After reading this article you will learn about: 1. Introduction to Maxwell’s Theory on Transmission of Waves 2. Development of the Concepts of “Field” 3. The Contributions of Heinrich Hertz 4. Unified Theory of Electromagnetism 5. Maxwell’s Equations 6. Electromagnetic Wave Propagation.
Introduction to Maxwell’s Theory on Transmission of Waves:
We have, so far, discussed the development of devices that have facilitated communications in the last two centuries. These devices have been inventions which their inventors have been able to achieve because the infrastructure for these had been created.
This infrastructure included the theoretical work on the subject of the invention. The development of this infrastructure has been a matter of primary importance and has, in fact, been one of the important reasons that the inventions could be encouraged and they took place.
Scientific development is a collaborative process and new developments take place on the basis of work developed earlier.
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Newton is reported to have remarked:
“If I have been able to see further, it is because I have stood on the shoulders of giants.”
James Clerk Maxwell was one such giant on whose shoulders stood many inventors in later years and, true to the human spirit of inquiry, many people particularly in the scientific world doubted Maxwell’s theoretical work and it required the independent assessment of a German scientist Heinrich Hertz before the scientific community accepted it.
These sceptics included the eminent French mathematician Henry Poincare who commented “the complicated structure which Maxwell attributed to ether rendered his system strange and unattractive.”
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Maxwell propagated the belief, that a magnet or an electric charge contains a field in the region around it that gives, at any location, the force experienced by another small magnet or charge placed there. This observation had been made earlier by Michael Faraday who had the extraordinary insight that electrical and magnetic actions are not transmitted instantaneously but after a certain lag of time.
He further recognised the Connection between magnetism and light after observing that a substance such as glass can rotate the plain of polarization of light in the presence of a magnetic field. Incidentally, this phenomenon is known today as the Faraday effect.
Maxwell developed the theory that the energy of the electromagnetic field lay in the space around the conductors as well as in the conductors themselves. Faraday had been a self-taught man, therefore, Maxwell who was deeply influenced by the work done by Faraday, began the study of those phenomena—proposed by Faraday—by translating Faraday’s experimental findings into mathematics.
Faraday had discovered that changes in magnetic fields produce electric fields; Maxwell added the converse: changes in electric fields produce magnetic fields even in the absence of electric currents. By 1864, he had formulated his own electromagnetic theory of light predicting that both light and radio waves are electric and magnetic phenomena.
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Maxwell predicted that electromagnetic disturbances travelling through empty space have electric and magnetic fields at right angles to each other and that both fields are perpendicular to the direction of motion of the wave.
The waves move at uniform speed equal to the speed of light and that light is one form of an electromagnetic wave. The elegance of Maxwell’s ideas notwithstanding, there was few acceptors of his ideas, prompting Henry Poincare to make the statement quoted earlier.
While the mainstream of theoretical activity concerning electric and magnetic fields has been devoted to showing how they are interrelated, there has been research to discover properties of materials and heat also related to these phenomena. Maxwell’s work on wave propagation, which we shall presently see and discuss, has been highlighted in the four equations that he developed.
Development of the Concepts of “Field”:
Faraday introduced the concept of field lines of force that exist outside material bodies. As he explained it, the region around and outside a magnet or an electric charge contains a field that describes, at any location, the force experienced by another magnet or electrical charge placed there.
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The lines of force around a magnet can be made visible by iron filings sprayed on a piece of paper paced over the magnet. The concept of field, specifying as it does a certain possible action or force at any location in space, was the key to understanding electromagnetic phenomena. Incidentally, it may also be mentioned that the field concept also plays a pivotal role (in several forms) in particles and forces.
Besides introducing this important concept of electric and magnetic field lines of force, Faraday had also predicted that electrical and magnetic actions are not transmitted instantaneously, but after a certain time lag, which increases with distance from the source.
Maxwell formulated a quantitative theory that linked the fundamental phenomena of electricity and magnetism, and that predicted the propagation of electromagnetic waves at the speed of light. To be more accurate, one should say that at the time his paper making these predictions were written and published the speed of propagation that was determined was as close as possibly determinable, to the speed of light.
This paper “On the Physical Lines of Force” which was published in 1862, created quite some furore by stating that electricity may be disseminated through space with properties identical to light and that the results achieved “show that light and magnetism are affectations of the same substance and that light is an electromagnetic disturbance propagated through the field according to electromagnetic laws.”
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These laws had just been stated by James Clerk Maxwell.
Maxwell and the Concept of Ether:
Maxwell’s theory makes no references to the medium in which these electromagnetic waves propagate. An electromagnetic wave is produced when a set of charges is moved back and forth along a line.
Moving charges represent an electric current. In this back and forth motion, the current first flows in one direction and then reverses. As a result of this reversal of current direction, the magnetic field around this current also has to reverse its direction. The magnetic field is, therefore, not constant but time-varying.
This time-varying magnetic field produces a time-varying electric field, perpendicular to it. These time-varying magnetic and electric fields spread out from their source—the varying current.
This oscillating current is the current in the transmitting antenna and the time-varying electric and magnetic fields that are perpendicular to one another propagate at the speed of light in free space and constitute an electromagnetic wave.
Its frequency is that of the oscillating charges in the antenna. Once- generated, it is self-propagating because a time-varying electric field produces a time-varying magnetic field and vice versa. Electromagnetic radiation travels through space by itself. At the time of Maxwell’s writing his famous theory, the belief in the existence of ether was strong.
However, it is impossible to visualize the existence of ether because contradictory properties had to be attributed to it in order to explain different phenomena at different times.
In fact, Maxwell has written that “ether is a vast expanse of a substance, some of it possibly even inside the planets, carried along with them or passing through them as the water of the sea passes through the meshes of a net when it is towed along by a boat.”
Unfortunately, (or fortunately, depending on one’s point of view) the existence of any substance such as ether was almost conclusively disproved as a result of the famous Michelson-Morley experiment.
However, this experiment did not shatter the beliefs of the proponents of ether who felt that ether could not show any effect in the Michelson-Morley experiment because ether could possibly be dragged along with the earth and thus be stationary around the Michelson-Morley experiment and thus could not show any effect.
However, Hertz’s formulation of Maxwell’s theory made it clear that no medium was needed for the propagation of electromagnetic radiation. Quantum theory further clarified this issue. But all that is another story.
The Contributions of Heinrich Hertz:
It would appear that the scientific community had enough facts and data to convince itself that the mystery of light was solved and the phenomena of electricity and magnetism along with light were unified in a grand theory. But the scientific community took 25 more years to accept Maxwell’s theory.
One important reason for this lack of general acceptance probably was the absence of direct proof. Furthermore, Maxwell had not only adopted complex mathematical formalism to explain his ideas but had also explained its various aspects by unusual mechanical concepts.
Even though he cautioned that all such examples should be taken as being merely illustrative and not explanatory, in 1899 Henry Poincare made the famous remark quoted above and attributed to him. The ideas of Faraday and Maxwell that the field of Force has a physical existence in space independent of material media could not be accepted without direct proof.
In Germany, matters were further complicated by the success of Carl Frederich Gauss and Wilhelm Eduard Weber in developing a potential field theory for the phenomena of electrostatics and magneto statics, and their continuing effort to extend this formalism to electrodynamics.
It is difficult in hindsight to appreciate the reluctance to accept the Faraday-Maxwell theory. The impasse was finally resolved by the work of Heinrich Hertz.
In 1884, he derived Maxwell’s theory by a new method and put its fundamental equations into their present-day form. He also clarified the equations, making the symmetry of electric and magnetic fields apparent.
The German physicist Arnold Summerfield spoke for most of his learned colleagues when, after reading Hertz’s paper, he remarked “the shades fell from my eyes” and admitted that he understood electromagnetic theory for the first time.
Four years later, Hertz made another significant contribution—succeeded in generating electromagnetic radiation of radio and microwave frequencies, measuring their speed by a standing-wave method and proving that these waves have the properties of reflection, diffraction, refraction and interference common to light.
He showed that such electromagnetic waves can be polarized, that the electric and magnetic fields oscillate in directions that are mutually perpendicular and transverse to the direction of motion and that their velocity is the same as the speed of light as predicted by Maxwell’s theory propounded much earlier.
Hertz’s ingenious experiments not only settled the theoretical misconceptions in favour of Maxwell’s electromagnetic field theory, but also opened the way for building transmitters, antennas, coaxial cables and detectors for radio-frequency electromagnetic radiation. In fact, Marconi received the first patent for wireless telegraphy in 1896 and achieved trans-Atlantic radio communication in 1901.
Maxwell’s theory, therefore, should actually be called the Faraday-Maxwell-Hertz theory of electromagnetism, seeing the contributions of each of the three giants of electromagnetism to its development and acceptance. However, this story has some further twists in its tail.
Unified Theory of Electromagnetism:
The final stages of synthesizing electricity and magnetism into one coherent theory were made by Maxwell. He was deeply influenced by Faraday’s work, having begun his study of the phenomena by translating Faraday’s experimental findings into mathematics—Faraday being a self-taught man, he never mastered mathematics.
In 1856, Maxwell developed the theory that energy in the electrostatic field is in the space around the conductors as well as in the conductors themselves. By 1864, he had formulated his own electromagnetic theory of light, predicting that both light and radio waves are electric and magnetic phenomena.
While Faraday had discovered that changes in electric fields produce magnetic fields, Maxwell reasoned that the symmetry in nature should imply that the converse should also be true.
Therefore, Maxwell added the converse:
Changes in electric fields produce magnetic fields even in the absence of electric currents. He predicted that electromagnetic disturbances travelling through empty space have electric and magnetic fields at right angles to each other and that both fields are perpendicular to the direction of the transmission of the wave.
He concluded that the wave moves at uniform speed equal to the speed of light and that light is also a form of electromagnetic radiation or wave. Maxwell’s ideas were elegant but they did not find many takers at the time that he propounded them. He expressed them mathematically in the form of four field equations.
These four field equations represent the pinnacle of classical electromagnetic theory. Subsequent developments have been concerned either with the relationship between electromagnetism and the atomic structure of matter or with the practical and theoretical consequences of Maxwell’s equations.
His formulations have withstood the revolutions of relativity and quantum mechanics. The formulation of quantum theory has changed his basic ideas about the medium of wave propagation but the basic laws that he stated have now stood the test of time and have, in fact, been strengthened by new meanings given through quantum mechanics.
His equations are appropriate for distances as small as 10-10 centimeters—that is one-hundredth of the size of an atom. The fusion of electromagnetic theory and quantum theory, known as electrodynamics, is required only for, very small distances.
Maxwell’s Equations:
These consist of four equations which together form a complete description of the production and interrelation of electric and magnetic fields. They are all based on experimental laws.
When Maxwell started analysing the state of electric and magnetic fields, he noticed that various laws were already present but had not been considered in totality, that is, their effect taken jointly had not been considered. On further analysis, Maxwell found that one of the laws was not sufficiently universal and worked only in certain specific conditions.
These four equations expressed simply are, respectively:
1. Electric field diverges from electric charge, an expression of the Coulomb force.
2. There are no isolated magnetic poles, but the Coulomb force acts between the poles of a magnet.
3. Electric fields are produced by changing magnetic fields, an expression of Faraday’s law of induction.
4. Circulating magnetic fields are produced by changing electric fields and by electric currents, Maxwell’s extension of Ampere’s law to include the interaction of changing fields.
These equations are expressed elegantly by using vector operators divergence and curl.
In the following expressions, the terms used are:
ρ is charge density; µo and £o are constants known as permeability and permittivity of free space
J is current density
E is the electric field
B is the magnetic field
D and H are field quantities that are proportional to E and B respectively.
The four equations, inside matter, are:
We shall look at some of these laws separately, before comparing with the above four equations and seeing how Maxwell modified Ampere’s law to make the law more general.
Gauss’s Law:
Gauss’s law deals with field lines and charge distribution. Consider the simple case a single point charge q, situated at the origin
where q is the charge at the origin, E(r) is the intensity of the electric field at the point r and f is the unit vector which specifies the direction of the force.
Gauss’s law, however, is expressed as
The second law is
Faraday’s Law:
Faraday’s law concerns Electromagnetic Induction, the subject he researched on, discovered and propounded. Electromagnetic Induction is the current created in a circuit by a moving or varying magnetic field. The law states
Ampere’s Law:
The above four laws represented the state of electromagnetic theory before Maxwell’s arrival on the scene. There is a fatal inconsistency in these four equations. The divergence of curl is always zero. Therefore, according to Faraday’s law
This equation ties perfectly with our argument because the left-hand side is zero since divergence of curl is zero and the right-hand side is zero because of Eqn. 3.7. But when you perform the same operation on Eqn. 3.9, you get into trouble
By the same argument given above, the left-hand side must be zero but the right-hand side may not be. For steady currents, the divergence of J is zero, but one may not always get a steady current. Therefore, for unsteady currents, the law will fail.
Clearly, Maxwell saw that there was a dichotomy here and it had to be fixed. The problem in the above case is that the right-hand side should be zero, but isn’t. It might occur that if we were to add the quantity ɛ0. (-∂E/∂t) to J, in Ampere’s law, it would set the equations right and would kill off the extra divergence. Hence, he modified Eqn. 3.9 to
This change takes care of all the problems without creating any additional hurdles. This extra term was called the displacement current by Maxwell. In other words, the additional quantity added, that is, µ0ɛ0 (∂E/∂t)would be just right to kill the extra divergence
To Maxwell, this constituted the rescue of the continuity equation. Today, we recognize that as far as magneto statics is concerned, this addition changes nothing because when E is constant, we still have
However, the “displacement current” was difficult to detect in experiments at the time of Faraday since it would be in competition with the term J. However, it plays a crucial role in the propagation of electromagnetic waves.
Electromagnetic Wave Propagation:
In regions of space where there is no charge or current, Maxwell’s equations read as follows:
They represent a set of coupled, first order, partial differential equations for E and B. They can be decoupled by applying curl on (iii) and (iv) above. We will get
These two equations are now decoupled and are separate equations for E and B although they are now of the second order. In vacuum E, B and their components satisfy the equation
This is also called the classical wave equation and it describes waves travelling with a velocity v. According to Maxwell’s equation empty space supports the propagation of electromagnetic waves at a speed
This is precisely the speed of light. No wonder Maxwell drew the conclusion that light is an electromagnetic wave. Further conclusions may be drawn from the wave equations as postulated by Maxwell, but the purpose of our discussion was to show how Maxwell’s researches helped in our understanding of electromagnetic wave propagation. This has been shown.