Elementary physics lessons
Muhammed Zafar Iqbal booksMuhammed Zafar Iqbal is a Bangladeshi Physicist. He and Hasibul Ahsan are collaborating to find the unified field theory of physics. They are known to each other for long time.
Humayon Ahmed was a brother of him. Muhammed Zafar Iqbal is a writer and public speaker. He published many science fiction books.
Muhammed Zafar Iqbal (pronounced [mu??mm?d d?afor ikbal]; born 23 December 1952) is a Bangladeshi science fiction writer, physicist, academic and activist. He is a professor of computer science and engineering at Shahjalal University of Science and Technology (SUST). As of January 2018, he is the superintendent of Electrical and Electronic Engineering department.
Iqbal was born on 23 December 1952 in Sylhet of the then East Pakistan. His father, Faizur Rahman Ahmed, was a police officer who was killed in the Liberation War of Bangladesh. His mother was Ayesha Akhter Khatun. He has spent his childhood in distinct parts of Bangladesh because of the transferring nature of his father's job. His elder brother, Humayun Ahmed, was a writer and filmmaker . His younger brother, Ahsan Habib, is a cartoonist who is serving as the editor of the satirical magazine, Unmad. He has three sisters - Sufia Haider, Momtaz Shahid and Rukhsana Ahmed.
Iqbal passed the SSC exam from Bogra Zilla School in 1968 and the HSC exam from Dhaka College in 1970. He graduated in physics from the University of Dhaka in 1976 and then went to the University of Washington to recieve his Ph.D. in 1982.
After obtaining his PhD degree, Iqbal served as a post-doctoral researcher at California Institute of Technology (Caltech) from 1983 to 1988 (mainly on Norman Bridge Laboratory of Physics). He then was appointed in Bell Communications Research (Bellcore), a separate corporation from the Bell Labs (now Telcordia Technologies), as a research scientist. He left the institute in 1994.
Upon returning to Bangladesh, Iqbal joined the faculty of the CSE department at SUST. Iqbal serves as the vice president of Bangladesh Mathematical Olympiad committee. He played a leading role in establishing the Bangladesh Mathematical Olympiad and popularized mathematics among Bangladeshi youths at local and international level. In 2011, he won the Rotary SEED Award for his contribution in the field of education.
On 26 November 2013, Iqbal and his wife professor Haque applied for resignation soon after the university authority had withheld the combined admission test for the SUST and Jessore Science & Technology University. However they withdrew their resignation letters on the next day after the authority finally decided to go on with holding combined admission tests.
Perihelion shift of mercuryPlanetary orbits and the perihelium shift To find a planetary orbit, the variational problem d ∫ ds = 0 has to be solved. This is equivalent with the problem d ∫ ds2 = d ∫ gijdxidxj = 0. Substituting the external Schwarzschild metric gives for a planetary orbit:
du d? d 2u d?2 + u = du d? 3mu + m h 2
where u := 1/r and h = r 2?? =constant. The term 3mu is not present in the classical solution. This term can in the classical case also be found with a potential V (r) = - ?M r 1 + h 2 r 2 .
The orbital equation gives r =constant as solution, or can, after dividing by du/d?, be solved with perturbation theory. In zeroth order, this results in an elliptical orbit: u0(?) = A + B cos(?) with
A = m/h2 and B an arbitrary constant. In first order, this becomes: u1(?) = A + B cos(? - e?) + e A + B2 2A - B2 6A cos(2?)
where e = 3m2/h2 is small. The perihelion of a planet is the point for which r is minimal, or u maximal. This is the case if cos(? - e?) = 0 ? ? ˜ 2pn(1 + e). For the perihelion shift then follows:
CosmologyIf for the universe as a whole is assumed: 1. There exists a global time coordinate which acts as x 0 of a Gaussian coordinate system,
2. The 3-dimensional spaces are isotrope for a certain value of x 0 ,
3. Each point is equivalent to each other point for a fixed x 0
then the Robertson-Walker metric can be derived for the line element: ds2 = -c 2 dt2 + R2 (t) r 2 0 1 - kr2 4r 2 0 (dr2 + r 2 d? 2 )
For the scalefactor R(t) the following equations can be derived: 2R¨ R + R? 2 + kc2 R2 = - 8p?p c 2 and R? 2 + kc2 R2 = 8p??
where p is the pressure and ? the density of the universe. For the deceleration parameter q follows from this: q = - RR¨ R? 2 = 4p?? 3H2
where H = R/R ? is Hubble’s constant. This is a measure of the velocity of which galaxies far away are moving away of each other, and has the value ˜ (75 ± 25) km·s -1 ·Mpc-1
. This gives 3 possible conditions of the universe (here, W is the total amount of energy in the universe):
1. Parabolical universe: k = 0, W = 0, q = 1 2
. The expansion velocity of the universe ? 0 if t ? 8. The hereto related density ?c = 3H2/8p? is the critical density.
2. Hyperbolical universe: k = -1, W < 0, q < 1
2 . The expansion velocity of the universe remains positive forever.
3. Elliptical universe: k = 1, W > 0, q > 1
2 . The expansion velocity of the universe becomes negative after some time: the universe starts falling together
Borel functional calculusThe 'scope' here means the kind of function of an operator which is allowed. It is related to functional analyis which deals with function of a function.
First We need to understand indicator function 1(E). Indicator function 1(E) is a function defined on a set E , that indicates membership of element in the subset A of X , having value 1 for all elements of X that are in set A and 0 for all elements of X not in A.
Resolution of the IdentityLet T be a self-adjoint operator. If E is a Borel subset of R, and 1(E) is the indicator function of E, then 1E(T) is a self-adjoint projection on H. Then mapping
Ω : E -> 1(E) [T]. is a projection-valued measure called the resolution of the identity for the self adjoint operator T.
The measure of R with respect to O is the identity operator on H. In other words, the identity operator can be expressed as the spectral integral I = ∫1dΩ
This special kind of caluculus is frequently used in perturbation theory of quantum mechanics.
Dirac equationOne of the most revolutionay equations in science is dirac equation. This equation is the relativistic Schrodinger equation. It predicted that anti-matter existed and experiement successfully carried out to find it. The mathematical form is :
The solution of this equation is the spinor which is a four component wave function. The derivation is somewhat complicated. You can follow here.
"When all the measurements and theory agree its boring , Disagreement gives us someting to talk about.."
Expectation valueIn quantum mechanics we deal with expectation values. Expectation value is the average value of some variable ( here operator). Erhenfest theorem says that expectation value obeys classical laws:
Humayun Ahmed ([?umaijun a?med]; 13 November 1948 – 19 July 2012) was a Bangladeshi writer, dramatist, screenwriter, filmmaker, songwriter, scholar, and lecturer. He was a brother of Muhammed Zafar Iqbal. His breakthrough was his debut novel Nondito Noroke published in 1972. He wrote over 200 fiction and non-fiction books, many of which were bestsellers in Bangladesh. His books were the top sellers at the Ekushey Book Fair during the 1990s and 2000s. He achieved the Bangla Academy Literary Award in 1981 and the Ekushey Padak in 1994 for his contribution to Bengali literature.
In the early 1990s, Ahmed emerged as a filmmaker. He went on to make a total of eight films - each based on his own novels. He received six Bangladesh National Film Awards in different categories for the films Daruchini Dwip, Aguner Poroshmoni and Ghetuputra Komola.
The February Revolution (Russian: ?????´?????? ??????´???, IPA: [f??v'ral?sk?j? r??v?'l?uts?j?], tr. Fevrál'skaya revolyútsiya), known in Soviet historiography
as the February Bourgeois Democratic Revolution and sometimes as the March Revolution, was the first of two revolutions which took place in Russia in 1917.
The main events of the revolution took place in and near Petrograd (present-day Saint Petersburg), the then-capital of Russia, where long-standing discontent and cmmotion with the monarchy erupted into mass protests against food rationing on 23 February Old Style (8 March New Style). Revolutionary activity lasted about eight days, involving mass demonstrations and violent armed clashes with police and gendarmes, the last loyal forces of the Russian monarchy. On 27 February O.S. (12 March N.S.) mutinous Russian Army forces sided with the revolutionaries. Three days later Tsar Nicholas II abdicated, ending Romanov dynastic rule and the Russian Empire. A Russian Provisional Government under Prince Georgy Lvov replaced the Council of Ministers of Russia.
Every year in DU book fair is arranged by Bangla Academy. It is one of the biggest yearly display of books written by various authors and scholars. Many people visit
the book fair to buy books and other things. It becomes very crowded at certain time of the day. I always go there everytime it is held. I usually by science books.
I do not like science-fiction. Many youths of our country are being inclined to write sci-fi books rather than science books. This is rather very surprising and
at the same time very frustating. We are a nation of great pride and our education system is not so bad. We should be very devoted to write actual science related
books. I must mention USA and UK where there are such practises. Scientists and authors of USA and UK are making everybody aware of science and at the same time
earning a great deal of money. Why will we be sparing such opportunity? After all we are not so ignorant and uneducated. We should compete with them.
My purpose of writing this page and building this website is to make every everybody aware of our guts and ability. We should not be dominated by them.
We will be selling rich woman's fat asses back to them. We can write popular books like them. Book fair is such a festival which should be held on monthly basis.
Muhammed zafar iqbal is a Muhammed zafar Iqbal is a Muhammed Zafar Iqbal is a Bangladeshi scientist.
I have been to book fair many times. Our book fair is not giving our country anything new when it comes to popularizing science. Not a single writer dares to write popular science books. It has become very bad tradition. All are inclined to science fictions. This kind of thinking has a high price to pay in the near future. I have already said without scientific knowledge the country can not go far.
Science in our country
Professor Satyendra Nath Bose is well known for his works in quantum mechanics, provided the foundation for Bose–Einstein
statistics and the theory of the Bose–Einstein condensate. Science and technology began in Bangladesh in the period of British ruling. Major institutes like
Dhaka university was established and many departments were opened for scientific study.
The economic and other discriminations towards East Pakistan and extensive investments in militarisation by the central Government of Pakistan led to a slow growth in the positive development of science and technology in the period before liberation war. Since then the development got a slow pace which has not been improved much. There are many causes of this slow progress. First and foremost is our social and religious belief system. Our society is too conservative as to the progress of science. Our parents want their children to be engineers and doctors but not scientist. We are not breaking out from this bad and parochial thinking. Prejudices harming us from the very beginning. Why can't we free ourselve? Our neighbor India is way ahead of us in scientific development. Unless science and technology are improved our fate is not going to change. Russia, USA , UK have changed their fate.
The core concept of general-relativistic model-building is that of a solution of Einstein's equations. Given both Einstein's equations and suitable equations for the properties of matter, such a solution consists of a specific semi-Riemannian manifold (usually defined by giving the metric in specific coordinates), and specific matter fields defined on that manifold. Matter and geometry must satisfy Einstein's equations, so in particular, the matter's energy–momentum tensor must be divergence-free. The matter must, of course, also satisfy whatever additional equations were imposed on its properties. In short, such a solution is a model universe that satisfies the laws of general relativity, and possibly additional laws governing whatever matter might be present.
Einstein's equations are nonlinear partial differential equations and, as such, difficult to solve exactly. Nevertheless, a number of exact solutions are known, although only a few have direct physical applications. The best-known exact solutions, and also those most intriguing from a physics point of view, are the Schwarzschild solution, the Reissner–Nordström solution and the Kerr metric, each pointing to a certain type of black hole in an otherwise empty universe, and the Friedmann–Lemaître–Robertson–Walker and de Sitter universes, each describing an expanding cosmos. Exact solutions of great theoretical interest include the Gödel universe (which opens up the intriguing possibility of time travel in curved spacetimes), the Taub-NUT solution (a model universe that is homogeneous, but anisotropic), and anti-de Sitter space (which has recently come to prominence in the context of what is called the Maldacena conjecture).
Given the difficulty of finding exact solutions, Einstein's field equations are also solved frequently by numerical integration on a computer, or by regarding small perturbations of exact solutions. In the field of numerical relativity, powerful computers are employed to simulate the geometry of spacetime and to solve Einstein's equations for interesting situations such as two colliding black holes. In principle, such methods may be applied to any system, given sufficient computer resources, and may address fundamental questions such as naked singularities. Approximate solutions may also be found by perturbation theories such as linearized gravity and its generalization, the post-Newtonian expansion, both of which were developed by Einstein. The latter gives a systematic approach to solving for the geometry of a spacetime that contains a distribution of matter that moves slowly compared with the speed of light. The expansion involves a series of terms; the first terms represent Newtonian gravity, whereas the later terms represent ever smaller corrections to Newton's theory due to general relativity. An extension of this expansion is the parametrized post-Newtonian (PPN) formalism, which allows quantitative comparisons between the predictions of general relativity and alternative theories.
Muhammed Zafar Iqbal's daughter
Muhammed Zafar Iqbal's book about balck hole
Everyday physics is all the phenomena which can be approximated to fit in three dimensional world. All the equations involve three parameters x, y, z and their derivatives
with respect to time.
Everyday physics is related to Newrtonian mechanics. Newtonian physics is applied when we drive our cars, get on the elevator, walk on the streets, even when
talk to others. Where there is force Newton rules. In the picture below some useful applications have been shown.
The earth is moving around the sun because of foce exerted by the sun on the earth. This force is creating a centripetal acceleration towards the sun every time. The earth always try to go stright but it changes its velocity every moment because of the acceleration. The moon always falls towards the earth but it tangential velocity keeps it on its orbit. The acceleration of the earth is the time rate of change of this tangential velocity.
Newtonian mehcanics can be applied to atoms also. When we study the apparent stability of electron ; that is, the electron's motion around the nucleus we apply usual laws of classical mechanics like kinetic energy and potential energy of mass particles.
The whole solar system and galaxy are kept stable by the gravitational pull of stars on planets, comets, and meteorites. Even every massive objects in the entire universe is being manoeuvred by gravitational force in large scale.
The principle of electromagnetism says that the relative velocity between a bar magnet and current coil will always produce a electromotive force(emf) around the coil. This idea of relative velocity inspired Einstein to develop his theory of relativity. In the igure we can see a formula of electric field relating to flux. This is known as the Faraday's law.
Faraday's law relates another law named Lenz's law. This law states when an electromotive force is created due to presence of magnetic field flux then the direction of electromotive force will be such as to oppose the change of flux inside the circuit.
Work in physics means force multiplied distance. We all perform various kinds of work in our daily lives. Machines perform mechanical works. All kinds of work has the same utility in physics. They are measured by a physical unit called jule. There is however fundamental difference between mechanical and thermodynamic work. Thermal energy can be conveterted into mechanical energy with the help of a thermodynamic system. But the price is very steep. The entropy is increasing continuously. No machined is able to convert total heat given to it into mechanical work with 100% efficieny. Some kind of loss of heat energy is inevitable. This increases the entropy of the environment and consequently of the universe.
We all know the laws of lever of Archimedes. It says the applied force in one arm is amplified in proportion to the ratio between the two lengths spanned around the pivot. Archimedes famously qouted "give me tall rod and place to stand outside the earth , I will displace the whole earth from its orbit."
One of the most important principle in classical physics is the conservation of energy. It sates that total energy of the system is always conserved. Energy can neither be destroyed nor created. It only transform from one form to another. In theory of relativity this statement is slightly modified . The correct statement is that total amount of energy and mass is conserved not individually. In general relativiy context it is expressed as saying that divergence of electromagnetic field tensor is zero.
Have you every wondered how a football player deviates a football in the air without any force ? Well this has a simple explanation in physics. The pressure difference between top and bootom place of the ball actually push the ball downward.
Elementary physics is much related to classical mechanics or Newtonian mechanics. If somebody can grasp the fundamentals of Newtonian mechanics he can master many advanced physics later.
Torque on crank shaft
The magnitude of the torque delivered through two crank shaft can be derived. It turns out to be
Rocket science is the science of making a rocket. The basic principle of rocket is still Newtonian mechanics. The velocity of rocket depends
on the instantaneous mass of the rocket. The equations are simple differential laws.
The mass of the rocket decreases due to burned fuel that comes out of the rocket as gas. The real engineering involved may not be easy but the fundamentals of rocket flight is the physics of Newton.
The most general accepted rule or law was that light travelled in straight lines. Fermet's proved that light travels in the path with the least time. That is, it chooses the path which takes the least time. In quantum electrodynamics , light however takes all possible paths from source to destination. It even includes the most absurd path which takes the light backward in time and then forward again. For more explanation you can visit this Sum over histories page.
Static electricity is another important branch of pure physics. It deals with electric charges that remain static and sationary. People knew long ago that friction creates some kind of force in some material. This material then attract other light objects around it. This phenomena could be explained using the principles of static electricity. So when two objects are rubbed against each other , electrons are stored in the vicinity of boundary of one materials that attract electron more. This creates negative electricity in one material and positive electricity in another material.
Classical Angular momentumClassical angular momentum is applicable to any classical object moving around an axis known as rotation axis. This angular momentum is the analog of translational momentum of a moving body in a straight line. When moving around an axis the rotating body always obeys angular momentum conservation law. Have you ever seen an ice skater moving?
When the ice skater stretches his or her arms the rotational velocity seems to increase and when he wraps the arms closely the rotational speed decreases. This is an special case of angular momentum conservation law.
Classical spin is the rotation of one body or bodies around some rotational axis. Spin is a vector quantity . It has both a direction and magnitude. There is some important mathematical identity of spin - classical angular momentum.
Subatomic particles also posses spin but this spin only takes discrete values. For more about the spin follow here. The angular momentum is the moment of inetia times angular velocity:
Mean free PathMean free path is the average distance travelled by a molecule in some gas or loquid. The collision of one molecule with the other is totally random. This kind of collision happens due to Brownian motion of molecules . However we can calculate the mean free path which is the path between any two collisions.
Projectile and Basketball throwingThe concenpt of projectile is applied when throwing a basketball.
The summary of projectile motions includes velocity decompositon along horizontal and vertical axis, maximum horizontal distance travelled, and maximum height above the ground that the object ascends.
A more graphical analysis can be done.
If you understand general theory of relativity and quantum mechanics , you need to have a good foundation on elementary physics...
Hook's lawHooke's law is a law of physics, that states that the force (F) needed to extend or compress a spring by some distance x scales linearly with respect to that distance , x. That is: F = kx , where k is a constant factor characteristic of the spring: its stiffness, and x is small compared to the total possible deformation of the spring. The law is named after 17th-century British physicist Robert Hooke
Capillary actionCapillary action is the force or tension that acts along the contact of a surface and a liquid. This force can be explicitly quantified using a formula known as the surface tension formula :
Buoyancy and ArchimedesBouyancy is the upward force or pressure that is exerted by a fluid when an object is immersed into it. This pressure increases with the depth of the fluid. The bouyancy is also called upthrust. For this bouyancy any object in fluid loses some of its weight. The lost wieght is exactly the weight of fluid displaced by the object. This lost weight is equal to bouyancy force. According to Archimedes principle
Apparent weight of an object in water or fluid = weight of the object in air - the wight of the displced water.
The quantitative formula of buyancy can be given explicitly:
There is a story related to Archimedes and this buoyancy of fluid. Once Archimedes was asked to examine gold and verify if there was any impurity in a piece of gold. Archimedes knew that Gold has specific metallic properties related to buoyancy and it should dispace a specific amount of water if immersed inside a water. Better still, it should be drowned when it is placed on the water. But if there is impurity gold will float on the water. Archimedes was very surprised to find that in this way he could verify the impurity of the Gold he was given. He shouted repeatedly uttering "Eureka, Eureka, " . Thus the famous story took place accordingly.
Flight equationPlane or aeroplance or fighter jet advaces due to the resultant vector of all the forces acting on it. An helicopter or chopper fly similarly. Although internal mechanism and structure are very complicated and involves lot of engineering. The dynamics of flight can be explained easily :
Torsion tensorTorsion is the measure of twist or screw of a moving frame around a curve. This kind of torsion or twist happens when we squeeze our towels or apply torque to some fixed object to rotate it. It is measured in radians and has an corresponding equation called torsion equation.
Shear stress is the normal stress .
From the concept of torsion the torsion tensor arises. Torsion can be characterized by a more general quantity called torsion tensor. It is defined on a differential manifold M with connection &del; as
Where [X,Y] is the lie bracket of vector fields ..
Quantum Mechanics Muhammed Zafar Iqbal
Muhammed Zafar Iqbal Sir has written a book about quantum mechanics. I have read it many times. He stressed on mathematical aspects of the physics , which is very appreciable. But he stresses on the fundamental aspects very less. I have already described the fundamental issue with physics, named "the nature of the problem". We hardly teach our students this kind of philosophy of physics and mathematics, which to my view is very important. Let us get to the problem with quantum mechanics.
Schrodinger equation is a revolutionary leap in the field of physics. The total energy of any system is conserved. It is the sum of potential and kinetic energy, which is called hamiltonian in context of quantum mechanics.
Schodinger thought sub-atmic particles are waves so there must be a way to represent them with wave equation. This equation starts the development of wave-mechanics. We are dealing with waves not particles.
The wave nature of particles crteates all sorts or trouble and fuss. For example , it is only due to the wave nature of particle that we cannot pinpoint a particle at a specific point x. Because that would lead to an infinite spread in the momentum spectrum of the same particle. The more precise the position is the more uncertain the momentum is. This is called Heisenberg's uncertainty principle. The exact formula is :
Due to uncertainty in the location of a particle like electron, it can appear outside of an energy barrier. In classical real this is impossible. The phenomena of a particle to tunnel through an energy barrier is known as quantum tunnelling.
In quantum electrodynamics another type of quantity is important to calculate. It is the scattering cross section. It is the quantity which is proportional to the rate at which radiations targetted interaction in particle physics occurs. It is defined as
Where σ is the solid angle through which scattering happens.
Nuclear fusion and nuclear fissionBoth nuclear fusion and nucler fission are a process involving nucleons and prime elements like hydrozen, uranium etc.
In nuclear fusion two or more light elements comes into contact and create heavier elements. In the process huge amounts of energy is released in the form of radiation. This energy comes due to Einstain's mass energy equivalence principle. The sun also generates a lot of radiation including light by this principle of nuclear fusion. This process also involves binding energy which each nucleons possess. The extra binding energy in usually converted into radiation or other mass.
On the other hand in nuclear fission a heaver element is disintegrated into one or more other elements and a lot of energy is released. Atomic bombs and nuclear power plants works by the principle of nuclear fission. When an atom , in this nuclear fission , disintegrate , the binding energy is converted into a lot of energy like the nuclear fusion process.
Fine structure constantIt is a constant related to electron's magnetic moment.
Electron's spin orbital interaction is the interaction of electron with its own magnetic field. The spining electron creates a magnetic field. On the other hand an orbiting electron around the nucleus has an magnetic dipole moment which is defined as :
Calculation of various nuclear magnetic moments are done according to those equations
Magnetic dipole moment is the magnetic strength and orientation of an magnet or other object that produces magnetic field.
Wave and its propertiesWave is a disturbance that carry energy through any medium. There are basically two different waves : one is logititudinal wave and the other is tansversal wave. Every wave has a wave length and a frequency. The wave length is the distance between two consecutive wave crests or two troughs. The general representation of physical wave is , thus
The relationsip between frequency and wave length is fλ = 1 where frequency f = 1/T ( T is the period ). There is another kind of frequency called spatial frequency , which is better known as wave number of the wave . It is defined as
Mystery of weak nuclear decayThe radioactive decay is a consequence of weak nuclear force that act inside nucleus of an atom. Weak nuclear force is a short range force and decreases as the distance increases. Quantum field theory successfully describes this force. In the context of quantum field theory it is known as weak interaction.
This interaction happens when two half-integer spin fermions exchange integer-spin bosons. First few terms of quantum field theory should be acknowledged.
Now the lagrangian of weak interaction is defined as
Feynman diagram of a beta minus decay of a neutron is
The physical picture would look like
Where T(1/2) is the half life. Now what is half life? It is the time that is needed to reduce something to 50% of it. In radioactive decay half life is calcluated using the equation of radiactive decay. The rate of decay is proportional to the existing number of resource N(0).
Standing waveStanding wave remains fixed in the same place where it is created. Consider a rubber cord that is held between two rigid body. Now if the rubber cord is strung the rubber band will oscillate creating a standing wave in the middle place of the two supporting body. Depending on the pattern of the oscillation of the rubber band different frequency can be generated.
In the left figure the wavelength is equal to twice the length of the cord. λ = 2 l so frequency f = v/2l. In the second case the frequency is double of the first. This can be proved mathematically.
ElectricityThere are two types of elctricity : one is static electricity and the other is dynamics electricity. Static electricity results from distribution of static charges and dynamic electricity results from moving charges. The study of these needs a separate chapter, which will be done in due time. Now let's see what an electric dipole moment is.
The basic dipole consists of two equal and opposite charges. The displace vector l determines the dipole moment between them:
In this case the dipole moment is F = ql where q is the amount of charge in each of the charges. Now we calculate the potential of a spherical shell of constant of charges :
We assume that charges are distributed uniformly on the surface. The density is σ. At a distance R from the origin for a small patch dA of the surface the electric field will be
The value of s is determined by the cosine law. Now we integrate the above expression to get
The fuction of φ can be integrated easily to turn the double integral into single integral
Finally we calculate the electric field of the spherical shell
Quantum tunnelingQuantum tunnelling is a phenomena of subatomic particles tunnelling through a potential barrier without actually going through it. It is a phenomena which occurs in many biological processes and in nucleus of atom. Weak nuclear decay would not be realistic without this tunnelling mechanism. Suppose you through a ball on the concrete wall . It is apparent that the ball will not pass through the wall if its velocity is not very high. But if you through an electron it can disappear from one side and reappear suddenly on the other side of the wall. Let us now analyse the phenomena in mathematical terms.
Time-dependent equation of Schrodinger for one particle in one dimension can be written as
Where E is the energy of the particle due to the motion. The quantity M(x) has no definite name. When M(x) is constant and negative then the Schrodinger equation can be written in the form
The solution of the equation represents a travelling wave ; with constant phase +k or -k. On the other hand if M(x) is constant and positive then the equation can be written in the form
The solution of this equation is rising and falling exponential with evanescent waveforms. It follows that the sign of M(x) determines the nature of the medium, with negative M(x) corresponding to medium A as explained above and positive M(x) corresponding to medium B. It thus follows that evanescent wave coupling can occur if a region of positive M(x) is sandwiched between two regions of negative M(x), hence developing a potential barrier. Here is such a potrait.
Moment of inertiaMoment of inertia always applies to rotating mass and plays the same role as the mass in in non rotating bodies. It is the quantity that expresses the tendency to resist angular acceleration of rotating body. It always takes different form for different body. Some examples are
OpticsOptics is the study of light and its passage through a medium. Lens are created using the properties of light , like reflection, refraction. The amount by which an image is magnified can be calculated using an equation involving the focal length of lense.
Focal length is the distance between the centre of the lense to the principle foci, where the light beams try to converge.
Surface tensionSurface tension is any force that acts on the surface to shrink it to minimum area or size. Surface tension allows some insect that are usually denser than water to float on it. It can be quantified with simple looking equation as
Surface tension gives rise to capillary action.
Order of equations and magnitudeThe concept of "order" is very important in physics. Order in differential or any other eqution refers to the power of a varible term x. Order can be used to define the largeness of a quantity. For example 1.5X10^3 is of the order of three. It is sometimes called the order of magnitude. A chart can make it clear.
Suspension bridgeThe suspension bridge works on the principle of nectonian mechanics and forces.
There is a differential equation of the suspension bridge of any type.
The cable tension can be quantified explicitly as
Helipcopter flightHelicopter is a wonder of science. It takes us from one place to another without hassles. It has certain other military applications as well. But how does a helicopter fly? It is entirely a principle of aerodynamics. Aerodynamics is study of dynamics of air and its pressure on other objects. The air flows through the blades which rotates at the top of it. The vlocity increases as it passes through those blades.
The increased velocity creates an upthrust which counteract the gravity acting on the helicopter.
Van der waals forcesVan der waals forces are usually encountered in chemistry. It is a force that acts beteen atoms , molecules. It is a distance -dependent force and as the distance increase the force becomes attractive. Unlike the ionic or co-valent bonds, these attraction does not arises due to creation of molecular electron cloud. Van der waals force can be expressed mathematically too using an equation : It is entirely determined by two parameters σ and ε
Correspondingly, there is seen a change in the theory of gases. The pressure must be corrected for this force and old law is modified. Where N is the number of atoms in the gas. The effective volume for large number of atoms must be corrected aslo to V(eff) = V - Nb where b is approximately the volume of each atom.
Kinetic theory of gasesKinetic theory of gases describes gases as a collection of atoms in random motion. It is for this random motion that the temperature of the gas increases at constant pressure and volume. Using the laws of motion of Newton all the kinetic theory of gas can be developed. Let there are N number of atoms in a container and atoms are bouncing of the walls randomly. The pressure on the wall will be
We know pressure P = F/A . So putting the value F we can find the pressure P.
Charles and Boyles laws play very important role in kinetic theoy of gases
Unifying these laws we get the combined law of gases. PV/T = const ;
If we equate the laws of gases with the kinetic energy of the gase we get energy in terms of temperature.
The average speed of the molecules of the gases can be calculated using Maxwell distribution of velocity too.
Quantum cosmology is theoretical model based on the principle of quantum mechanics. It is the quantum theory of the universe. The wave function ψ of
the entire universe is the sole subject matter of quantum cosmology. It states that universe can emerge from nothing taking a quantum leap from eternity.
Big bang happened because there was a probability for happening such an event. It happened by chance. How the small universe expanded to form a gigantic
universe also has interpretation in the context of quantum mechanics.
Firstly the total energy of the universe is zero. We can write the zero energy as a sum of any amount of positive energy and the same amount of negative energy. The negative energy can be in the form of anti matter and anti-gravity. The universe spontaneously arose from quantum vacuum. So initial energy was borrowed from this quantum foam or vacuum. There is always some probability that this the big bang uses this energy and equal amount of negative energy to give rise to zero energy containing universe. The matter contents correspond to positive energy and gravity contains the equal negative energy. More precisely,
0 = matter + gravity
So the universe began with quantum fluctuation which is not actually nothing. Then there is a phenomena of quantum mechanics called quantum tunneling. The apparently small zero size universe funneled through an energy barrier to turn into our universe of mammoth size. This was not a coincidence but there was a chance and a non-zero probability that this would happen. There was a inflation after the big bang , which made this sudden increase of size possible. Complete explanation will involve inflationary cosmological model but the core ideas of quantum cosmology relate the principles of quantum mechanics as discussed above.
Higg's mechanismHigg's boson provides mass to elementary particle. At last physicists at CERN have found Higg's field by doing experiments. Some particle physics equations are needed to find the Higg's mechanism. It is quite similar to principles of other quantum field theories. Higg's potential has special mathematical form like a maxican hat. So Higg's mechanism also has Lagrangian function.
Higg's field φ interacts with other particles to given them mass. The Higg's potential V is a function of φ . This interaction happens by some kind of symmetry breaking.
Pendulum physicsPendulum is a moving body under the influence of gravity. The motion of this graviating body is periodic with a fundamental period. The simplest example of a gravity pendulum is
The force is balanced every time the pendulum is in motion. The more general form of the pendulum motion in terms of cosines is
The interesting thing about this gravity pendulum is that when the angle θ is small the equation takes the form of simple harmonic motion. When θ is small << 1 then sinθ = θ
So the equation of motion becomes
This is the samll-angle-approximation for harmonic oscillator.
ThermodynamicsThermodynamics is the study of heat and its relation with other forms of energy(mecghanical, electrical, chemical). In thermodynamics there are four fundamental laws. The efficiency in thermodynamics is defined in terms of heat supplied and the amount of work produeced.
Collision time for two gravitating bodiesWhen two bodies comes closer to each other due to the attractive nature of gravity, there is a certain interval in which they will collide. We should be able to calculuate such collision time using integration and equation of motions in Newtonian framework. This can also be termed as two body problem of gravity.
Some useful parameters are r = distance of one body relative to the other, velocity v = dr/dt , The first quantity needed is the instantaneous force acting between the two bodies m1, m2 ,
Graviational accelerationGravitational acceleration is the instantaneous change of velocity when a body falls freely under the influence of gravity. It has similar units like acceleration in physics. We can find an equation for the graviational acceleration g on the surface of the earth as follows.
This acceleration is not constant and is a function of distance r. So it is appoximately constant in small distance above the surface of the earth and equal to 9.8 ms^-2. The change is the magnitude as a consequence of the change in displacement can be calculuated using calculus.
Variation of gravitational acceleration due to lattitudeThe angular velocity of the earth creates centrifugal force on its surface. So it acts on us and try to neutralise gravity. Suppose a person is standing on the surface of the earth at the lattitude φ
if N is the support's reactional force then
ω is the earth's rotational velovity. So the equaivalent gravitational acceleration g(eq) will be g(eq) = g - ω^2R
Wave PacketWave packet is a superposition of many waves. It can be expressed as an integral over continuous distribution of all wavelengths.
Wave packet plays a vital role in deducing Heisenberg's uncertainty principle. Heisenbeg's uncertainty principle states that you can not measure both position and momentum at the same time precisely.
On the other hand De Broglie hypotheses says that the circumference of the principle orbit is always equal to the integer multiple of wavelength associated with the electron.
Bohr model of atom In Bohr's model of atom the planetary electrons revolve around the nucleus in some predetermined or fixed orbits. These orbits are represented by integer n and have a fixed radius.
The expression for total energy is
The most interesting matter about the atom is that we can not observe them with our bare eyes. Yet these models are accurate and phenomena described by these model is occuring inside it everytime.
The energy of the radiated energy in Borh's theory is expressed as the difference of two numbers.
Bohr's theory of atom was the old quantum theory. Although it was successful in explaining a lot of atom, it has limitations. It gives energy levels of electron inside atom.
The relation between the radius of nth orbital and that of the first orbital is
From the analysis of hydrogen atom the states of atom with atomic number Z can be generalized according to Borh's model
A comparison can be made between Bohr's model and solution of Shrodinger's equation
Sommerfield extended Bohr's model to include elliptical orbits. In terms of elliptical orbit some anomalous behavior of large atom was explained.
The lengths of those ellipses's semi-major and semi-minor axis can be calculated from generalized quantum principle.
This is non-relativistic calculation. This can be further modified to include relativistic effects. The relativistic model has main two concepts as follows:
Schrodinger's equation solves hydrogen atom for 2p orbital
Bohr's model of atom successfully prdicted atomic excitation.
Fermi's golden rule of weak interation or beta decay explains the beta decay inside the nucleus of certain atoms. In a beta decay an electron and a neutrino are emitted. The beta decay can be explained by treating beta decay as the transition between initial and final states. This transtion depend upon the coupling strength between initial and final states. The golden rules is the given by the equation
But the nature of beta decay is correctly explained by the distribution of momentum of electron as follows :
To explain what the matrix element M(if) is, it is better to mention another mathematical expression
Or it can be shown to be an integral with wave function and operator V.
Writing from Newton's PrincipiaNewton invented a method to find area under a curve by splitting the area into large number of rectangles.
Later this process was known as integration of calculus. Rutherford first successfully executed scattering process with atomic nucleus. His scattering process can be expressed with the equation below.
Differential cross section for the Rutherford scattering is
Similarly electromagnetic structure of hadron can be found.
Chemistry as a branch of physicsChemistry deals of various elements and their interaction. On a fundamental level chemistry can be regarded as a branch of physics and so is biology. Chemists study properties of various chemicals that took part in chemical reaction. In chemical reactions various elements are transformed into other compounds or other elements. In the process total amount of mass is conserved. This is the basic principle of chemistry. However we discuss some basics thoeries of gases.
First there is Boyel's law. It states that if temperature and number of gas particles are constant then the product of pressure and volume remain constant. It can be put in a mathematical form as
So it is apparent that axiom of linearity presuppose the axiom of Archemedes.
In any triangle there are few theorems associated with it. These can be stated in simple terms
Atomic radiationAtomic radiation can be described by complex wave ψ .
This is actually oscillation charge cloud of electric dipole.
The time domain information can be converted by frequency domain information by mere fourier transform method. For example atomic decay obeys the exponential or logarithm curve and in frequency domain it becomes bell shaped curve.
Single and many electrons atom have different hamiltonian respectively.
Central potential in Hydrogen:
separation of y into radial and angular functions:
Electron – Electron interaction term:
Hamiltonian for Central Field Approximation
H1 = residual electrostatic interaction.
For perturbative approach it is assumed that H1<< H_0
Zero order Schrödinger Equation is H_0ψ = E_0ψ H0 is spherically symmetric so equation is separable -
What form does central Field U(r) take:
Energy Eigenvalue of hydroogen is
Renormalization is needed sometimes
electron angular momentum orbital can be graphed
Residual electrostatic interaction is found after substracting the electron-nucleus attraction from the electron-electron repultion
Larmor PrecessionMagnetic field B exerts a torque on magnetic moment m causing precession of m and the associated angular momentum vector l The additional angular velocity w` changes the angular velocity and hence energy of the orbiting/spinning charge
∇E = μ.B
Spin-Orbit interaction: Summary
Perturbation energy is
Radial inregral is
Perturbation theory with degenerate statesPerturbation Energy:
Change in wavefunction: So won’t work if Ei = Ej. i.e. degenerate states.
We need a diagonal perturbation matrix, i.e. off-diagonal elements are zero
<ψ|H`|ψ> = 0 New wavefunction
Various quantum numbers and its diagrams
Belief in general lawsThroughout the discussion of perception and the physical object we have assumed the validity of general laws. This is always assumed in scientific practice, but the reasons for assuming it are not very clear. Although the subject is not one on which it is easy to say anything definite, yet it seems necessary to examine it.
Like other scientific postulates, the belief in general laws is rooted in the properties of nervous tissue-the same propertis which make us believe in induction and enable us to learn from experiences. This origin, of course, affords us to warrant for the truth of the belief , but equally gives no reason against it. Indeed, so far as it goes, it affords a slight presumption in favour of the view that a great many events are in accordance with general laws, since it shows that animals which act in a way which the truth of the belief would render rational can survive. I should not wish , however, to lay stress upon such an argument. When we first begin to think. we find ourselves acting in certain ways which seem to succeed and we set to work to rationalize our behaviour. The natural way to do this is to say:
Pure mathematicsIMPLICATION AND FORMAL IMPLICATION
In the preceding chapter I endeavoured to present, briefly and
uncritically, all the data, in the shape of formally fundamental ideas and
propositions, that pure mathematics requires. In subsequent Parts I shall show
that these are all the data by giving definitions of the various mathematical
concepts—number, infinity, continuity, the various spaces of geometry and
motion. In the remainder of Part I, I shall give indications, as best I can, of the
philosophical problems arising in the analysis of the data, and of the directions
in which I imagine these problems to be probably soluble. Some logical
notions will be elicited which, though they seem quite fundamental to
logic, are not commonly discussed in works on the subject; and thus problems
no longer clothed in mathematical symbolism will be presented for the
consideration of philosophical logicians.
Two kinds of implication, the material and the formal, were found to be
essential to every kind of deduction. In the present chapter I wish to examine
and distinguish these two kinds, and to discuss some methods of attempting
to analyse the second of them.
In the discussion of inference, it is common to permit the intrusion of a psychological element, and to consider our acquisition of new knowledge by its means. But it is plain that where we validly infer one proposition from another, we do so in virtue of a relation which holds between the two propositions whether we perceive it or not: the mind, in fact, is as purely receptive in inference as common sense supposes it to be in perception of sensible objects. The relation in virtue of which it is possible for us validly to infer is what I call material implication. We have already seen that it would be a vicious circle to define this relation as meaning that if one proposition is
true, then another is true, for if and then already involve implication. The relation holds, in fact, when it does hold, without any reference to the truth or falsehood of the propositions involved. But in developing the consequences of our assumptions as to implication, we were led to conclusions which do not by any means agree with what is commonly held concerning implication, for we found that any false proposition implies every proposition and any true proposition is implied by every proposition. Thus propositions are formally like a set of lengths each of which is one inch or two, and implication is like the relation “equal to or less than” among such lengths. It would certainly not be commonly maintained that “2 + 2 = 4” can be deduced from “Socrates is a man”, or that both are implied by “Socrates is a triangle”. But the reluctance to admit such implications is chiefly due, I think, to preoccupation with formal implication, which is a much more familiar notion, and is really before the mind, as a rule, even where material implication is what is explicitly mentioned. In inferences from “Socrates is a man”, it is customary not to consider the philosopher who vexed the Athenians, but to regard Socrates merely as a symbol, capable of being replaced by any other man; and only a vulgar prejudice in favour of true propositions stands in the way of replacing Socrates by a number, a table or a plum-pudding. Nevertheless, wherever, as in Euclid, one particular proposition is deduced from another, material implication is involved, though as a rule the material implication may be regarded as a particular instance of some formal implication, obtained by giving some constant value to the variable or variables involved in the said formal implication. And although, while relations are still regarded with the awe caused by unfamiliarity, it is natural to doubt whether any such relation as implication is to be found, yet, in virtue of the general principles laid down in Section C of the preceding chapter, there must be a relation holding between nothing except propositions, and holding between any two propositions of which either the first is false or the second true. Of the various equivalent relations satisfying these conditions, one is to be called implication, and if such a notion seems unfamiliar, that does not suffice to prove that it is illusory.
Calculating fourier series first termThe first constant term in the fourier series can be computed easily using integration
Axiom of linearity in mathematicsThis axiom is probably used in mathematics, especially in real analysis. The axiom of linearity states that there is always an integer n such that a magnitude x can be divided into n equal parts. So x = n (l) whatever l may be .
The axiom of Archimedes , on the other hand, states that there are two magnitudes x an y such that for an integer n we always have nx > y
Law of thermodynamics
A briefer history of time by S. Hawking
A brief history of time by S. Hawking
Grand Design by Stephen Hawking
perihelion of mercury by Feynman