Category: Physics Notes PPT

The concept of Angular Displacement teaches the students the concept of Displacement in the way it is discussed here later. It is specifically related to the Physics  subject. has provided the students with information on this topic to help them understand the same easily. The site also makes available many more study resources to let the students practice, learn, and enhance their level of preparation. The resources available on ’s website include revision notes, textbooks from various boards and institutes, worksheets, sample papers, previous year’s question papers, and others to facilitate improved exam preparation.

What Is Angular Displacement?

The motion of the body along a circular path is known as Rotational Motion. The Displacement done through such a type of motion is different from the Displacement done on linear motion; it is usually in the form of an angle, and hence it is known as Angular Displacement. Below we discuss Angular Displacement along with the formula, let us define it with the help of examples.

While moving in a circular path, the angle made by the body is known as Angular Displacement. Before discussing further on the topic, we have to understand what rotational motion is. The motion ceases to become a particle when a rigid body is rotating about its axis. Due to the motion in the circular path, change in the acceleration and velocity can happen at any time. Rotational motion is defined as the motion of the rigid bodies that will remain constant throughout the rotation over a fixed axis.

Angular Displacement Definition 

To define Angular Displacement, let’s suppose a body is moving in a circular motion, the angle made by a body from its point of rest at any point in rotational motion is known as Angular Displacement.The shortest angle between the initial and the final position for an object in a circular motion around a fixed point is known as the Angular Displacement; it is considered a Vector quantity.

 

Unit of Angular Displacement

The unit of Angular Displacement is Radian or Degrees. 360o is equal to two pi radians. Meter is the SI unit for Displacement. Since Angular Displacement involves the curvilinear motion, the SI unit for Angular Displacement is Degrees or Radian.

 

Angular Displacement Formula

The Formula of Angular Displacement

For a point the Angular Displacement is as follows:

Angular Displacement = θf-θi The Displacement will have both magnitudes as well as the direction. The circular arrow pointing from the initial position to the final position will indicate the direction. It can either be clockwise or anticlockwise in direction.

It can be measured by using a simple formula. The formula is:

θ=s/r, where,

θ is Angular Displacement,

s is the distance traveled by the body, and

r is the radius of the circle along which it is moving.

Simplistically, the distance traveled by an object around the circumference of a circle divided by its radius will be its Displacement.

 

Derivation

The Angular Displacement can be calculated by the below formula when the value of initial velocity, acceleration of the object, and time are shared.

[theta = wt + 1/2alpha t^{2}]

Where,

θ- Angular Displacement of the object

t- Time

α- Angular acceleration

Now, the formula for Angular Linear is

In Rotational, the kinetic equation is

[omega = omega 0 + alpha t]

[triangleomega = omega_{0}t+1/2alpha t^{2}],

[omega^{2} = omega_0^2 + 2alphatheta],

 

In translational, the kinetic equation is

v=u+at

or [s = ut+1/2at^{2}]

v2 = vo2 + 2ax

 

Where,

ω- Initial Angular velocity

 

Considering an object having a linear motion with initial acceleration a and velocity u, when time t and the final velocity of the object is with the total Displacement s then,  

a = dv​/dt

The change in velocity

 The rate which can be written as 

dv = a dt

Integrating both the sides, we get,

∫uv​dv=a∫0t​dt

v – u = at

Also,

a=dv/dt​

a=dxdv​/dtdx​

As we know v=dx/dt​, we can write,

a=vdv/dx​

v dv=a dx

The equation we get after integrating both sides

∫uv​vdv=a∫0s​dx

[V^{2}-u^{2} = 2as]

From the equation -1 into equation – 2 by substituting the value of u, we get

[V^{2}−(v−at)^{2}=2as]

[2vat–a^{2}t^{2}=2as]

By dividing the equation of both sides by 2a, we have

[s=vt–1/2at^{2}]

And at last, the value of v being substituted by u, we will get.

[s=ut+1/2​at^{2}]

 

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Any fluid applies equal pressure in every direction. This pressure is the result of the weight of the fluid. When an object is partially or completely submerged in a fluid, it exerts an upward force on the object. This upward force is called the buoyant force. Due to the buoyant force, there is an apparent decrease in the weight of the object. The decreased weight is equal to the weight of the fluid, displaced by the object. This relationship was invented by Archimedes. From large ships to small boats, aircraft, submarines all of these operate according to the principle of buoyancy.  Archimedes’s principle is extremely important in the study of buoyancy in physics, this principle is explained very briefly in Chapter 10 called mechanical properties of fluids of Class 11 in the NCERT book, this chapter is prescribed by the Central Board of secondary education as it includes various important topics whose understanding is extremely important for higher studies.

The chapter mechanical properties of fluids mainly deal with the common physical properties of liquids and gases.

The study of fluid is extremely important as fluids cover a significant part of the earth and they are used in our day to day lives as an essential component. Every mammal’s body on the earth constitutes mostly water. All the processes that are responsible for the growth of living beings are mediated by fluids.

The topics related to Archimedes principle have taught us that volume of solids can be changed by stress. The volume of all the three forms that Is solid liquid or gas is dependent on the stress or pressure acting upon it. Fixed volume in physics means the volume under atmospheric pressure. Solids and liquids as compared to gases have much lower compressibility.

Fluids offer very little resistance to shear stress. The shear stress of fluids is about a million times smaller than the shear stress of solids.

Archimedes’s principle is also known as the physical law of buoyancy; it was discovered by the Asian Greek mathematician Archimedes who was a Greek philosopher, scientist and engineer. Eureka is a word popularised by Archimedes. He exclaimed Eureka when he realised he had invented the method of detecting if something is made of pure or impure gold. In the widespread tale, Archimedes didn’t use his principle he only used displaced water to measure the volume of the crown, an alternative approach is applied with the use of this principle – A scale has to be balanced after placing a crown on one side and pure gold on the other, submerge the scale in water, According to Archimedes principle, if the crown’s density differs from pure gold’s then the scale will get out of balance underwater.

Apparent Weight

The original weight of an object acts downwards through its centre of gravity. When an object is immersed in a fluid, an upward thrust namely buoyant force is exerted on the object. Due to this upward force, the resultant downward force decreases and the object feels lighter. If the object floats on the surface, it is effectively weightless. The apparent decrease in the weight is equal to the magnitude of the upward buoyant force.

()

The apparent weight of an object is given by the difference between the actual weight and the buoyant force.

Archimedes Principle Derivation

The principle is based on the buoyancy principle, which states that a gas or liquid can exert an upward force on any object, fully or partially immersed in it. The upward thrust is called the buoyant force. 

()

In the diagram above, a cylinder of height h and radius r is immersed vertically in a liquid such that its flat surfaces are at depths h₁ and h₂ with h₁ <  h₂. The liquid exerts a perpendicular thrust (pressure) at each point on the surface of the cylinder. Due to the axial symmetry, the net thrust on the curved surface is zero. The downward pressure on the upper flat surface is h₁ρg, where ρ is the density of the liquid and g is the gravitational acceleration. The upward pressure on the lower flat surface is h₂ρg. If the atmospheric pressure isP[_{atm}], the downward force on the upper surface,

F[_{1}] = [left ( P_{atm}+h_{1}rho g right )][pi] r[^{2}]

The upward force on the lower surface is,

F[_{2}] = [left ( P_{atm}+h_{2}rho g right )][pi] r[^{2}]

Since, h₁ <  h₂ the resultant force on the cylinder is upward and the magnitude is,

F[_{b}] = F[_{2}]-F[_{1}]

F[_{b}] = [left ( h_{2}-h_{1} right )][rho] g [pi] r[^{2}]

Since h = h₂ – h₁ is the height of the cylinder and V = πr²h is its volume, the upward thrust can be expressed as,

F[_{b}] = [rho] Vg

The right-hand side of this upthrust formula is nothing but the weight of liquid of equal volume V as the submerged object. The magnitude of the buoyant force is, however, equal to the apparent decrease in the weight of the object. Therefore,

Apparent decrease of the object’s weight = weight of the fluid displaced by the object

Law of Floating

Whether an immersed object will float or sink, depends on the magnitudes of the actual weight W₁ of the object and the buoyant force W₂ exerted by the fluid.

• W₁ > W₂:  The resultant force on the object is downwards, causing it to sink. When the density of the object is greater than that of the fluid, this condition arises.

• W₁ = W₂: When the densities of the object and the fluid are equal, the actual weight and the buoyant force become equal. The object can float at any depth in a fully submerged state.

• W₁ < W₂: The net force acts in the upward direction leading to a partially submerged condition of the object. The density of the object is less than the fluid in such cases.

Application of Archimedes Principle

• Using Archimedes law, the volume or density of any rigid body can be computed. The proportions of the constituent metals of an alloy can be easily calculated using this principle.

• Submarines operate using the Archimedes theory. It has a large ballast tank, which controls the depth of the marine. By adjusting the quantity of water in the ballast tank, the actual weight of the submarine is varied and thus the desired depth can be achieved.

• Ships are made hollow such that the effective density is less than the density of water. Due to the buoyant force having greater magnitude than the ship’s weight, the ship can float in a partially submerged state. Aircraft are made using the same concept.

• The densities of liquids are computed using hydrometers which work according to the Archimedes principle of buoyancy.

• Hot air balloons can float in the air because the density of hot air is less than the density of ambient cool air.

Key Concepts related to Archimedes Principle are as Follows-

• Pressur
  e

• Streamline flow

• Bernoulli’s principle

• Viscosity

• Reynolds number

• Surface tension

Did You know?

• Archimedes of Syracuse introduced the theory of buoyancy in his book On Floating Bodies (written in the Greek language) around 250 BC. This theory is considered to be the cornerstone in the study of hydrostatics.  

• It is reported that Archimedes called out “Eureka”, meaning “I have found (it)” when he finally comprehended how to detect if a crown is made of impure gold using the theory of buoyancy.

• A floating body does not have any apparent weight.

• Surface tension or capillarity effect is not incorporated with the Archimedes principle.

• A large lunar impact crater is named after Archimedes.

• A portrait of Archimedes is engraved on the prestigious “Fields Medal”. 

Sound is a form of energy which is generated by a vibrating body. It requires a medium for its propagation. The transmission medium could be gaseous, solid or liquid. The waves in which the direction of propagation of the wave is the same as the direction of vibration of the particles of the medium are known as longitudinal waves. The waves in which the direction of propagation of the wave is perpendicular to the direction of vibration of the particles of the medium are known as transverse waves. In order for a sound to be generated, a source is required. An example of a source of sound is a speaker in which the diaphragm of the speaker vibrates to generate sound.

Now let us see how a source of sound works. When a sound source vibrates, then the particles of the medium surrounding it vibrate. The vibrating particles move further away from the source of sound as the vibration of the medium due to the vibrating particles continues. The propagation of the vibrating particles away from the source takes place with the speed of the sound. This is how a sound wave is formed. The velocity, displacement and even the pressure of the medium vary in time at a distance. One thing which should be kept in mind is that the vibrating particles of the medium do not actually travel along with the sound waves. The vibration of those vibrating particles further passes on the vibrations and make other particles vibrate and the process continues. There are three phenomena that can take place during the propagation of the sound wave. Either the sound waves will be reflected or attenuated or else refracted by the medium. 

There are Three Factors Which Will Affect the Nature of the Propagation of Sound. These Factors are

The Relationship Between Density and Pressure: the relationship between density and pressure will affect the speed of sound in the medium. This relationship is further affected by temperature.

State of the Medium Through which Sound is Propagating: If the medium through which sound is propagating is moving, then the speed of sound will increase if the motion of the medium is in the same direction as that of propagation of sound and it will decrease if the motion of the medium is in the opposite direction of propagation of sound. 

Medium Viscosity: The rate at which the quality of sound will be attenuated is determined by the viscosity of the medium through which it is propagating.

 

Properties of a Sound Wave

There are some key features of a sound wave. They are frequency, wavelength, intensity, the pressure of the sound, amplitude, direction of propagation and speed of sound.

Underneath is the description of each characteristic of the sound wave. 

1. Frequency – The total number of waves that are produced in one sec is known as frequency. It is also defined as the total number of vibrations counted in one sec. The frequency is obtained when the velocity of the wave is divided by the wavelength of the wave. It is measured in hertz (Hz).

2. Wavelength – The distance between the adjacent same parts that is the distance between two consecutive troughs and crests of a sound wave is known as the wavelength. “Metre” is the SI unit of wavelength. The wavelength of the audible frequency range lies between 17mm to 17m.

3. Intensity – The power that is carried or produced by the sound wave per unit area is known as the intensity of the sound wave. The power carried is in the perpendicular direction of that area. The SI unit of the sound intensity is watt per metre square (W/ m^2). 

4. Pressure – The deviation of the local pressure from the equilibrium or the average atmospheric pressure is known as the pressure of the sound wave. It is also known as acoustic pressure. The SI unit of this pressure is Pascal (Pa).

5. Amplitude – The maximum distance travelled by the vibrating particles from their mean position when a sound wave is propagating through a particular medium is called the amplitude of a sound wave. It can also be described as the loudness of the sound wave. It is measured in decibels (dB).

6. The Direction of Propagation – The direction of propagation of the sound wave is decided by the direction of vibration of the vibrating particles. The sound wave shows to and fro motion. When sound travels, the vibrating particles form the regions of compression and rarefaction. The sound wave propagates as a longitudinal wave in the air or other mediums.

7. Speed – The distance travelled by the sound wave per unit time is known as the speed of the sound. In other words, the speed of the sound wave is equal to the product of the frequency and the wavelength. The SI unit of speed is metres per sec (m/ s). The speed of sound in the air is 343 metres per sec.

Basically, Based on the Information About the Frequency, Sound Can be Classified Into Two Categories. They are

Audible Sound: All the frequencies residing between the limit of 20Hz and 20KHz can be perceived by human beings. Therefore, these sound waves having frequencies within the range of 20Hz and 20KHz are known as audible sounds. But the frequency that we can hear is often dependent on several other factors like our environment. Frequent exposure to loud noises can affect the hearing capacity of an individual. The frequencies in the higher frequency range get difficult to perceive. This often happens with old age.

Inaudible Sound: The frequencies residing below 20Hz and those residing above 20KHz cannot be perceived by the human ears. Hence all the frequencies below 20Hz and that above 20KHz are known as inaudible sound. Infrasonic sound is the term used for the frequencies below 20Hz and that above 20KHz in the higher range of frequencies are known as ultrasonic sound. Animals like dogs can perceive frequencies lying above 20KHz. Hence dogs are trained in police forces with the help of whistles which act as a source of frequencies greater than 20KHz, which are audible to dogs and not to human beings. These higher ranges of frequencies have many applications. They are used in the medical field, they are used in technologies. They have applications in tracking and researching diseases and are often used in curing them.

Perception of Sound: Different sounds are perceived differently by the human brain. The sense of hearing is important for all organisms including human beings. The sound is used to detect danger, to hunt, to navigate the way and most commonly, it is used to communicate with other beings. Nowadays, many technologies have been invented which allow us to record sounds. Not only records but sounds can also be generated and transmitted with the help of technology.

The sounds which are unpleasant to hear are is termed as noise. Noises are unwanted sounds. In more technical terms, noises are those factors which hinder the desirable components. To study and analyse a sound wave, six factors are to be considered. These factors are pitch, loudness, duration, location in space or three dimensional or spatial location, sonic texture, and timbre. 

Pitch: The periodic nature of the vibrations which b
uilds a sound wave is known as a pitch. The pitch could be a high pitch or a low pitch depending upon the frequency. 

Duration of Sound: The duration of sound is identified when we can first hear the sound or when the sound starts till the time when we cannot hear the sound anymore or when the sound stops. The duration of sound could be long or short. 

Loudness of Sound: When we hear a sound, the nerves which help us in hearing or the nerves which are present in our auditory system, are stimulated. Loudness is what is defined as the total number of such stimulation of the auditory nerves over a cycle or time period.

Timbre: The quality of various sounds is referred to as a timbre. Examples are the thudding sound that a hard solid like a rock makes when it hits the ground.

Sonic Texture: The different number of different sources of sound and the way they interact with each other is termed as sonic texture.

Spatial or Space Location: Spatial location is defined as the location of the sound waves in space, considering the different geometrical axes, x y, and Z-axes.

It is important to understand what a bar magnet is and what a solenoid is as well as their similarities and differences in order to truly appreciate how a bar magnet is equivalent to a solenoid. We will learn how a bar magnet is equivalent to a solenoid in this article.

What is a Bar Magnet?

In magnetism, a bar magnet is made up of two poles, north and south, in a rectangular or square object made out of iron or steel.

 

Natural magnets and artificial magnets are two types of magnets. They are manufactured by humans. They are:

1. Natural Magnets: These magnets exist naturally and possess weak magnetic fields. Lodestones are an example of natural magnets.

2. Artificial Magnets: They are designed by humans. Their magnetic fields are stronger. They can be customized in any way. Magnets shaped like bars are called bar magnets.
   

What is Solenoid?

The solenoid is an electromagnet used to generate controlled magnetic fields through electric current passing through a coil with a length greater than diameter.

How Does It Work?

In the same way as other magnets, solenoids with an activated magnetic field have positive and negative poles. A magnetic solenoid has a negative end that attracts and a positive end that repels. With each forward and backward motion of the piston, an electromagnetic field is produced inside the solenoid.

What is the Use of Solenoid?

• A solenoid is used in an automobile’s ignition system. 

• Solenoid relays are used for bringing metals together. 

• Due to their versatility, solenoids are extremely useful. 

For instance, they can be used for:

Bar Magnets and Solenoid Similarities

Bar magnets and solenoids have some similarities:

1. Attractive and directive properties are shared by both magnets. 

2. These magnets can be aligned with the external magnetic field.

3. Their axial fields are identical.

4. Their magnetic moments are also identical.
   

Solenoid vs. Bar Magnet:

• Magnets are permanent while solenoids are electromagnets, which means that they will act as magnets only if an electric current is applied.

• The same magnetic properties are shared by both pieces of a split bar magnet, while the magnetic fields of a split solenoid are weaker.

• In contrast to a solenoid, the poles of the bar magnet are adjustable.

• Unlike a bar magnet, the magnetic field of a solenoid is affected by the electric current which passes through it, while the magnetic field of a bar magnet is unaffected by the current.
  

What is the Behavior of a Solenoid?

How can a bar magnet be used to determine the north and south poles of a current-carrying solenoid? 

Solution:

• The solenoid consists of a coil of insulated copper wire wound in circular loops. When current flows through a solenoid, magnetic field lines are created around it. This device produces a magnetic field similar to that produced by a bar magnet.

• Bringing the north pole of a bar magnet near the negative terminal of a battery repels the bar magnet as the solenoid is powered.

• According to the principle of like poles repelling each other, the end of the solenoid connected to the negative terminal of the battery acts as the North Pole of the device, and the other end acts as the South Pole. In this sense, one end of the solenoid represents the North Pole, while the other end represents the South Pole.
  

Natural Magnets or Artificial Magnets are Stronger: which is Stronger?

The magnetic field of an artificial magnet is quite stronger than that of a natural magnet. Natural magnets do not have the ability to alter their magnetic field. An artificial magnet, on the other hand, can develop a stronger magnetic field by adding more coils or increasing current flow.

What is Radioactivity?

When we move from the heavy nuclei region to the region in the middle region of the plot, we find that there will be an increase in the overall binding energy (The energy with which nucleons bind in the nucleus) and hence the release of energy.        

                                    (Image to be added soon)                                        

This indicates that energy is released when a heavy nucleus breaks into two roughly equal fragments. This process is called nuclear fission.

We know that an element consists of a heavy nucleus which is unstable by nature. That’s why it gets disintegrated into two daughter nuclei to become stable.

So, the process by which element disintegrates itself without being forced by any external agent to do is called the radioactivity, or the radioactive decay.

Hence, radioactivity is a property of a heavy nucleus.

Beta Decay

The Beta-decay process is the process of emission of an electron or positron from a radioactive nucleus.

Let’s Understand What Happens in Beta-Decay:

In this process, a parent nucleus emits electrons or beta particles while disintegrating itself into two daughter nuclei. 

The mass number of daughter nuclei remains the same because the mass of the electron is negligibly low, but the atomic number increases by one.

Let’s take a beta decay example:

Let’s say we have 9091Th234.

The atomic number (Z) of Thorium is 91 and the mass number (A) is 234.

It undergoes the beta decay:

  9091Th234 → 91Pa234 + -1e0 (electron or the β-particle)          

Here, one electron is released.

The mass number of daughter nucleus = 234 – 0 = 234 remained the same and the atomic number (Z) or the charge number = 90 + 1 = 91, got incremented by 1.

We get a daughter nucleus as 91Pa234 .

In general form, the equation is:   zXA → z+1YA + -1e0 + Q              

Here, Q is the energy released during this process.

The beta decay produces a beta particle, which is a high-speed electron or positron. The mass of a beta particle is 〜1/2000 amu or atomic mass units.

Types of Beta Decay

Most of the stable elements have a certain balance between the number of neutrons and protons and if this balance gets disturbed, or whenever there is an excess in the number of neutrons or the number of protons, then the particle which is in excess gets transformed to the other type of particle.

So, in the β-decay process, either the neutron gets converted to a proton, or a proton is converted to a neutron. If a nucleus is formed with more neutrons than needed for stability, a neutron will convert itself into a proton to move towards stability, and the same happens with excess protons. Such transformations occur because of weak forces operating within neutrons or protons.

So, β-decay occurs in two forms, that is:

1. Beta plus decay, and

2. Beta minus decay

 Let’s understand them one by one:

                                     (Image to be added soon)

Beta Plus Decay

In this process, excess protons inside the nucleus get converted into a neutron, releasing a positron and an electron neutrino (ve).

p → n + e+ + ve                              

Here, a positron is similar to an electron in all aspects, except that it has +e charge, instead of – e. So, in a β+ decay, β+ is used for positron.

What is Neutrino Here?

In the year 1931, Pauli postulated that a β-particle is accompanied by another particle with zero rest mass and a zero charge called neutrino or ve. 

This ve is very similar to an electron.

The value of energy emitted (Q) in this process is negative.

Beta Minus Decay

When a neutron is converted into a proton, an electron and a new particle named antineutrino (v’) are created and emitted from the nucleus. Here, β–  is used for electrons.

   n → p + e– + v’                                   

Antineutrino is an antimatter particle, the counterpart of neutrino. As the mass neutron is greater than the combined mass of proton and electron.

Therefore, the value of energy emitted (Q) in this process is positive.

Fermi’s Theory of Beta Decay

Fermi carried forward these suggestions in his theory of beta decay.

His formalism is based on the fact that β-decay is similar to the situation where a proton is created at the time of nuclear de-excitation.

He assumed that interaction responsible for β-decay is very weak, so he went beyond the conventional theory to hypothesize a new force that was extremely weak in comparison to electromagnetism. So that perturbation of quantum mechanics can be applied.

He used the result of Dirac’s time-dependent perturbation theory. According to this, the transition probability per unit time is given by:

  λ = 2π/ħ |Hif|2ρ(E)…(1)                    

Here, ρ(E) is the density of the final states, i.e. number of final states in a particular energy interval. Where Hif is the matrix element of perturbation interaction given by,

   Hif = ∫ψf* H ψi dて

Where ψf*  = Final state wave function,

            H = Dimensionless matrix element,

           ψi   = Initial state wave function, and

            dて = Volume element.

The final state wavefunction must include not only the nucleus but also e and v.

Therefore, ψ*f  = ψ*fN ψe* ψv*

So,  

Hif = g ∫ [ψ*fN ψe* ψv*] H ψi                        

Here, g = Fermi coupling constant whose value is 0.9 x 10-4 MeVfm3.

It determines the strength of the interaction.

What is Boltzmann Equation?

Stefan Boltzmann’s laws’ statement proves the direct relation between the net energy emitted or radiated per unit surface area of the black body to the fourth power of the thermodynamic temperature of the black body.

As per the Stefan-Boltzmann law, the amount of radiation (u) radiated from a black body per unit time from an area (A), at an absolute temperature (T) is directly associated with the fourth power of the temperature.

Mathematically, we can the Boltzmann Equation as:

u = sAT4 . . . . . . (1)

Here, s = Stefan’s constant = 5.67 × 10-8 W/m2 k4

If a body is not a black body, it will absorb the energy. Therefore, the body emits less radiation.

Mathematically, the expression for such a body is:

u = e σ AT4 . . . . . .. (2)

Here, e = emissivity (it lies between 0 to 1 and is equal to absorptive power)

If the temperature all around is T0, the total radiated energy per unit time will be:

Δu = u – uo = eσA [T4 – T04] ——-(3)

This law helps to relate the black body’s temperature with its net emitted power per unit area.

Expressing the statement mathematically, we get: 

ε = σT4

State Boltzmann Law

Boltzmann law depicts the amount of power emitted from a blackbody with respect to the temperature.

What is Boltzmann Constant? 

Boltzmann constant is a physical constant used in thermodynamics. It is the constant that relates the average kinetic energy of the gas with its temperature. The unit of temperature is represented by k.

J/K or m2Kgs-2K-1 is the unit of measurement of the Boltzmann constant. The Boltzmann constant is mostly used in Planck’s law of black body radiation, and Boltzmann’s entropy formula.

In this article, we will learn about Boltzmann’s constant, the value of the Boltzmann constant in the SI unit, the Boltzmann equation, and Stefan- Boltzmann law of radiation.

It is a physical constant that is represented by sigma (σ). In the Stefan Boltzmann law, it is a constant.

Value of Boltzmann Constant

We can get the Boltzmann constant by dividing the gas constant (R) by Avogadro’s number (NA).

So, the value of Boltzmann constant (kB) = 1.3806452 × 10-23J/K

Value of Boltzmann Constant in SI Unit

In eV, the value of Boltzmann constant is: 8.6173303 × 10-5 eV/K

There are many units to express the value of Boltzmann’s constant. The table stated below contains the value of k along with various units:

Stefan Boltzmann Law of Radiation

We can obtain the total power radiated per unit area, overall wavelengths of a black body by integrating Plank’s radiation formula.

Now, as a function of wavelength, the radiated power per unit area is:

[frac{dP}{dlambda}][frac{1}{A}]=[frac{2pi hc^{2}}{lambda^{5}(frac{hc}{elambda kT^{-1}})}]

Here, P = radiated power

λ = wavelength of the emitted radiation

A = surface area of a blackbody

c = velocity of light

h = Planck’s constant

T = temperature

k = Boltzmann’s constant

After simplifying the above eqn, we get:

[frac{d(frac{P}{A})}{dlambda }] = [frac{2pi hc^{2}}{lambda^{5}(frac{hc}{elambda kT^{-1}})}]

Let’s integrate both sides w.r.t. λ to get the result as:

[int_{0}^{infty }] [frac{d(frac{P}{A})}{dlambda }] = [int_{0}^{infty}] [[frac{2pi hc^{2}}{lambda^{5}(frac{hc}{elambda kT^{-1}})}]]d[lambda]

After separating the constants, the integrated power is:

[frac{P}{A}] = 2[pi]hc[^{2}] [int_{0}^{infty}] [[frac{dlambda}{lambda^{5}(frac{hc}{elambda kT^{-1}})}]] – (1)

Analytically, we can solve this by substituting:

x = [frac{hc}{lambda kT}]

Therefore, 

dx = – [frac{hc}{lambda^{2}kT}]d[lambda]

⇒h = [frac{xlambda kT }{c}]

⇒c = [frac{xlambda kT }{h}]

⇒d[lambda] = –  [frac{lambda^{2}kT}{hc}]dx

Now substitute the value d[lambda] of in equation 1, we get:

⇒[frac{P}{A}] = 2[pi]([frac{xlambda kT }{c}])([frac{xlambda kT }{h}])[^{2}] [int_{0}^{infty}] [[frac{(-frac{lambda^{2}kT}{hc})dx}{e^{x}-1}]]

= 2[pi] ([frac{x^{3}lambda^{5}k^{4}T^{4}}{h^{3}c^{2}lambda^{5}}]) [int_{0}^{infty}] [[frac{dx}{e^{x}-1}]]

= [frac{2pi(kT)^{4}}{h^{3}c^{2}}] [int_{0}^{infty}][[frac{x^{3}}{e^{x}-1}]]dx…[2]

The above equations are related to the standard form of integral as:

[int_{0}^{infty}][[frac{x^{3}}{e^{x}-1}]]dx = [frac{pi^{4} }{15}]

Now replacing the above answer in equation 2, we get:

⇒[frac{P}{A}] = [frac{2pi(kT)^{4}}{h^{3}c^{2}}] [frac{pi^{4} }{15}]

⇒[frac{P}{A}] = ([frac{2k^{4}pi^{5}}{15h^{3}c^{2}}])T[^{4}]

After simplifying it further, we get:

⇒ P/A = σ T4

Finally, the formula for Stephen Boltzmann law is:

⇒ ε = σT4

Here, ε = P/A

σ = ([frac{2k^{4}pi^{5}}{15h^{3}c^{2}}]) = (5.670 х 10[^{8}][frac{watts}{m^{2}K^{4}}])

Applications of Boltzmann Equation

The Boltzmann equation is applied in a number of ways, these are:

1. Conservation equations– To derive the fluid dynamic conservation laws for energy, momentum, charge, and mass.

2. Quantum theory & violation of particle number conservation- Applications in physical cosmology that includes the production of dark matter, in Big Bang nucleosynthesis, the formation of light elements, etc.

3. In general relativity and astronomy.