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 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 reviewed 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 of a 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 B 1 1 f qE 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 4E r e r dV r r lj B 1 2 In the rationalized MKS system of units the permittivity eo of free space is qo 8 854 x 10 12 367r X 10 9 F m B 1 3 Note that the integration of B 1 2 is carried out over all the charge distribution 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 U h U dA s an exampp I suppose tlat iLe cargeL qQI 1 W 4Xeo r 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 0 0 r r S6 r dV 1 0 Fig B 1 2 Coulomb s law for point charges Q at the origin and q at the position r B 1 5 For the charge distribution of B 1 4 integration of B 1 2 gives Qr E r 4rreo Ir B 1 6 Hence the force on the point charge q due to the point charge Q is from B 1 1 f 4 qQr ore 0 Irl B 1 7 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 19 C The charge density p introduced with B 1 2 is defined as Pe r lim I q av o 61 i B 1 8 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 6V 0 must represent a volume large enough to contain a large number of charges q1 yet small enough to appear infinitesimal when compared with the significant dimensions of the system being analyzed This condition is met in most electromechanical systems For example in copper at a temperature of 20 0 C the number density of free electrons available for carrying current is approximately 1023 electrons cm3 If we consider a typical device dimension to be on the order of 1 cm a reasonable size for 6V would be a cube with 1 mm sides The number of electrons in 6 V would be 10 which certainly justifies the continuum model The force as expressed by B I 1 gives the total force on a single test charge in vacuum and as such is not appropriate for use in a continuum model of electromechanical systems It is necessary to use an electricforce density F newtons per cubic meter that can be found by averaging B 1 1 over a small volume F lim lim I qjEj B 1 9 av o 6V 6v o 6V Here q represents all of the charges in 6V E is the electric field intensity acting on the ith charge and f is the force on the ith charge As in the charge density defined by B 1 8 the limit of B 1 9 leads to a continuum model if the volume 6V can be defined so that it is small compared with macroscopic dimensions of significance yet large enough to contain many electronic charges Further there must be a sufficient amount of charge external to the volume 6V that the electric field experienced by each of the test charges is essentially determined by the sources of field outside the volume Fortunately these requirements are met in almost all physical situations that lead to useful electromechanical interactions Because all charges in the volume 6 V experience essentially the same electric field E we use the definition of free charge density given by B 1 8 to write B 1 9 as F p E B 1 10 Although the static electric field intensity E can be computed from B 1 2 it is often more convenient to state the relation between charge density and field intensity in the form of Gauss s law soEE n da Pe dV B 1 11 In this integral law nis the outward directed unit vector normal to the surface S which encloses the volume V It is not our purpose in this brief review to show that B 1 11 is implied by B 1 2 It is helpful however to note that Review of Electromagnetic Theory Fig B 1 3 A hypothetical sphere of radius r encloses a charge Q at the origin The integral of eoE over the surface of the sphere is equal to the charge Q enclosed in the case of a point charge Q at the origin it predicts the same electric field intensity B 1 6 as found by using B 1 2 For this purpose the surface S is taken as the sphere of radius r centered at the origin as shown in Fig B 1 3 By symmetry the only component of E is radial E7 and this is constant at a given radius r Hence B 1 11 becomes 47rrEEo Q B 1 12 Here the integration of the charge density over the volume V enclosed by S is the total charge enclosed Q but can be formally taken by using B 1 4 with the definition provided by B 1 5 It follows from B 1 12 that E 4rEr B 1 13 a result that is in agreement with B 1 6 Because the volume and surface of integration in B 1 …
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