tailieunhanh - Electromagnetic Field Theory: A Problem Solving Approach Part 25

Electromagnetic Field Theory: A Problem Solving Approach Part 25. Electromagnetic field theory is often the least popular course in the electrical engineering curriculum. Heavy reliance on vector and integral calculus can obscure physical phenomena so that the student becomes bogged down in the mathematics and loses sight of the applications. This book instills problem solving confidence by teaching through the use of a large number of worked problems. To keep the subject exciting, many of these problems are based on physical processes, devices, and models. This text is an introductory treatment on the junior level for a two-semester electrical engineering. | Fields and Their Forces 215 sheet x x O Eln Ex x 5 E2n 5 so that the electric field is Ex E in Ein Ein 6 d As the slab thickness 8 becomes very small we approach a sheet charge relating the surface charge density to the discontinuity in electric fields as hm p0S a7 e E2 - iB PO- S- 0 7 Similarly the force per unit area on the slab of volume charge is s s f x 1 l Pol .Ein ln lnl d Jq L d J r 2 11s PoCE2n- ln 7 lnX L 4Ô Jlo ln 2n 8 In the limit of 7 the force per unit area on the sheet of surface charge agrees with 3 lim Fx ln 2n J 1n- n 0 Po8-trf 2 4 3-9-2 Forces on a Polarized Medium a Force Density In a uniform electric field there is no force on a dipole because the force on each charge is equal in magnitude but opposite in direction as in Figure 3-34a. However if the dipole moment is not aligned with the field there is an aligning torque given by t p X E. The torque per unit volume T on a polarized medium with N dipoles per unit volume is then T Art NpXE PxE 10 216 Polarization and Conduction Unifprm field Nonuniform field a b Figure 3-34 a A torque is felt by a dipole if its- moment is not aligned with the electric field. In a uniform electric field there is no net force on a dipole because the force on each charge is equal in magnitude but opposite in direction b There is a net force on a dipole only in a nonuniform field. For a linear dielectric this torque is zero because the polarization is induced by the field so that P and E are in the same direction. A net force can be applied to a dipole if the electric field is different on each end as in Figure 3-346 f -ff E r -E r d 11 For point dipoles the dipole spacing d is very small so that the electric field at r d can be expanded in a Taylor series as d d d E r d E r dx E r d E r dz E r dx dy dz E r d V E r 12 Then the force on a point dipole is f qd V E r p V E r 13 If we have a distribution of such dipoles with number density N the polarization force density is F Nf Np V E P- V E 14 Of course if there is any