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"Everything has got an equation"


Albert Einstein popularizes the name 'Maxwell's Equations'. Albert Einstein said: "The special theory of relativity owes its origins to Maxwell's equations of the electromagnetic field. Einstein also said: Since Maxwell’s time, physical reality has been thought of as represented by continuous fields, and not capable of any mechanical interpretation. This change in the conception of reality is the most profound and the most fruitful that physics has experienced since the time of Newton."

Maxwell equations explained

Maxwell equations created a great revolution in physics. But it was not totally advantageous as it might sound. It could not explain the ultraviolate catastrophy which means that a heated object radiates infinte amount of energy at thermal equilibrium. This was a theoretical loophole of electromagnetism. Planck later gave right solution to avoid the energy problem and revolutionized physics in another direction.


pre-relativity physics

The physics of Newton, considered as deductive system, had a perfection which is absent from the physics of the present day. Science has two purposes, each of which tends to conflict with the other. On the one hand, there is a desire to know as much as possible of the facts in the region concerned; on the other hand, there is attempt to embrace all the known facts in the smallest possible number of general laws. The law of gravitation accounted for all the facts about the motions of the planets and their satellites which were known in Newton's day; at the time, it exhibited the ideal of science. But facts and theories seem destined to conflict sooner or later. When this happens , there is a tendency either to deny the facts or to despair of theory. Thanks to Einstein, the minutes facts which have been found incompatible with natural philosophy of Newton have been fitted into a new natural philosophy; but there is not yet the complete theoretical harmony that existed while Newton was disputed.
A serious change was introduced by Faraday and Maxwell. Light had never been treated on the analogy of gravitation, but electrically appeared to consist of central forces varying inversely as the square of the distance , and therefore confidently fitted into the Newtonian scheme. Faraday experimentally and Maxwell theoretically displayed the inadequacy of this view; Maxwell, moreover, demonstrated the identity of light and electromagnetism. The aether required for two kinds of phenomena was therefore the same, which gave it a much better claim to be supposed to exist. Maxwell proof , it is true, was not conclusive , but it was made so by Hertz when he produced electromagnetic waves artificially and studied their properties experimentally. It was clear that Maxwell equations, whic parctically contained whole of his system, besides the theory of relativity accounted as affording explanation for vast range of phenomena.
Suppose you are walking or travelling inside your car with your cellphone. Suddenly your phone rings up and you receive the call. How the hell did it happen? Maxwell equations has carried the information from another cellphone of your acquaintance. There is telecommunication engineering involved here also. The important factor is the electromagnetic wave. But when you receive the call, is there a breaking of the flow of physical laws? Your brain is also interpreting the same information sending from your friend's cellphone. We are also connected to the same physical universe where Maxwell equations apply. This brings us the interconnectedness of all things in the universe. You are connected with everything. Universe is an inseparable whole. Fritz of Capra has explained this fact in his book "the tao of physics". Space, time , matter , energy , stars and subatomic particles all have deep connection with themselves. Physical laws bears this truth. But where is the theory of everything? It is definitely string theory.

"Gentlemen, you had my curiosity , now you have my attention"
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Fields and waves

Fields and waves are important concepts in both physics and engineering. In physics, field is a physical quantity that has a value or a set of values at each point of space or space-time. So field is a function of coordinates. Scalar field is single-valued function and vector field is multi-valued function. For example we can think of gravitational field which exerts a force on every object placed in space. This is valid in newton's framework and the field has a single value at each point of space. In General relativity , gravitational field at each point in space-time has ten quantities.This gravitational field is called metric tensor. The apparent equivalence between metric tensor and gravitaional field had brought a revolutioin in physics, which we will discuss when explaining theory of general relativity. Electric field and magnetic field are vector quantities themselves. Electric field is the rate of change of potential with respect to position.


maxwell equations explained

Electric field is the force which a unit change experiences. It can be defined using electric force lines that permeate some region of space. Electromagnetic field is a combination of electric field and magnetic field. In fact it is the cross product of those . So the direction of propagation is transverse to both. Waves are disturbances in some medium, which carry energy through that medium. Sound and light are transmitted as a form of wave. Sound wave is a longitudinal wave whereas light is a transverse wave. In case of longitudinal wave the direction of the propagation is parallel to the vibration in the medium. In case of transverse wave the direction of propagation is perpendicular to the vibration in the medium. Propagation of light is transverse to the plane of electric and magnetic field. Electric field creates magnetic field and pushes it. Magnetic field, in turns, creates electric field and pushes it. By their mutual interaction a disembodied wave is formed and it travels at the speed of light. For mathematical analysis , we will discuss the electromagnetic wave theory which was developed by James Clerk Maxwell . His theory of electromagnetism revolutionized the field of physics and engineering.The four maxwell's equations related to electric and magnetic field are

maxwell equations explained

Maxwell's equations are a perfect embodiment (example) of differential laws. Differential laws are very powerful and essential tools for describing continuous processes that occur in nature. Differential laws must be described by differential equation. In quantum mechanics where the processes are discrete such laws seems scarcely satisfactory.
The four equations ( in either differential or integral form) are coupled by the electric field and magnetic field (E, B). So They are not independent as changing electric and magnetic field are dependable with each other.

Maxwell actually compactified the equation of electromagnetism that a scientist named Oliver Heaveside developed. There were a total of twenty four equations in the original theory of electromagnetism. These are :
Heaviside equations explained

From the first equation we can easily say that f, g and h are functions of three coordinate variables x,y and z. Here Heaviside made analogy with e and the charge density ρ in Maxwell's equations.
In the second set of equations equivalent of Gauss's law for magnetism has been formulated. Here , again F , G and H has been given as a function of three coordinate variables. Here we are considering static fields which are not functions of time t.
Next set of equations are the equations of Faraday's law of electromagnetism. Here we need time derivative of all the functions ψ, F , G and H.

Divergence and Curl

In the figure above maxwell's equations are written in both differential an integral form. Differential form and integral form are known as microscopic and macroscopic form also. There are four equations that perfectly describes the nature of electromagnetic waves and their propagation. Four maxwell's equations must not be thought independent rather they are coupled. The first equation is basically the Gauss's law of electric charge. It says that the total outward flux throughout a closed surface is equal to the total charge enclosed by the surface multiplied by a constant. Flux is any physical quantity that flows through a medium or surface and is defined as the product of any field and component of an area that the field intersect perpendicularly. As an example, the electrical flux is the electric field multiplied by a small planar area perpendicular to field lines. When the area is not planar then evaluation of the flux requires area or surface integral to be taken over the area as the angle will be always changing. Some examples of flux are electric and magnetic flux. Flux can be vector quantity also. And example of flux integral can be seen when we derive Gauss's law from Culomb's law

Gauss's law

Some charge q is assumed to exist at a point at space and we want to find the flux that it creates around it. So we can imagine a spherical surface around the charge and take surface integral. The surface is curved (Gaussian) so we are bound to take surface integral. As the angle is zero between infinitesimal area ds and E , the surface integral get simplified as given in the second equation. It is just the integral of (Ecos(0)dA= EdA). Finally we can take E out of integral as E does not depends on ds. We know that total surface area of sphere of radius r is 4(3.14)r(squared). Thus we can arrive at the equation for flux , which corresponds to Gauss's original law.

Definitions of μ0 and ε0

Ampere uncovered that two parallel wires carrying electric currents in the same direction attract each other magnetically, the force in newtons per unit length being given by F=2(μ04π)(I1)(I2)/r,
for long wires a distance r apart. We are using the standard modern units (SI). The constant μ0/4π that appears here is exactly 10^(-7), this defines our present unit of current, the ampere(A). To repeat: μ0/4π is not something to measure experimentally, it's just a funny way of writing the number 10^(-7) That's not quite fair ; it has dimensions to ensure that both sides of the above equation have the same dimensionality. (Of course, there's a historical reason for this strange convention, as we shall see later). Anyway, if we bear in mind that dimensions have been taken care of, and just write the equation
F= 2X10^(-7)I1XI2/r
it's clear that this defines the unit current(one ampere) as that current in a long straight wire which exerts a magnetic force of 2X10^(−7) newtons per meter of wire on a parallel wire one meter away carrying the same current.


The second maxwell's equation says that changing magnetic field creates electric field. If the magnetic field were not changing the electric field would not have been created. The differentiation of a constant function is zero. But this formula tells something more: the curl of electric field is the time rate of change of magnetic field.
The third maxwell's equation relates magnetic field with its divergence. It is apparent from the equation that the divergence of the magnetic field is zero. From this we can infer there is no isolated magnetic mono pole. The magnetic field lines always close on in itself. This is hard to explain without mathematics but it is apparent that some properties are fundamentally different in case of magnetic field.

The fourth formula is actually the modified version of Ampere's circuit law. Maxwell added an extra term to accommodate for the displacement current, which is the integral of time rate of change of electric field, multiplied by constant term. We can visualize this displacement current total charge that is deposited on the capacitor plate. It is not actual current but kind of electric field as no current flows through two capacitor plates.


Problem with Ampere's Law

A simple example to see that Ampere's Law must be faulty in the general case is given by Feynman in his Lectures in Physics. Suppose we use a hypodermic needle to be inserted a spherically symmetric blob of charge in the middle of a large vat of solidified jello (which we assume conducts electricity). Because of electrostatic repulsion, the charge will dissipate, currents will flow outwards in a spherically symmetric fasion. Question: does this outward-moving current distribution create a magnetic field? The answer must be no , because since we have a completely spherically symmetric case, it could only generate a spherically symmetric magnetic field. But the only possible such fields are one pointing outwards everywhere and one pointing inwards everywhere, both corresponding to non-existent monopoles. So, there can be no magnetic field.

ampere circuit law

However, imagine we now consider checking Ampere's law by taking as a path a horizontal circle with its center above the point where we injected the charge (think of a halo above someone’s head.) Surely, the left hand side of Ampere's equation is zero, since there can be no magnetic field's existence. (It would have to be spherically symmetric, meaning radial.) On the other hand, the right hand side is most definitely not zero, since some of the outward flowing current is going to go through our circle. So the equation must be erroneous.
Ampere's law was established as the result of large numbers of careful experiments on all kinds of current distributions. So how could it be that something of the kind we discussed above was overlooked? The reason is really similar to why electromagnetic induction was missed for so long. No-one thought about looking at dynamic fields, all the experiments were done on steady situations. With our ball of charge spreading outward in the jello, there is surely a changing electric field. Imagine yourself in the jello near where the charge was injected: at first, you would feel a strong field from the nearby concentrated charge, but as the charge spreads out spherically, some of it going past you, the field will weaken with time.

Maxwell's reasoning

Maxwell himself gave a more practical example: consider Ampere's law for the usual infinitely long wire carrying a steady current I, but now break the wire at some point and put in two large circular metal plates, a capacitor, maintaining the steady current I in the wire everywhere else, so that charge is simply accumulating on one of the plates and draining off the other.
Looking now at the wire some distance away from the plates, the situation appears normal, and if we put the usual circular path around the wire, application of Ampere's law tells us that the magnetic field at distance r, from


ampere circuit law

Recall, however, that we defined the current threading the path in terms of current punching through a soap film spanning the path, and said this was independent of whether the soap film was flat, bulging out on one side, or whatever. With a single infinite wire, there was no escape— no twisting of this covering surface could wriggle free of the wire going through it (actually, if you distort the surface enough, the wire could penetrate it several times, but you have to count the net flow across the surface, and the new penetrations would come in pairs with the current crossing the surface in opposite directions, so they would cancel).


ampere circuit law

Once we bring in Maxwell's parallel plate capacitor, however, there is a way to deform the surface so that no current penetrates it at all: we can run it between the plates!
The question then arises: can we salvage Ampere's law by adding another term just as the electrostatic version of the third equation was rescued by adding Faraday's induction term? The answer is of course yes: although there is no current crossing the surface if we put it between the capacitor plates, there is definitely a changing electric field , because the capacitor is charging up as the current I flows in. Assuming the plates are close together, we can take all the electric field lines from the charge q on one plate to flow across to the other plate, so the total electric flux across the surface between the plates,


ampere circuit law
We finally arrive at maxwell's fourth equation.

Finally putting everything together we get :

maxwell equation

An analysis or study of vector calculus is necessary for the full treatment of Maxwell's equations. The divergence and curl used in the equations are simply measure of spacial variation of vector field. Divergence finds any source or sink from which something flows out. Divergence is calculated by finding dot product of any vector with an operator usually denoted by del(∇). So it is a scalar quantity. It can be expressed in any coordinate system and in Cartesian coordinate it has the simplest expression. In similar way curl operator can be expressed in any coordinate system. All are equivalent. Curl operation on vector field gives the infinitesimal rotation of that vector field in three dimensional euclidean space. Thus the curl of any vector is also a vector quantity.

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Consequences

The coupling of those equations gives rise to wave equation. The wave represented by the wave equation is the electromagnetic wave which is what we know as light or other electromagnetic radiation. Mathematician Jean-Baptiste le Rond d'Alembert first found the solution to wave equation. The wave equation is also called thed'Alembert's equation. That the solution to Maxwell's equations is electromagnetic wave should make sense because a wave is itself a function. We always expect solution to differential equation to be some kind of function. This wave is a function of space and time.

wave equation in 3D
Fig: wave equation

We first , as given in the figure, take curl of Maxwell's second equation. In the second equation we substitute B by H with a constant. Then using slightly modified form of Maxwell's fourth equation we get the penultimate(just before the last ) equation. Assuming constant sigma(σ) to be zero in free space we get the last equation which is the wave equation for electromagnetic radiation. The last equation is the wave equation for electric field. Similarly we can derive another wave equation for magnetic field. This is the wave equation in three dimensional euclidean space. We can find out the speed of the wave from the very equation. This wave equation is the standard form of all kind of transverse waves that exist in the physical world. Later we will use the same equation describing water waves, gravitational waves and others.

Wave is unchanging shape that travel across space. To be theoretic, wave is a function of space and time. But this is not to be confused with four dimensional space-time which is a continuum of events. The four equation of Maxwell are partial differential equations and they are linear. More discussion about the partial differential equation and their classifications deserve another mathematical treatise(essay). The physical world required for electromagnetic theory is three-dimensional but the theory is consistent with the theory of relativity . The four coupled equations of Maxwell can be written in tonsorial form which reduces the four equation to just two. electromagnetic field tensor captures all the information about the electromagnetic force. One of the two tensor equations says something like this:

Divergence of four dimensional electromagnetic field tensor in empty space is zero. ( the value of J is zero is empty space or vacuum).
There is a maxim like this : "the nature lay hidden and God says let there be light. After that there was light".
electromagnetic field tensor

The last equation in the figure above, is the divergence of electromagnetic field tensor, which is identically zero in empty space. Although the derivation is cumbersome but this is the compact and simplified version of Maxwell's equations. The equation is derived using Euler-Langrange equation which involves Langrangian (L). The Langrangian for electromagnetic field is given in the first equation. Lagrangian of a system is difference of the kinetic energy and potential energy. A system of equations involving Lagrangian is a more elegant and rigorous formulation of classical mechanics. It is widely known as Lagrangian mechanics. Other tensor equation of electromagnetism is :



electromagnetic field tensor

The electromagnetic field tensor is :
electromagnetic field tensor

Summation of all covariant derivatives of electromagnetic field tensor is zero.

Tensor is a geometric objects that shows relationship between coordinate frames , coordinates and a transformation rule between them. Vectors and scalars are also tensors. Scalars are tensors of rank zero and vectors are tensors of rank one. Before the advent of theory of relativity space and time were considered independent and absolute. Classical world is four dimensional but not the one needed for electromagnetism. The theory of relativity is still a classical theory whereas quantum theory is a branch of modern physics. Although these theories are the two main strands of physics, electrical engineering can go without those as long as the application is concerned.

Maxwell's equation in other formalism

Maxwell's equations can be expressed in curved spacetime also. It can be expressed using differential forms and exterior algebra too. In vector calculus the fomulation is quite similar:
maxwell equations

In Tensor calculus the equations are as follow :
maxwell equations


maxwell equations

In differential form the equations are :
maxwell's equations

maxwell equations and science

Maxwell equations were a triumph in theoretical physics history. It was the explanation of one of the four fundamental forces of nature aka electromagnetic force. The apparent simplicity of nature was expressed in mathematical language. A set of four simple equations describe the nature of electromagnetic force completely. But it was a classical theory and force was explained in terms of field lines. Later quantum mechanics as as developed to explain electromagnetism as an exchange of photon between two electric charges. The other two forces strong and weak nuclear forces can also be explained in terms of exchange of particles. To explain gravity hypothetical particle graviton has been proposed.
What we call seeing is a complicated process involving light. The light wave emited by the sun's surface travel through the intervening space before reaching our eye. The electrical impulses created in our retina go to the brain and create the picture of the sun. The first part of the process is related to physiology while the second part is related to psychology. Without troubling ourselves about what the physicist has to say is destructive of the common-sense noton of "seeing". It makes no difference, in this matter, which of the possible theories we adopy as to the physical character of light, since all equally make it something utterly different from what we see. The data of sight, clearly analyzed , resolves it inot color shapes. The physical counterpart of color is a periodic process of a certain frequency relative to the eye of the observer. Correspondence between colors and their counterpart is peculiar: colors are qualities whereas their counterpart is periodic process , which are in between our eye and the object which we see. Now when an object shines by its own light , according to Bohr's theory, electron jumps from one orbit to another. This is very unlike a sensation of (red) say. And what looks to the eye like a continuous red surface is supposed to be really a volume whose apparent color is due to the fact that some of the electrons in it are jumping in a certain way. Wghen we say "jumping" we are saying something too pictorial. What we mean is that they possess an unkwown quality called "energy" which is a known function of a certain number of small integers. Seeing would not be possible without Maxwell equations too.
What is called frequency of light wave is its frequency with respect to axes fixed relatively to the emitting body. Its frequency relative to axes travelling with it is zero; this is only the extreme of doppler effect. There is perhaps certain inconsistency in the practice of studying bodies by means of axes which move with them., while light is always treated with reference to matrial axes. If we want to understand light in itself, not inits relation to matter, we ought to let our axes travel with it. In that case , its periodicity is spatial, not temporal; it is like that of corrugated iron.


maxwell's equations

From the standpoint of the light itself, each part of a light wave is a steady event in the sense defined above. Though light wave is treated with reference to material axes its speed does not depend it. No matter which axes you choose the maxwell equations will remain invariant. That is consequence of the postulates of theory of special relativity.
The importance of maxwell equations can be seen in phenomena inside atom too. First there is rythmic process , which means a recurring cyclke of events, in which there is a qualitative similarity between corresponding members of different periods ( first period, second period and so on). A rythm may have a period consisting of finite number of events or one consisting of an infinite number; it may be discrete or continuous. If it is discrete , the proper time of one period is measured by the number of events in the period and the frequency of the process is reciprocal of this number.
If we adhere to Bohr's theory , what can be supposed to be happening inside an atom? If relative motion were all that was taking place, we should have either to find an interpretation for the spinnnig electron, or elseto say that , taking axes fixed relatively to any large body, the line joining the electron and the proton rotates rapidly; any large body will do since none rotates with an angular velocity comparable to that of the electron. But why should the electron be interested in this fact? Why should its capacity to emit light be connected with it? There must be something happening where the electron is, if the process is to be intelligible. This brings us back to Maxwell's equations, as govering what is occurring in the medium. And there must be a rythmic character in the events occurring where the electron is, if we are to avoid all the troubles of "ACTION AT A DISTANCE". We suppose therefore, that throughout an electromagnetic field there are events whose formal properties we know more or less, and they , not the spatial configurations, are the immediate causes of what takes place.

Matter and space

Common sense starts with the notion that there is mattter where we can get sensation of touch, but not elsewhere. Then it gets puzzles by winds, breath, clouds and etc., whence it is led to the conception of " spirit" has been replaced by "gas" there is a further stage, that of aether. Assuming the continuity of the physical process, there must be happening something between the earth and the sun when light travels from the sun to the earth; assuming the midieaval metaphysic of "substance" as all physicists did until recently, what is happening between the earth and the sun must be happening "in" or "to" a substance, which is called aether.
Apart from metaphysical interpretation, what we may be said is that processes occur where there is no gross matter (taking the word literally) and these processes travel, at least approximately , according to maxwell equations. There does not seem any necessity to interpret these processes in terms of substance; indeed , I shall argue that processes associated with gross matter should also be interpreted so as not to involve substance. There must, however, remain a difference , expressible in physical terms, between regions where there is matter and other regions. In fact, we know the difference. The law of gravitation is different, and the laws of electromagnetism suffers a discontinuity when we reach the surface of an electron or proton. These difference , however , are not of a metaphysical kind. To the philosopher, the difference between "matter" and "empty space" is , I believe, merely a difference as to the causal laws governing successions of events, not a difference expressible as that between the absence of substance, or as that between one kind of substance and another.
I now come to the difficulties of reconciling the laws governing the propagation of light with those governing interchanges of energy between light and atoms. On this subject the present position of physics is one of perplexity, aptly summarized by Dr Jeans in Atomicity and Quanta. The wave theory of light accounts adequately for all phenomena in which only light is concerned, such as inrterference and diffraction; but it fails to account for the quantm phenomena seem unable to account for the very things which the wave theory explains perfectly.
After setting forth the difficulties encountered by wave-theory in regard to interference and diffraction , Dr Ellis proceeds to the very interesting suggestion made by Professor G. N Lewwis in "nature" "it is a striking fact" says Dr Ellis, summarizing this suggestion, "that while all the theories are directed towards explaining the propagation of light, one theory suggesting that it occurs in the form of waves, the other in the form of corpuscles, yet light has never been observed in empty space. It is quite impossible to observe light in course of propagation; the only events that can ever be detected are the absorbtion and emission. Until there is some atom to absorb the radiation we must be unaware of its existence. In other words , the difficulty of explaining the propagation of light may be because we are endeavouring to explain something about which we have no experimental evidence. It might be more correct to interpret the experimental facts quite directly and to say that one atom can transfer energy to another atom although they may be far apart, in a manner analogous to the transference of energy between two atoms which collide.

"Remember , All I am offering is the truth nothing more.."

Professor's Lewis theory suggests that we should take seriously the fact that interval between two events that light passes through is zero. So the point of departure and point of arrival may be regarded, in some sense, in contact. In a passage he seems to have qouted :
"I shall make the contrary assumption that an atom never emits light except to another atom, and that in this process, which may rather be called a transmission than an emission, the atom which loses energy and the atom which gains energy play co-ordinate and symmetric parts."
Professor Lewis also suggested that light is carried out by corpuscles of new sort, which he calls "photons". He supposes that when light radiates, what happens is that photons travel; but at other time photon is a structural element in an atom. The photon, he says, "is not light , but plays an essential role in every radiation.". He assigns to the photon the following properties : " 1. In any isolated system the total number of photon is constant. (2) All radiant energy is carried out by photons,the only difference between the radiation from wireless tv station and from an X-ray tube being that the formeremits a vastly greater number of photons , each carrying a very much smaller amount of energy. 3) all photons are intrinsically similar 4) the energy of an isolated photon divided by planck constant gives the frequency of the photon 5) all photons are alike in one property which has the dimension of action or of angular momentum, and is invarant to a relativity transformation. 6) the condition that the frequency of the photon being emitted by the system is equal to some physical frequency existing within the system, is not in general fulfilled , but comes nearer to the fulfilment the lower the frequency is" professor Lewiss promises to deal with the difficulties in the way of his hypothesis on a future occasion.

"It is just a game, just a game, we are just players. So try to give your best.."

Professor Lewi's view is perhaps less radical than the view which it suggests- namely, that nothing whatever happens between emission of light from one atom and absorption of light by another atom. Whether this view is Professor Lewis's or not , it deserves to be considered, for although it is revolutionary, it may well prove to be right. If so, "empty space" is practically abolished. There will be need of a considerable labour if physics is to re-written in accordance with this theory, but what is said about the absence of evidence concerning light in transit is a powerful consideration.

On differential laws

The differential laws are the same as the causal laws. The view of causal law is absent from quantum theory, from the ideas of savages and uneducated persons, and from the works of philosophers , including Bergson, and J. S Mill. In quantum theory , we have a discrete series of possible sudden changes and a certain statistical knowledge of the proportion of cases in which each possibility is realized; but we have no knowledge as what determine the occurances of a particular change in a particular case. Morever, the change is not of the sort that can be expressed by differential equations; it is a change from one integer or a set of integers to a state expressed by another. This kind of change may turn out to be physically ultimate, and to mark out at a part of physics as governed by laws of a new kind. But we are nor likely to find science returning to the crude form of causality believed in by Fijians and phiosophers, of which the type is "lightning causes thunder". It can never be a law that fiven A at one time, there is sure to be B at another time, because something might intervene to prevent B. We do not derive such laws from quantum phenomena, because we do not , in their case , know that A will not continue throughout the time in question. The natural view to make at present is that quantum phenomena have to do with the interchange of energy between matter and surrounding medium, while continuous change is found in all processes which involve no such interchange. There are , however , difficulties in any view at present, and it is not for a layman to venture an opinion. It seems not improbable that , as Heisenberg suggests , our views of space-time may have to be modified profoundly before harmony is achieved between quantum phenomena and laws of transmission of light in vacuum. For the moment, however, I wish to confine myself to the standpoint of relativity theory.
Wherever mathematics works in a continuous medium with relations which may be loosely described as next-to-next, there must be other relations, holding between points at finite distance from each other, and having next-to-next relations as their limits. Thus, when we say that laws have to expressed by differential equations, we are saying that finite relations which occur cannot be brought under accurate laws, but only their limits as the physical realities; on the contrary, the physical realities continue to be the finite relations. And if our theory is to be adequate , some way must be found of so defining the finite relations as to make the passage to the limit possible.

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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. 38. At this point, it is necessary to consider a very difficult logical problem, namely, the distinction between a proposition actually asserted, and a proposition considered merely as a complex concept. One of our indemonstrable principles was, it will be remembered, that if the hypothesis in an implication

Producing electromagnetic waves

Herzt first produced electromagnetic waves artifially. He had the following setup. How is the waves produced? the first necessary condition is that charge needs to be accelerated.
producing electromagnetic wave

Five ways to think like a mathematician

A real mathematician develops their own examples, whether standard examples, extreme examples or non-examples! Let’s look at worked examples (i.e., examples of processes, algorithms, etc.). Consider the standard example of maxima and minima of functions in the study of calculus. We define first how a function is to be differentiated. Then singular points are defined as points where the derivative becomes zero. Next we are told that there are 3 types of singular extemum point: maxima, minima and inflection. It is then shown that the second derivative of the function determines the type. After this examples are shown: here’s a function, here is where it has singular points, this is the type of singular point. The process is simple, differentiate f , solve f`(x) = 0, differentiate to get f``(x) and use the sign of f``(x) to find the type of singular point. This is the standard process of using worked examples. If you learn the method, then given a function you can easily find the maxima and minima. But what if I reverse engineer it and ask you to create a function f of the variable x with a maximum at x = 2 and with a minimum at x = 􀀀6 . This is a far greater test of understanding. It is a lot harder. But in attempting to do it you can learn a lot of mathematics.
So, given a method for worked examples you should reverse it to create new problems. Furthermore, if you create these problems with your friends, then you can exchange them (the problems, not the friends) and get even more practice. You can also set a competition: see who can set the hardest - yet manageable - problem.
` is here having the meaning of differentiation. 'What about the converse?'

Statements of the form A =) B are at the heart of mathematics. We can also state this as ‘If A is true, then B is true.’ The converse of the statement ‘A =) B’ is ‘B =) A.’ For example, the converse of ‘If I am Winston Churchill, then I am English’ is ‘If I am English, then I am Winston Churchill.’ This simple example illustrates that even if a particular statement is true, then its converse need not true. It may be true or it may not be true. Investigation is required before we can say. A good mathematician, when presented with an A implies B type statement, will ask ‘Is the converse true?’ Internalize this question and make it part of your tool kit for doing mathematics. Whether the converse is true or not is not too important, the point is that the exercise sharpens mathematical ability. [By the way, as an aside, a big mistake people make when A => B is that they think that if A is not true, then B is not true either. That’s not right, the statement only says what happens when A is true. It says nothing about what happens when A is false. Now think like a mathematician and give an example!]

Magnetic Field in Vacuum

The Earth’s magnetic field and the magnetism of some natural ores or iron rods that have been stroked by a magnet, have been known in the Middle East and China since antiquity. In 1821 Oersted discovered that an electric current produces a magnetic field. This effect was studied by Ampère, Biot, Savart, and others. Ampère assumed that permanent magnetism is due to microscopic currents in matter; this idea is retained in modern physics. Conversely, Faraday discovered in 1831 that a variable magnetic field induces an electric current in circuits. In 1888, Maxwell unified electricity and magnetism in a single theory, called electromagnetism. Currently, magnetism has many technological applications: magnets and electromagnets are used in generators and motors, instruments, computers, telecommunications, etc. In this chapter, we introduce the concept of magnetic field and we study its action on magnetic currents. Then we study the creation of magnetic fields by moving charges and currents, magnetic energy and the interactions of circuits.
The magnetic field is defined by its action on a charged particle in motion (Figure 6.1a in the case of positive charge).
F = qvXB
B is the magnetic field or, more precisely, the magnetic induction field. The magnetic force FM vanishes if the particle is at rest or if its velocity is oriented in the direction of the field B. The SI unit of magnetic field is the kg/s2. A called tesla (T).
Magnetic moment
A) Moment of the magnetic forces on a circuit A magnetic field B may exert a moment of force on a closed circuit carrying a current I. This moment may provoke a rotation of the circuit. Consider for instance a rectangular circuit MNPQ free to rotate about the axis Oz that joins the mid points of MP and NQ (Figure 6.5a). The forces F1 and F2 that a uniform field B parallel to Ox exerts on the sides MQ and PN are opposite and oriented in the direction of Oz. Thus, they produce no moment with respect to O. The sides MN and PQ of length L are orthogonal to B. They are subject to opposite forces F3 = −ILB ey and F4 = ILB ey. Let n be the unit vector normal to the circuit and oriented according to the right-hand rule and let θ be the angle that n forms with B measured algebraically about Oz. The total moment of the magnetic forces with respect to O is the vector sum of the moments of F3 and F4
ΓM = − LL′IB sin θ ez = − SIB sin θ ez ,
We define the magnetic moment of the circuit as the vector
M = IS n.

Magnetism in Matter

Before the discovery of the magnetic effects of electric current and charges, the understanding of magnetism pertained to permanent magnets. Even today, some of the magnetic properties of matter remain little understood and other properties remain to be explored. This does not prevent magnetism from underlying many applications, ranging from the magnetic compass to measurement instruments, electric generators and motors, magnetic tapes for sound and video recording and for computer data storage, magnetic levitation, etc. The purpose of this chapter is to introduce some basic elements of magnetism in matter. 7.1. Types of magnetism Some materials, said to be ferromagnetic, become magnetized if they are exposed to a magnetic field and they remain permanently magnetized if the magnetic field is removed. A magnetized body is equivalent to a magnetic moment M in a characteristic direction SN. An external magnetic field acts on this body and orients it in such a way that the field B enters the body at S and leaves it at N (Figure 7.1a). Particularly, in the Earth’s magnetic field, N points approximately toward the geographic North and S toward the South . However, contrary to the electric charges, which constitute an electric dipole, the “magnetic poles” cannot be separated and the concept of magnetic pole is simply an analogy with electric charges. Similar to dielectrics, which polarize if they are placed in an electric field,
all materials become magnetized to some extent if they are submitted to the magnetic field B of an electric current or another magnetized body. Some materials (such as aluminum, chrome, platinum, etc.) acquire a magnetic moment in the direction of B, they are said to be paramagnetic. Other materials (such as silver, gold, copper, mercury, lead, etc.) acquire a magnetic moment in the oppositedirection to B, they are said to be diamagnetic. A magnetized body produces its own magnetic field, which leaves the body near N and enters the body near S. It acts on nearby magnets in such a way that like poles repulse each other while unlike poles attract each other.
The macroscopic magnetic properties of materials have their origin in their atomic structure. An electron in orbital motion in the atom is equivalent to a microscopic electric circuit, which is subject to the magnetic field of other systems and which produces a magnetic field exactly like a magnetic moment Mo = −eLe/2me, where Le is the orbital angular momentum of the electron. The magnetic moments of the various electrons of the atom add up vectorially to form the magnetic moment of the atom Ma = −eLa/2me, where La is the total angular momentum of the atom. We must add also the intrinsic magnetic moments of the electrons and the nuclear magnetic moment. The magnetic properties of materials may be explained only by using quantum mechanics. In this theory, the three components of angular momentum Lˆ cannot be determined simultaneously. It is possible to determine only the squared magnitude and one component of Lˆ , in the z direction, for instance. 2 Lˆ takes the values 2l(l+1), where h(cut) = h/2π is the reduced Planck’s constant and the quantum number l takes the values 0, 1, 2, etc. For a given l, Lz takes the values ml, where ml may take one of the values −l, −l + 1, … l −1, l. It is convenient to express the angular momentum in unit of and write the orbital magnetic moment of the electron as Mo = − μB e Lˆ , where μB = (e/2me) is Bohr magneton. On the other hand, the electron also has an intrinsic angular momentum or spin s = ½, thus two states of polarization ms = −½ and ms = +½. The spin corresponds to an intrinsic magnetic moment Ms = gμB ˆse where g is the gyromagnetic ratio of the electron that is very approximately equal to −2. Similarly, the proton has an orbital magnetic moment Mo = μN p Lˆ and an intrinsic magnetic moment Ms = gpμNˆsp , where gp = 2.793 and μN = e/2mp (about 1839 times smaller than μB). Although the neutron is neutral, it has an intrinsic magnetic moment Ms = gnμNˆsn , where gn = −1.913. Although electrons have individual magnetic moments, they are often paired in such a way that the magnetic moment of the atom is zero. On the other hand, in solids, the magnetic moment of the tightly packed atoms is not the same as that of the free atoms. The most important contribution to magnetism comes from the electron spin.

Special Relativity and Electrodynamics

Until the end of the 19th Century, classical mechanics was confirmed by all experiments and nobody dared to think that this might not be the case in electromagnetism. However, several experiments have shown some contradictions between classical mechanics and electromagnetic phenomena, especially the propagation of light. In fact, as we shall see in this chapter, Maxwell’s equations, which are the basic laws of electromagnetism, are not in accordance with the Galilean invariance, which is one of the basic principles of classical mechanics. Several attempts have been made, without success, to modify Maxwell’s equations in order to make them agree with classical mechanics. Lorentz adopted the opposite strategy and proposed to modify classical mechanics by replacing the Galilean transformation by the now-called Lorentz transformation. In 1905, Einstein analyzed the basic concepts of space and time, and formulated the special theory of relativity. The Lorentz transformation resulted straightforwardly from this analysis. Up to now, all the consequences of this theory have been verified experimentally. The special theory of relativity and the general theory of relativity, both formulated by Einstein, are new perceptions of physics and the Universe with very important consequences. Special relativity is used to study high-velocity (thus highenergy) phenomena. All fundamental physical theories must be formulated in accordance with relativity in order to be covariant (that is, independent of the observation frame). In this chapter we introduce the basic ideas of this theory and analyze some of its consequences in mechanics and in electromagnetism

Modern physics and maxwell equations

I referred, at the beginning of this chapter, to the fact that a profound shift in Newtonian foundations had already begun in the 19th century, before the revolutions of relativity and quantum theory in the 20th. The Wrst hint that such a change might be needed came from the wonderful experimental findings of Michael Faraday in about 1833, and from the pictures of reality that he found himself needing in order to accommodate these. Basically, the fundamental change was to consider that the ‘Newtonian particles’ and the 'forces' that act between them are not the only inhabitants of our universe. Instead, the idea of a 'field', with a disembodied existence of its own was now having to be taken seriously. It was the great Scottish physicist James Clark Maxwell who, in 1864, formulated the equations that this ‘disembodied Weld’ must satisfy, and he showed that these Welds can carry energy from one place to another. These equations uniWed the behaviour of electric fields, magnetic Welds, and even light, and they are now known simply as Maxwell's equations, the Wrst of the relativistic field equations.
From the vantage point of the 20th century, when profound advances in mathematical technique have been made (and here I refer particularly to the calculus on manifolds.) Maxwell's equations seem to have a compelling naturalness and simplicity that almost make us wonder how the electric/magnetic fields could ever have been considered to obey any other laws. But such a perspective on things ignores the fact that it was the Maxwell equations themselves that led to a very great many of these mathematical developments. It was the form of these equations that led Lorentz, Poincare´, and Einstein to the spacetime transformations of special relativity which, in turn, led to Minkowski's conception of spacetime. In the spacetime framework, these equations found a form that developed naturally into Cartan’s theory of differential forms ; and the charge and magnetic flux conservation laws of Maxwell's theory led to the body of integral expressions that are now encapsulated so beautifully by that marvellous formula referred to, as the fundamental theorem of exterior calculus.
The electromagnetic field tensor has already been explained. Now there is another concept named Hodge dual. It is defined as

hodge dual tensor notation

hodge dual
Hodge dual of F and J is 2-form F* and 3-form J*.
hodge dual tensor notation

hodge dual tensor notation
The bi-vector is an entity of exterior calculus.
bi-vector
In 4-space, a simple bivector H (Hab) represents the same 2-plane element as its 'dual' 2-form H# 1/2 e(abcd)H(cd) . But the index-lowered version of H, the simple 2-form Hab, which is equivalent to its 'dual' bivector 1/2 E(abcd)H(cd) , represents the orthogonal complement 2-plane element. Hence it is the index raising/lowering in the Hodge dual that leads to the passage to the orthogonal complement.
Having set up this notation, we can now write Maxwell’s equations very simply as
dF = 0; d*F = 4πJ ;

Conservation and flux laws in Maxwell's theory

The vanishing divergence of the charge-current vector provides us with the equation of conservation of electric charge. The reason that it is referred to as a 'conservation equation' comes from the fact that, by the fundamental theorem of exterior calculus , we have
bi-vector
integrated over any closed 3-surface Q in Minkowski space M. (Any closed 3-surface in M is the boundary qR of some compact 4-dimensional region R in M.) See Figure below. The quantity *J can be interpreted as the 'flux of charge' (or ‘flow’ of charge) across Q = dR. Thus, what the above equation tells us is that the net flux of electric charge across this
conservation law
boundary has to be zero; i.e. the total coming into R has to be exactly equal to the total going out of R: electric charge is conserved.{note} We can also use the second Maxwell equation d*F = 4π *J to derive what is called a 'Gauss law'. This particular law applies at one given time t = t0, so we are now using the three-dimensional version of the fundamental theorem of exterior calculus. This tells us the value of the total charge lying within some closed 2-surface S at time t0 (see Fig. 19.4), by expressing this charge as an integral over S of the dual of the Maxwell tensor *F — which amounts to saying that we can obtain the total charge surrounded by S if we integrate the total flux of electric field E across S.
More generally, this applies even if S does not lie in some Wxed time t = t0. Suppose that S is the spacelike 2-boundary of some compact 3-spatial region A. Then the total charge X in the region A, surrounded by S (or, in spacetime terms, 'threaded through' S—see Figure), is given by
conservation law
Within the 3-surface of constant time t = t0, Maxwell’s d*F = 4&pi*J gives us the Gauss law, whereby the integral of electric flux (integral of *F) over a closed spatial 2-surface measures the total charge surrounded (by the fundamental theorem of exterior calculus). In fact, this is not restricted to 2-surfaces at constant time, and the Gauss law is thereby generalized

conservation law
We can also obtain a related kind of conservation law from the first Maxwell equation dF = 0. This has just the same form as the second Maxwell equation, except that F replaces *F and the source corresponding to *J is now zero. Thus, for any closed 2-surface in Minkowski space,2 we always have the flux law
∫F = 0; Note that in passing from *F to F (or from F to *F) we simply interchange the electric and magnetic Weld vectors (with a change of sign for one of them). The absence of a source for F is an expression of the fact that (as far as is known) there are no magnetic monopoles in Nature. A magnetic monopole would be a magnetic north pole or a magnetic south pole on its own—rather than north and south poles always appearing in pairs, which is what happens in an ordinary magnet. (These poles are not independent physical entities, but arise from the circulation of electric charges.) It appears that in Nature there is never a net 'magnetic charge' (non-zero 'pole strength') on a physical object. From the point of view of the Maxwell equations alone, there does not seem to be any good reason for the absence of magnetic monopoles, since we could simply supply a right-hand side to the Wrst Maxwell equation dF = 0 without any loss of consistency. In fact, from time to time, physicists have contemplated the possibility that magnetic monopoles might actually exist and have tried to look for them. Their existence would have important implications for particle physics but there is no indication, as of now, that there are any such monopoles in the actual universe. {note}
Although correct, this argument has been given somewhat glibly. Spell out the details more fully, in the case when R is a spacetime 'cylinder' consisting of some bounded spatial region that is constant in time, for a fixed finite interval of the time coordinate t. Explain the diVerent notions of 'flux of charge' involved, contrasting this for the spacelike 'base' and 'top' of the cylinder with that for the timelike 'sides'. Spell out why this is just the electric flux

Maxwell field as Gauge curvature

Maxwell field as Gauge curvature

The Wrst Maxwell equation dF = 0 also has the implication that
F = 2dA
for some 1-form A. (This is taking advantage of the ‘Poincare´ lemma’, which states that, if the r-form a satisfies da = 0, then locally there is always an (r - 1)-form b for which a = db; ) Moreover, in a region with Euclidean topology, this local result extends to a global one. The quantity A is called the electromagnetic potential. It is not uniquely determined by the Weld F, but is fixed to within the addition of a quantity dY, where Y is some real scalar field:

conservation law
This 'gauge freedom' in the electromagnetic potential tells us that A is not a locally measurable quantity. There can be no experiment to measure ‘the value of A’ at some point because A þ dY serves exactly the same physical purpose as does A. However, the potential provides the mathematical key to the procedure whereby the Maxwell field interacts with some other physical entity C. How does this work? The speciWc role of Aa is that it provides us with a gauge connection (or bundle connection).
gauge potential
where e is a particular real number that quantiWes the electric charge of the entity described by ψ. In fact, this 'entity' will generally be some charged quantum particle, such as an electron or proton, and C would then be its quantum-mechanical wavefunction. All that we shall need to know about it now is that ψ is to be thought of as a cross-section of a bundle , a bundle describing charged fields, and it is this bundle on which ∇ acts as a connection The electromagnetic Weld quantities F and A are uncharged (e = 0 for them), so that all our Maxwell equations, etc., are undisturbed by having this new definition for ∇a i.e. we still have ∇a = d/dxa in those equations, in that Minkowski coordinates—or the appropriate generalization if we are considering curved spacetime. What is the geometrical nature of the bundle that this connection acts upon? One possible viewpoint is to think of this bundle as having fibres that are circles (S1s), over the spacetime M, where this circle describes a phase multiplier eiy for C. (This is the kind of thing that happens in the ‘Kaluza–Klein’ picture referred but where in that case the entire bundle is thought of as 'spacetime'.) More appropriate is to think of the bundle as the vector bundle of the possible C values at each point, where the freedom of phase multiplications make the bundle a U(1) bundle over the spacetime M. (This kind of issue was considered ) For this to make sense, ψ must be a complex Weld whose physical interpretation is, in some appropriate sense, insensitive to the replacement ψ -> e^(iθ)ψ(where θ is some real-valued field on the manifold M). This replacement is referred to as an electromagnetic gauge transformation, and the fact the physical interpretation is insensitive to this replacement is called gauge invariance. The curvature of our bundle connection then turns out to be the Maxwell Weld tensor F(ab). Before exploring with these ideas further, it is appropriate to make some brief historical comments. Shortly after Einstein introduced his general theory of relativity in 1915, Weyl suggested, in 1918, a generalization in which the very notion of length becomes path-dependent. (Hermann Weyl, 1885–1955, was an important 20th-century mathematical figure. Indeed, among the work of those mathematicians who wrote entirely in the 20th century, his was, to my mind, the most inXuential—and he was important not only as a pure mathematician but also as a physicist.) In Weyl's theory, the null cones retain the fundamental role that they have in Einstein's theory (e.g. to deWne the limiting velocities for massive particles and to provide us with the local 'Lorentz group' that is to act in the neighbourhood of each point), so a Lorentzian (say +---) metric g still is locally required for the purpose of defining these cones. However, there is no absolute scaling for time or space measures, in Weyl’s scheme, so the metric is given only up to proportionality. Thus, transformations of the form
g -> λg
for some (say positive) scalar function l on the spacetime M, are allowed, these not aVecting the null cones of M. (Such transformations are referred to as conformal rescalings of the metric g; in Weyl's theory, each choice of g provides us with a possible gauge in terms of which distances and times can be measured.) Although Weyl may have had spatial separations more in mind, it will be appropriate for us to think in terms of time measurements . Thus, in Weyl's geometry, there are no absolute 'ideal clocks'. The rate at which any clock measures time would depend upon its history.
The situation is 'worse' than in the standard 'clock paradox'. In Weyl's geometry, we can envisage a space traveller who journeys to a distant star and then returns to Earth to Wnd not just that those on the Earth had aged much more, but also that the clocks on Earth are now found to run at a diVerent rate from those on the rocket ship! See Figure below. Using this very striking idea, Weyl was able to incorporate the equations of Maxwell's electromagnetic theory into the spacetime geometry
gauge potential
In Weyl's original gauge theory of electromagnetism, the notion of time interval (or space interval) is not absolute but depends on the path taken. (a) A comparison with the 'clock paradox' in Weyl's theory we find that the space traveller arrives home (world-line ABC) to find not only diVering clock readings between those on Earth (direct route AB) and those on the rocket ship, but also differing clock rates! (b) Weyl's gauge curvature (giving the Maxwell field F) comes about from this (conformal) time scale change as we go around an infinitesimal loop (diVerence between two routes from p to neighbouring point p`).
The essential way that he did this was to encode the electromagnetic potential into a bundle connection, just as I have done above, but without the imaginary unit 'i' in the expression for ra. We can think of the relevant bundle over M as being given by the Lorentzian metrics g that share the same null cones. Thus, the Wbre above some point x in M consists of a family of proportional metrics (where we can, if desired, choose the proportionality factors to be positive). These factors are the possible 'λs' in g -> λg above. For any particular choice of metric, we have a gauge whereby distances or times along curves are deWned. But there is to be no absolute choice of gauge, and so no preferred choice of metric g from the equivalence class of proportional ones. There is some structure additional to that of the null cones (i.e. to the conformal structure), however, namely a bundle connection—or gauge connection— which Weyl introduced, in order to have Maxwell's F (i.e. F(ab)) as its curvature. This curvature measures the discrepancy in the clock rates when the world-lines differ only by an infinitesimal part;. (This may be compared with the 'strained bundle' BC, over C, the basic bundle concept is very similar.)
When Einstein heard about this theory, he informed Weyl that he had a fundamental physical objection to it, despite the mathematical elegance of Weyl's ideas. Spectral frequencies, for example, appear to be completely unaffected by an atom's history, whereas Weyl's theory would predict otherwise. More fundamentally, although not all the relevant quantum-mechanical rules had been fully formulated at the time Weyl's theory is in conflict with the necessarily exact identity between different particles of the same type. In particular, there is a direct relation between clock rates and particle masses. As we shall see later, a particle of rest-mass m has a natural frequency mc^2h^(-1), where h is Planck's constant and c the speed of light. Thus, in Weyl's geometry, not just clock rates but also a particle’s mass will depend upon its history. Accordingly, two protons, if they had different histories, would almost certainly have different masses, according to Weyl's theory, thereby violating the quantum-mechanical principle that particles of the same kind have to be exactly identical .

Aharonov–Bohm effect

A beam of electrons is split into two paths that go to either side of a collection of lines of magnetic flux (achieved by means of a long solenoid). The beams are brought together at a screen, and the resulting quantum interference pattern depends upon the magnetic Xux strength—despite the fact that the electrons only encounter a zero Weld strength (F = 0). (b) The effect depends on the value of ∫A, which can be non-zero over the relevant topologically non-trivial closed path despite F vanishing over this path. The quantity ∫A is unchanged for continuous deformations of the path within the field-free region.
gauge potential

Maxwell distribution

Maxwell also worked on the kinetic theory of gases. He discovered a formula to calculate the number of molecules that have an speed between v and dv as
maxwell distribution
The horizontal axis represents the magnitude of velocity and the vertical line is the frequency , which is the usual explanation of probability distribution
maxwell probability distribution
The definitions of electric susceptibility and magnetic permitivity are
electrical susceptibility

Uncommon Wisdom

What is consciousness? It is the awareness of our mind to take right decision. Human mind is place of vast electrical activity. When we take decision lot of neuron fire simultaneously. Electrical signals pass from one neuron to another with the speed of light. Brain is a complicated electrical circuit. The complete scientific model of human mind is still unknown. We can only explain it partially. Human mind has two sides : one is left hemisphere and other is right hemisphere. These two parts are responsible for certain voluntary and involuntary actions. For example our logical and analytic skills are processed in left hemisphere. How consciousness arises is really a difficult problem. It is mysterious and simillar to quantum mechanics. May be quantum mechanics will explain consciousness completely one day.
Carl Sagan has tried to explain consciousness in his famous book "cosmos". He argues that fourteen billions of years of universe's evolution has turned matter into counsciousness. The laws of physics have played the vital role in this evolution.
But is there anything special about consciousness? Uncommon wisdom says a piece of rock is as conscious as us. The whole universe is conscious according to some basic interpretation. We are part of the universe and nature. Gia hypotheses regards the whole earth as an organism. Life form is nothing but some combination of specific compounds named organic compound. Life emerged on earth four billions years ago according to scientific evidences. First a biological cell which is the basic component of life was formed by some natural event. This cell was transformed and various other living organisms were created afterwards. That is what modern life sciences have claimed to be the fact. If a planet has been left for millions of years, some kind of life form will emerge.

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