tailieunhanh - Electromagnetic Field Theory: A Problem Solving Approach Part 14

Electromagnetic Field Theory: A Problem Solving Approach Part 14. 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. | The Method of Images with Point Charges and Spheres 105 Since 4 must be true for all values of 0 we obtain the following two equalities 02 2 I 2 Z 2 D2 q2b q 2D Eliminating q and q yields a quadratic equation in b-. We take the lower negative root so that the image charge is inside the sphere with value obtained from using 7 in 5 R2 R b q q D q D 8 remembering from 3 that q and q have opposite sign. We ignore the b D solution with q -q since the image charge must always be outside the region of interest. If we allowed this solution the net charge at the position of the inducing charge is zero contrary to our statement that the net charge is q. The image charge distance b obeys a similar relation as was found for line charges and cylinders in Section . Now however the image charge magnitude does not equal the magnitude of the inducing charge because not all the lines of force terminate on the sphere. Some of the field lines emanating from q go around the sphere and terminate at infinity. The force on the grounded sphere is then just the force on the image charge q due to the field from q f - r- qZR - q2RD lx 4ire0 D-b 2 4ire0D D-b 2 4ite0 D2-R2 2 v 106 The Electric Field The electric field outside the sphere is found from 1 using 2 as E -W - cos 0 ir D sin 0ia 4ire0 J r cos 0 ir i sin 10 On the sphere where s RID s the surface charge distribution is found from the discontinuity in normal electric field as given in Section r r - R - e0ET r -R - R R2 D _2RD cos ID The total charge on the sphere tr r R 2ttR2 sin 0d0 qR D2 Sing f 12 2 D Jo R2 D2-2RZ cos 0 s 2 2 can be evaluated by introducing the change of variable u R2 D2 2RD cos 0 du 2RD sin 0 d0 13 so that 12 integrates to _ q D2 R2 f D R 2 du qT 4D V q D2-R2 t 2 D R 2 _ R 4D u1 2 D-Kf D 14 which just equals the image charge q . If the point charge q is inside the grounded sphere the image charge and its position are still given by 8 as illustrated in Figure 2-276. Since D R the image charge is now outside