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Electromagnetic Field Theory: A Problem Solving Approach Part 33

Electromagnetic Field Theory: A Problem Solving Approach Part 33. 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. | Product Solutions in Spherical Geometry 295 which gives us a window for charge collection over the range of angle where cos0 -in 35 12ireE0R Since the magnitude of the cosine must be less than unity the maximum amount of charge that can be collected on the sphere is Qs l2treE0R2 36 As soon as this saturation charge is reached all field lines emanate radially outward from the sphere so that no more charge can be collected. We define the critical angle 0C as the angle where the radial electric field is zero defined when 35 is an equality cos 0C QIQS. The current density charging the sphere is Jr pop.Er r R 3poP-Eo cos 0 QIQs 0c 0 ir 37 The total charging current is then i Jt2itR2 sin 0d0 dt 6TrpopEoR2 i cos 0 QI Qi sin 0 d0 Je ec 6irp0P E0Ri -4 cos 20 - QIQS cos 0 1 e s -6irpoM of 2 -4 l cos 20J QIQs 1 cos 0C 38 As long as Q Qs 0C is defined by the equality in 35 . If Q exceeds Qt which can only occur if the sphere is intentionally overcharged then 0C ir and no further charging can occur as dQldt in 38 is zero. If Q is negative and exceeds Q in magnitude Q Q then the whole sphere collects charge as 0C 0. Then for these conditions we have r-i q q. cos 0C -QIQ .1 -Q Q Qt Q -Qs 39 cos 20c 2 cos2 0C - I I 2 jQ 40 Q QJ2- Q QS 296 Electric Field Boundary Value Problems so that 38 becomes 2 a di Q Q 41 Pop Q e Q Q -a with integrated solutions Qo Q Q Q Qo t to . Qo Qs 4t qJ -Q Q Q 42 4r QJ Qo -n Q.e Q -Qs where Qo is the initial charge at t 0 and the characteristic charging time is T poM 43 If the initial charge Qo is less than Q the charge magnitude decreases with the exponential law in 42 until the total charge reaches Qs at 1 i0. Then the charging law switches to the next regime with Qo Qs where the charge passes through zero and asymptotically slowly approaches Q Qs. The charge can never exceed Q. unless externally charged. It then remains constant at this value repelling any additional charge. If the initial charge Qo has magnitude less than Q then t0 0. The time .