The luminosity of the screen depends upon the stopping power of different parts of the body. Thus a depe shadow of the bones and a light shadow of the flesh will be obtained. It is called radiograph. If the various parts do not show sufficient contrast, artificial means are adopted e.g., in radiographing the portions of the alimentary tract, the patient is fed with a meal of barium sulphate.

2) Radiotherapy - X-Rays kill the diseased tissues of the body. Hence they are used to cure intractable skin diseases, malignant tumours etc. If the affected part is superficial soft rays are applied and for deep seated organs hard rays are used.

3) Engineering - Because of their high penetrating power, they are used to investigate the structure of metals, testing of castings for flaws and gas pockets, causes of weakness in structures and to detect cracks and blow holes in metal plates.

The method has the advantage that unlike some mechanical tests, it does not destroy the piece. Moreover, it often suggests alterations i mould design and heat treatment to avoid a recurrence of the detected faults.

4) Industry - X-Rays are employed in industries to detect defects in motor tyres, golf and tennis balls, wood and wireless valves. They are used in testing the uniformity of insulating material and for detecting the presence of pearls im oysters.

5) Detective Department - X-Rays are used at customs post for detection of Contraband goods like explosives, opium, etc. Concealed in leather or wooden cases, in examining of parcels without opening them and in the detection of precious metals like gold and silver in the body of smugglers. They are also used in distinguishing real diamonds, gems and pearls from artificial ones, pure ghee from vegetable ghee and real documents from forged ones.

6) Scientific Research - X-Rays have even been employed to investigate the structure of crystals, structure and properties of atoms and arrangement of atoms and molecules in matter.

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BSc first year, Statistical Physics- Basics

2020-06-16T17:24:00.000+05:30

STATISTICAL MECHANICS: A BRIEF HISTORY

Statistical mechanics was initiated in 1870 with the work of Austrian physicist Ludwig Boltzmann, much of which was collectively published in Boltzmann's 1896 lectures on gas theory. Statistical mechanics provided a molecular level interpretation of microscopic thermodynamic quantities such as eork, heat, free energy and entropy.

The study of statistical physics is mainly classified into categories. These are:

Classical statistics or Maxwell-Boltzmann statistics.
Quantum statistics or Bose -Einstein and Fermi-Dirac statistics.

STATISTICAL BASIS

Statistics is that branch of science which deals with the collection, classification and tabulation of numerical data as the basis of explanation, description and comparison of various phenomenon. When statistical concepts are applied to physics, the new branch that emerges is called statistical physics.

Statistical physics deals with macroscopic systems. i.e. the system consisting of a large number of individual particles such as atoms, molecules etc. It is not concerned with the behaviour of the individual particles of the system but takes into consideration the average or most probable properties of the system. As a system is a collection of a large number of particles of one type, we may have a system composed of atoms, molecules, protons, neutrons or electrons.

In certain cases, particularly where the system contains a large number of particles, ordinary laws of mechanics could not be used, as it is impossible to follow the motion of each particle. It is impossible to apply the ordinary laws of mechanics to a physical system containing large number of particles, particularly that of electrons. Such problems are however, successfully solved by statistical mechanics.

There are three statistics depending upon three different kinds of particles:

Maxwell-Boltzmann statistics: This is applicable to the identical, distinguishable particles of any spin stop the molecules of a gas are the particles of this kind.
Bose-Einstein statistics: This is applicable to the identical, indistinguishable particles of zero or integral spin. These particles are called bosons. Examples of bosons are Helium atoms at low temperature and the photons.
Fermi-Dirac statistics: This is applicable to the identical, indistinguishable particles of half integral spin. These particles obey Pauli exclusion principle and are called fermions. Examples of fermions are electrons, protons, neutrons etc.

SOME BASIC TERMS OF STATISTICAL MECHANICS

(1) System

In statistical mechanics, the term system is used to represent system (or collection) of large number of small particles( such as atoms or molecules) with one or more than one independent part.

For example, if we have to distribute some particles in the boxes or to distribute some energy(or momentum) in some states, then the particles and the boxes or the energy(or momentum) and the states together, is called the system.

(2) Statistical Ensembles

Statistical mechanics, Gibbs introduce the concept of ensemble. An ensemble is defined as a collection of large number of macroscopically identical but essentially independent systems. Macroscopically identical means that each of the systems constituting an ensemble satisfies the same macroscopic conditions e.g., volume, energy, pressure, temperature, total number of particles etc. Independent systems means that the system constituting an ensemble are mutually non- interacting.

Kinds of Ensembles - According to Gibbs, there are three standard ensembles to which real experiments could be approximated. These are:
1. The Microcanonical ensemble
2. The canonical Ensemble, and
3. The Grand-canonical ensemble.

Microcanonical Ensemble- The micocanonical ensemble is a collection of systems having the same energy E, volume V and number of particles N.

Microcanonical Ensemble

Canonical Ensemble- The canonical Ensemble is a collection of systems( independent) having the same temperature T, volume V, and number of identical particles N.

In canonical ensemble, systems can exchange energy but not the particles. There is an impermeable but conducting wall between systems.

Canonical Ensemble

Grand-Canonical Ensemble- The Grand-canonical ensemble is a collection of systems having the same temperature T, volume V and chemical potential µ.
The walls are permeable and conducting. So both energy and particles can be exchanged.

Grand-canonical Ensemble

(3) Macrostate

In statistical mechanics, the macrostate of a system is the state which can be experimentally observed. Consider 4 different particles. Let them be a,b,c and d.
We have two similar open boxes(compartments).

The possible ways by which 4 particles can be distributed in two compartments are shown in the table.

Distribution of 4 particles in 2 compartments

Thus, there are 5 different distributions (0,4), (1,3), (2,2), (3,1), and (4,0).
Each compartment wise distribution of a system of particles is known as a Macrostate.
In above example there are 5 macrostates - (0,4), (1,3), (2,2), (3,1), and (4,0).

In above example, we came to know that there are 5 macrostates for distribution of 4 particles in 2 boxes.
So if there are 'n' particles, which are to be distributed in 2 boxes, then there will be 'n+1' macrostates.

(4) Microstate

Let us take one Macrostate (1,3) from the above exmaple. It means that there is 1 particles in box 1 and 3 particles in box 2. The four particles are named as a,b,c and d.

Now the one which is in the box 1 can be a, b, c or d.

And the three particles which are in box 2 can be abc, bcd, abd and acd.

Hence microstate of the system is the state in which we consider the arrangement of the individual particles of the system. It can not be experimentally observed.

Possible arrangements for the above exmaple is given below in tables-

Table for number of microstates

Now from above table we can say that the total number of microstate for the system that we have taken in the above exmaple is 1+4+6+4+1=16

If there are 'n' particles , distributed in 2 boxes, then the number of microstates will be 2n.

(5) Thermodynamic Probability

The number of microstates corresponding to any given macrostate is called its thermodynamic Probability. It is denoted by W(or Ω).

Let us take the above given exmaple. We have to calculate thermodynamic probability for the Macrostate (1,3). In the above table we can observe that the number of microstates corresponding to the macrostate (1,3) is 4.

Hence the thermodynamic probability for the macrostate (1,3) is 4.

Therefore, W(1,3) = 4

If the total number of particles is 'n' and the macrostate is (r, n-r), then we will use combination to find the number of arrangements. i.e. ,

Number of meaningful arrangements

Now, the number of meaningful arrangements is equal to the number of microstates and the number of microstates is equal to thermodynamic Probability.

Therefore the thermodynamic Probability for the macrostate (r, n-r) can be given by

Thermodynamic Probability for Macrostate (r, n-r)

(6) Constraints

A set of conditions or restrictions that must be obeyed by a system are known as constraints.

Some typical constraints are-

1) Constant number of particles

2) Constant total energy

Which can be written in mathematical form as:

Constraints

(7) Accessible and Non- accessible Microstates

The micro States which are allowed under the given restrictions or constraints of the system are called accessible microstates.

On the other hand, the microstates which do not obey the given constraints are not allowed and they are called the non-accessible microstates.

PRINCIPLE OF EQUAL A PRIORI PROBABILITY

Suppose we toss a coin, then the Probability of head and tail is equal. We can assume before the experiment that the probability of coming head is equal to that of coming a tail.

The principle of assuming equal probability for events which are equally likely to occur is known as the Principle of equal a priori probability.

The Probability of each of the accessible microstates corresponding to a given macrostate is same.

PROBABILITY OF A MICROSTATE

To find the probability of a microstate, we will use the principle of equal a priori probability. Hence probability of each accessible microstate is equal.

Probability of a microstate= 1/(total number of microstate)

Let it be p, therefore
p= 1/(total number of microstates)

PROBABILITY OF A MACROSTATE

The probability of a macrostate is defined as the ratio of (number of accessible microstates corresponding to the given macrostate) thermodynamic probability to the total number of microstates.

Let it be P, therefore

P = (thermodynamic probability)/(total number of microstates)

Or

P = W/p
where W is thermodynamic probability of that macrostate and p is the probability of a microstate.

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What is force? Types of forces in nature

2020-06-13T17:50:00.000+05:30

What is force?

Force is a word which we have all heard about. When you push or pull some object you exert a force on it. If you put a body you exert a force away from yourself; when you pull, you exert a force towards yourself. When you hold a heavy block in your hand you exert a large force; when you hold a light block, you exert a small force.

Can non living bodies exert a force? Yes non living bodies can also exert force. If we stand in a great Storm, we feel that the wind is exerting a force on us. When we suspend a heavy block from a rope, the rope holds the block just as a man can hold it in air.

When we comb our dry hair and bring the comb close to small pieces of paper, the pieces jump to the comb. The comb has attracted the paper pieces i.e. the comb has exerted force on the pieces.

Force is an interaction between two objects.The SI unit for measuring the force is called a newton. Approximately, it is the force needed to hold a body of mass 102 gram near the Earth's surface.

Force is a vector quantity and if more than one forces act on a particle we can find the resultant force using the laws of vector addition.

Different forces in nature

In nature, we observe different kinds of forces such as muscular force, force of tension, force of reaction, force of friction, gravitational force, elastic force, wait electrostatic force, magnetic force, nuclear force etc. All these forces can be classified under the following four kinds:

Gravitational force
Electro magnetic force
Nuclear force
Weak force

1) Gravitational force

In universe, each particle attracts the other particle due to their masses. Moon revolves around Earth due to gravitation.

The force of attraction between two particles is directly proportional to the product of their masses and inversely proportional to the square of distance between them. The direction of this force is along the line joining the two masses. This force is called the gravitational force.

If two particles of masses m1 and m2 are kept at a a separation r, the gravitational force acting between them is given as:

Where G is the universal gravitational constant.

G = 6.67

\times 10^{-11} N \cdot m^{2}/ kg^ {2}

Some properties of this force are-

1) It is the weakest force.

2) It is always attractive in nature.

3) It is a long range force.

4) It does not depend on the medium between the particles.

2) Electromagnetic force

The electromagnetic force, also called the Lorentz force, explains how both moving and stationary charged particles interact. It's called the electromagnetic force because it includes the formerly distinct electric force and the magnetic force; magnetic forces and electric forces are really the same fundamental force. The electromagnetic force is one of the four fundamental forces.

The electric force acts between all charged particles, whether or not they're moving. The magnetic force acts between moving charged particles. This means that every charged particle gives off an electric field, whether or not it's moving. Moving charged particles (like those in electric current) give off magnetic fields.

Some properties of this force are-

1) The force can be attractive as well as repulsive.

2) The force is a long range force.

3) The force depends on the medium between the charged particles. The force decreases in presence of dielectric medium between the particles.

Some examples of Electromagnetic forces are - Attraction between two charges, magnetic forces and also friction. Friction is also an Electromagnetic forces.
To know about friction read this article-CLICK HERE TO KNOW MORE ABOUT FRICTION

3) Nuclear Force

Each atom has a nucleus of diameter nearly 1/10¹⁵m to 1/10¹⁴ m at its centre inside which neutrons and protons are bonded Together by the attractive forces. Although neutron is an uncharged particle and proton is a charged particle, therefore there must be no force between neutron-neutron or proton-neutron and an electric force of repulsion between proton and proton, but there is an equal attractive force between Proton -proton, proton -neutron and neutron-neutron due to which they remain bounded together inside the nucleus. This force is called the nuclear force.

Some properties of this force are-

1) This force is charge-independent.

2) It is a short range force.

3) It is a saturated force that is, the force needed to isolate a proton neutron from the nucleus is same whatever may be the number of proton and neutron inside it.

4) Weak force

Yet another kind of forces is encountered when reactions involving protons, electrons and neutrons take place. Neutron can change itself into a Proton and simultaneously emit an electron and a particle called Antinutrino. This is called beta minus decay.

Never think that a neutrino is made up of a Proton, an electron and an Antineutrino. A Proton can also change into neutron and simultaneously emi a positron( and a neutrino). This is called beta plus decay.

The forces responsible for the changes are different from gravitational, electromagnetic or nuclear forces. Such forces are called weak forces. The range of weak forces is very small, in fact but smaller than the size of a proton or a neutron. Thus, its effect is experienced inside such particles only.

Relative magnitude of all four fundamental forces

In increasing order of magnitude, these forces are: gravitational, week electromagnetic and nuclear force.

They are in ratio given as: 1: 10²⁶: 10³⁶: 10³⁹

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Latest Physics news 2020!, Black Holes, Quantum fifth state of matter, Dark Energy

2020-06-12T01:09:00.001+05:30

Here are some of the latest researches , experiments and observations in the field of Physics. Enjoy!

1) Quantum 'fifth state of matter' observed in space for first time!

JUNE 11, 2020|A team of NASA scientists unveiled the first results from Bose-Einstein condensates experiments aboard the International Space Station, where particles can be manipulated free from the constraints of gravity.

Scientists have observed the fifth state of matter in space for the first time, offering unprecedented insight that could help solve some of the quantum universe's most intractable conundrums.

2) Black holes? They are like a hologram!

JUNE 4, 2020|According to new research by SISSA(International School of Advanced Studies), ICTP and INFN, black holes could be like holograms, in which all the information to produce a three-dimensional image is encoded in a two-dimensional surface.

What researchers have done is apply the theory of the holographic principle to black holes.

As affirmed by quantum theories, black holes could be incredibly complex, and concentrate an enormous amount of information in two dimensions, like the largest hard disks that exist in nature. This idea aligns with Einstein's theory of relativity, which describes black holes as three dimensional, simple, spherical and smooth, as depicted in the first-ever image of a black hole that circulated in 2019. In short, black holes appear to be three dimensional, just like holograms. The study, which unites two discordant theories, has recently been published in Physical Review X.

The holographic principle applied to black holes

To study black holes, the two authors of the new study, Francesco Benini (SISSA Professor, ICTP scientific consultant and INFN researcher) and Paolo Milan (SISSA and INFN researcher), used a 30-year-old idea called the holographic principle.

The researchers write, "This revolutionary and somewhat counterintuitive principle proposes that the behavior of gravity in a given region of space can alternatively be described in terms of a different system, which lives only along the edge of that region and therefore in a one less dimension. And, more importantly, in this alternative description (called holographic), gravity does not appear explicitly. In other words, the holographic principle allows us to describe gravity using a language that does not contain gravity, thus avoiding friction with quantum mechanics."

3) Titan is migrating away from Saturn 100 times faster than previously predicted!

JUNE 8,2020| By Earthly standards, Saturn's moon Titan is a strange place. Larger than the planet Mercury, Titan is swaddled in a thick atmosphere (it is the only moon in the solar system to have one) and covered in rivers and seas of liquid hydrocarbons like methane and ethane. Beneath these is a thick crust of water ice, and beneath that may be a liquid water ocean that could potentially harbor life.

Now, decades of measurements and calculations have revealed that Titan's orbit around Saturn is expanding—meaning, the moon is getting farther and farther away from the planet—at a rate about 100 times faster than expected. The research suggests that Titan was born much closer to Saturn and migrated out to its current distance of 1.2 million kilometers (about 746,000 miles) over 4.5 billion years.

The findings are described in a paper that appears in the journal Nature Astronomy on June 8.

4) Researchers discover a new type of matter inside neutron stars!

JUNE 2, 2020| A Finnish research group has found strong evidence for the presence of exotic quark matter inside the cores of the largest neutron stars in existence. They reached this conclusion by combining recent results from theoretical particle and nuclear physics to measurements of gravitational waves from neutron star collisions.

All normal matter surrounding us is composed of atoms, whose dense nuclei, comprising protons and neutrons, are surrounded by negatively charged electrons. However, inside neutron stars, atomic matter is known to collapse into immensely dense nuclear matter in which the neutrons and protons are packed together so tightly that the entire star can be considered one single enormous nucleus.

Up until now, it has remained unclear whether nuclear matter in the cores of the most massive neutron stars collapses into an even more exotic state called quark matter, in which the nuclei themselves no longer exist. Researchers from the University of Helsinki now claim that the answer to this question is yes. The new results were published in Nature Physics.

"Confirming the existence of quark cores inside neutron stars has been one of the most important goals of neutron star physics ever since this possibility was first entertained roughly 40 years ago," says Associate Professor Aleksi Vuorinen from the University of Helsinki's Department of Physics.

5) New test of dark energy and expansion from cosmic structures!

JUNE 3, 2020| A new paper has shown how large structures in the distribution of galaxies in the Universe provide the most precise tests of dark energy and cosmic expansion yet.

The study uses a new method based on a combination of cosmic voids—large expanding bubbles of space containing very few galaxies—and the faint imprint of sound waves in the very early Universe, known as baryon acoustic oscillations (BAO), that can be seen in the distribution of galaxies. This provides a precise ruler to measure the direct effects of dark energydriving the accelerated expansion of the Universe.

This new method gives much more precise results than the technique based on the observation of exploding massive stars, or supernovae, which has long been the standard method for measuring the direct effects of dark energy.

The research was led by the University of Portsmouth, and is published in Physical Review Letters.

The study makes use of data from over a million galaxies and quasars gathered over more than a decade of operations by the Sloan Digital Sky Survey.

The results confirm the model of a cosmological constant dark energy and spatially flat Universe to unprecedented accuracy, and strongly disfavour recent suggestions of positive spatial curvature inferred from measurements of the cosmic microwave background (CMB) by the Planck satellite.

Source - PHYS ORG

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What is surface tension?

2020-05-28T20:43:00.001+05:30

Liquids have a property that their free surface behaves like a stretched elastic membrane and has a natural tendency to contract, i.e , the free surface of a liquid tends to occupy a minimum surface area.
This property of liquids is called as surface tension.

Some experiments and examples related to Surface Tension-

Let us consider a frame. There is a closed loop thread attached with the frame loosely, so that it can change its shape. Now dip the frame in soap solution. Break the film created inside the loop. After breaking the film, it is observed that the thread occupies a shape of a circle.

REASON- The surface tension tends to make the surface area of the film as minimum as possible. And for a given length of thread, the surface area of circle is maximum, which eventually decreases the surface area of the film to minimum.
Brush in water- When we put brush in water, the brush hair can be seen separately in water. But when we take out the brush out of the water, the water film between the hair gets contracted due to surface tension.

So when we take out the brush out of water , the hair of the brush stick together.
The shape of rain drops is spherical.
Insects can float on water surface.

Definition- The surface tension of a liquid is defined as the force per unit length in the plane of the liquid surface, acting normally on either side of an imaginary line drawn in that surface.

T= F/L

where T is surface tension F is force and L is length of the imaginary line.

Surface Energy- When the molecules from the interior of the liquid rise to the surface the surface area is increased.
Now this work is stored in the molecules in the form of potential energy.
The potential energy stored per unit area of the surface is called the surface energy.

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Photoelectric Effect

2020-04-30T00:41:00.001+05:30

Basic Definition

PHOTOELECTRIC EFFECT| Photoelectric effect is the emission of electrons or other free carriers when electromagnetic radiation, like light, hits a material. Electrons emitted in this manner can be called photoelectrons. This phenomenon is commonly studied in electronicphysics and in fields of chemistry such as quantum chemistry and electrochemistry.

A Brief History -

This effect was observed earlier before Albert Einstein gave the theory for it.

In 1839, Alexandre Edmond Becquerel discovered the photovoltaic effect while studying the effect of light on electrolytic cells. Though not equivalent to photoelectric effect.

The first of these phenomena was discovered by Heinrich Hertz and Wilhelm Hallwachsin 1887.

In 1887, Heinrich Hertz observed the photoelectric effect and the production and reception of electromagnetic waves.

The second was first announced by Philipp Lenard in 1900.

Albert Einstein's 1905 research paper

ALBERT EINSTEIN| In 1905, Albert Einstein solved this apparent paradox by describing light as composed of discrete quanta, now called photons, rather than continuous waves. Based upon Max Planck's theory of black-body radiation, Einstein theorized that the energy in each quantum of light was equal to the frequency multiplied by a constant, later called Planck's constant. A photon above a threshold frequency has the required energy to eject a single electron, creating the observed effect.

This discovery led to the quantum revolution in physics and earned Einstein the Nobel Prize in Physics in 1921.

Particle Nature of Light

The photoelectric effect supports the particle modelbecause the effect depends on the frequency and not the intensity of the light. This means that particles of certain energy were creating the effect, not the absorption of a wave.

Applications of photoelectric effect

Image sensors

Video camera tubes in the early days of televisionused the photoelectric effect, for example, Philo Farnsworth's "Image dissector" used a screen charged by the photoelectric effect to transform an optical image into a scanned electronic signal.

Some other applications are -

Photomultipliers, Night vision devices, Spacecrafts, Gold-Leaf Electroscope , Photoelectron spectroscopy, moon dust etc.

KNOW PHYSICS

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

Planck Blackbody Radiation Formula

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Planck's Radiation Law

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Action of the Number Operator

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Determination of the Coefficients

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Normalized Coherent State

Checking the Eigenvalue Equation

Probability Distribution

Classical Electromagnetic Wave

Quantum Mechanical Analog

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

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Some important terms-

Carnot's Cycle

1) Isothermal expansion-

Efficiency of heat engine

Friction in solids

For an example -

Types of friction

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Huygens wave model

The struggle: Newton or Huygens

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