"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.
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.
Power Engineering | Telecommunication | Control system Engineering | Electronics | Differential equation
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.
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'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.
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
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
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
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.
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 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
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).
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,
We finally arrive at maxwell's fourth equation.
Finally putting everything together we get :
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.
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.
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:
There is a maxim like this : "the nature lay hidden and God says let there be light. After that there was light".
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 :
The electromagnetic field tensor is :
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 formalismMaxwell'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:
In Tensor calculus the equations are as follow :
In differential form the equations are :
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.
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
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.
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.
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.