By Harry F. Tiersten
This tract relies on lecture notes for a direction in mechanics that has been provided at Rensselaer Polytechnic Institute off and on for the prior two decades. The path is meant to supply graduate scholars in mechanics with an knowing of electromagnetism and get ready them for stories at the interplay of the electrical and magnetic fields with deformable sturdy continua. As such, it really is principal that the excellence among particle and continuum descriptions of subject be conscientiously made and that the excellence among that that's inherently linear and that that's intrinsically nonlinear be essentially delineated. each attainable attempt has been made on my half to accomplish those ends. I desire to recognize the contributions of a few scholars and college who attended the lectures through the years and who, via their questions and proposals, considerably superior a number of the sections. This preface wouldn't be entire if i didn't indicate that my curiosity in electromagnetism used to be initiated and my perspective in the direction of the improvement of the equations used to be motivated via lectures given by means of the overdue Professor R.D. Mindlin at Columbia college within the overdue nineteen fifties. i need to thank Professor C. Truesdell for his beneficial feedback, which i believe considerably more desirable the readability and clarity of the creation, and Dr. M.G. Ancona for his remark in regards to the readability of an immense element in Sec. 1.1.
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Extra resources for A Development of the Equations of Electromagnetism in Material Continua
8) where V is the volume enclosing the current. The physical limitations on the distributed field description in the electric case naturally exist in the magnetic case also. Hence, we are subject to the limitations, as regards rate of change of variable with distance, of any continuum description of matter utilizing macroscopic laws. 10) where P must be a point outside the region containing current density J. Clearly then, we also have the relations Vp·V P A = 0, Vp· B = 0 except where J"# O. 11) We have now discussed the magnetic field and magnetic vector potential at points outside the region containing current density for a volume distribution of current density.
The field energy density at a localized region of the space has no meaning. On the other hand, the situation is quite different in the field theoretic approach, in that then the energy density of any localized portion of the space is meaningful and important, in fact it has to be in order to have a sensible field theory. Two other differences between the descriptions are: 1. In the point charge description, the differential equations are meaningful only outside all point charges while in the field approach, the differential equations exist and are valid within the regions of charge and polarization density, and in fact the differential equations relate the field variables to the charge and polarization densities.
This is a Poisson equation and to this we have to adjoin the boundary conditions. However, before we do this we observe that in actuality we have no particular interest in the linear constitutive equations we have written in this section, only in the development and the equations up through Sec. 3. 10). 1 0) can be written in the equivalent form v x E=O. 1 ) Now, in any continuum field theory the truly fundamental equations must always be integral statements, from which the governing differential equations may be derived when suitable differentiability and continuity conditions are satisfied.
A Development of the Equations of Electromagnetism in Material Continua by Harry F. Tiersten