Appendix B

# Appendix B - MIT OpenCourseWare http/ocw.mit.edu...

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MIT OpenCourseWare http://ocw.mit.edu Electromechanical Dynamics For any use or distribution of this textbook, please cite as follows: Woodson, Herbert H., and James R. Melcher. Electromechanical Dynamics . 3 vols. (Massachusetts Institute of Technology: MIT OpenCourseWare). http://ocw.mit.edu (accessed MM DD, YYYY). License: Creative Commons Attribution-NonCommercial-Share Alike For more information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms

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Appendix B REVIEW OF ELECTROMAGNETIC THEORY B.1 BASIC LAWS AND DEFINITIONS The laws of electricity and magnetism are empirical. Fortunately they can be traced to a few fundamental experiments and definitions, which are re- viewed in the following sections. The rationalized MKS system of units is used. B.1.1 Coulomb's Law, Electric Fields and Forces Coulomb found that when a charge q (coulombs) is brought into the vicinity ofa distribution of chargedensity p,(r') (coulombs per cubic meter), as shown in Fig. B.1.1, a force of repulsion f (newtons) is given by f = qE, (B. 1.1) where the electricfield intensity E (volts per meter) is evaluated at the position = qE Fig. B.1.1 The force f on the point charge q in the vicinity of charges with density Pe(r') is represented by the electric field intensity E times q, where E is found from (B.1.2).
Review of Electromagnetic Theory r of the charge q and determined from the distribution of charge density by E(r) = e(r') - r) dV'. (B.1.2) 4E(r) e = r - r'lj In the rationalized MKS system of units the permittivity eo of free space is qo = 8.854 x 10 - 12 _ -- X 10 - 9 F/m. (B.1.3) 367r Note that the integration of (B.1.2) is carried out over all the charge dis- tribution (excluding q), hence represents a superposition (at the location r of q) of the electric field intensities due to elements of charge density at the positions r'. A I- U h U d s an exampp , suppose tl at iLe cargeL distribution p,(r') is simply a point charge Q (coulombs) at the origin (Fig. B.1.2); that is, p,= Q 6(r'), (B.1.4) where 6(r') is the deltafunction defined by qQI, .1__W 4Xeo-]-r 0(r') = 0, r' # 0, Fig. B.1.2 Coulomb's law for point charges Q (at the origin) and q (at S6(r') dV' = 1. (B.1.5) the position r). For the charge distribution of (B.1.4) integration of (B.1.2) gives E(r) = Qr (B.1.6) 4rreo Ir " Hence the force on the point charge q, due to the point charge Q, is from (B. 1.1) f = qQr (B.1.7) 4 ore 0 Irl " This expression takes the familiar form of Coulomb's law for the force of repulsion between point charges of like sign. We know that electric charge occurs in integral multiples of the electronic charge (1.60 x 10 - 1 9 C). The charge density p., introduced with (B.1.2), is defined as Pe(r) = lim - I q,, (B.1.8) av-o 61 i

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Appendix B where 6V is a small volume enclosing the point r and Z, q, is the algebraic sum of charges within 6V. The charge density is an example of a continuum model. To be valid the limit 6 V -- 0 must represent a volume large enough to contain a large number of charges q 1 ,yet small enough to appear infinitesimal when compared with the significant dimensions of the system being analyzed.
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Appendix B - MIT OpenCourseWare http/ocw.mit.edu...

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