tailieunhanh - Electromagnetic Field Theory: A Problem Solving Approach Part 37

Electromagnetic Field Theory: A Problem Solving Approach Part 37. 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. | Divergence and Curl of the Magnetic Field 335 5-3-3 Currents With Cylindrical Symmetry a Surface Current A surface current Xoiz flows on the surface of an infinitely long hollow cylinder of radius a. Consider the two symmetrically located line charge elements di Ko adcf and their effective fields at a point P in Figure 5-1 la. The magnetic field due to both current elements cancel in the radial direction but add in the t direction. The total magnetic field can be found by doing a difficult integration over . However i B i Bi 4 82 A fraction of the current c a b Figure 5-11 a The magnetic field of an infinitely long cylinder carrying a surface current parallel to its axis can be found using the Biot-Savart law for each incremental line current element. Symmetrically located elements have radial field components that cancel but t field components that add. b Now that we know that the field is purely f directed it is easier to use Ampere s circuital law for a circular contour concentric with the cylinder. For r a no current passes through the contour while for r a all the current passes through the contour c If the current is uniformly distributed over the cylinder the smaller contour now encloses a fraction of the current. 336 The Magnetic Field using Ampere s circuital law of 19 is much easier. Since we know the magnetic field is 4 directed and by symmetry can only depend on r and not or z we pick a circular contour of constant radius r as in Figure 5-116. Since dl r d f is in the same direction as B the dot product between the magnetic field and dl becomes a pure multiplication. For r a no current passes through the surface enclosed by the contour while for r a all the current is purely perpendicular to the normal to the surface of the contour f B f 2irrB K02ira I r a p dl I r dtp ---------- JlMo Jo Mo Mo I 0. r a 20 where I is the total current on the cylinder. The magnetic field is thus _ noKoa r mo 2ut r a r a 2l Outside the cylinder the magnetic field is the .