Chapter 2 Maxwells theory of electromagnetism 2.1 The postulate In 1864, James Clerk Maxwell proposed one of the history of science. In a famous memoir to the royal B02 ul theories in the ented nir equations summarizing all known laws on electricity and magnetism. This was more than a mere cataloging of the laws of nature. By postulating the need for an additional term to make the set of equations self-consistent, Maxwell was able to put forth what is still considered a complete theory of macroscopic electromagnetism. The beauty of Maxwell's equations led Boltzmann to ask, Was it a god who wrote these lines.? Since that time authors have struggled to find the best way to present Maxwells theory. Although it is possible to study electromagnetics from an"empirical-inductive viewpoint(roughly following the historical order of development beginning with static fields), it is only by postulating the complete theory that we can do justice to Maxwell,s vision. His concept of the existence of an electromagnetic "field"(as introduced by Faraday) is fundamental to this theory, and has become one of the most significant principles of modern scie We find controversy even over the best way to present Maxwells equations. Maxwell worked at a time before vector notation was completely in place, and thus chose to use scalar variables and equations to represent the fields. Certainly the true beauty of Maxwell's equations emerges when they are written in vector form, and the use of tensors reduces the equations to their underlying physical simplicity. We shall use vector otation in this book because of its wide acceptance by engineers, but we still must decide whether it is more appropriate to present the vector equations in integral or point On one side of this debate, the brilliant mathematician David Hilbert felt that the fundamental natural laws should be posited as axioms, each best described in terms of integral equations [154]. This idea has been championed by Truesdell and Toupin [199. On the other side, we may quote from the great physicist Arnold Sommerfeld "The general development of Maxwells theory must proceed from its differential form; for special problems the integral form may, however, be more advantageous"(185, p 23). Special relativity flows naturally from the point forms, with fields easily converted between moving reference frames. For stationary media, it seems to us that the only difference between the two approaches arises in how we handle discontinuities in sources and materials. If we choose to use the point forms of Maxwells equations, then we must also postulate the boundary conditions at surfaces of discontinuity. This is pointed out @2001 by CRC Press LLCChapter 2 Maxwell’s theory of electromagnetism 2.1 The postulate In 1864, James Clerk Maxwell proposed one of the most successful theories in the history of science. In a famous memoir to the Royal Society [125] he presented nine equations summarizing all known laws on electricity and magnetism. This was more than a mere cataloging of the laws of nature. By postulating the need for an additional term to make the set of equations self-consistent, Maxwell was able to put forth what is still considered a complete theory of macroscopic electromagnetism. The beauty of Maxwell’s equations led Boltzmann to ask, “Was it a god who wrote these lines ... ?” [185]. Since that time authors have struggled to find the best way to present Maxwell’s theory. Although it is possible to study electromagnetics from an “empirical–inductive” viewpoint (roughly following the historical order of development beginning with static fields), it is only by postulating the complete theory that we can do justice to Maxwell’s vision. His concept of the existence of an electromagnetic “field” (as introduced by Faraday) is fundamental to this theory, and has become one of the most significant principles of modern science. We find controversy even over the best way to present Maxwell’s equations. Maxwell worked at a time before vector notation was completely in place, and thus chose to use scalar variables and equations to represent the fields. Certainly the true beauty of Maxwell’s equations emerges when they are written in vector form, and the use of tensors reduces the equations to their underlying physical simplicity. We shall use vector notation in this book because of its wide acceptance by engineers, but we still must decide whether it is more appropriate to present the vector equations in integral or point form. On one side of this debate, the brilliant mathematician David Hilbert felt that the fundamental natural laws should be posited as axioms, each best described in terms of integral equations [154]. This idea has been championed by Truesdell and Toupin [199]. On the other side, we may quote from the great physicist Arnold Sommerfeld: “The general development of Maxwell’s theory must proceed from its differential form; for special problems the integral form may, however, be more advantageous” ([185], p. 23). Special relativity flows naturally from the point forms, with fields easily converted between moving reference frames. For stationary media, it seems to us that the only difference between the two approaches arises in how we handle discontinuities in sources and materials. If we choose to use the point forms of Maxwell’s equations, then we must also postulate the boundary conditions at surfaces of discontinuity. This is pointed out