tailieunhanh - Electromagnetic Field Theory: A Problem Solving Approach Part 49

Electromagnetic Field Theory: A Problem Solving Approach Part 49. 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. | Energy Stored in the Magnetic Field 455 Remember that in 16 and 17 the currents and vector potentials are all evaluated at their final values as opposed to 11 where the current must be expressed as a function of flux. 6-5-4 Magnetic Energy Density This stored energy can be thought of as being stored in the magnetic field. Assuming that we have a free volume distribution of current Jy we use 17 with Ampere s law to express Jy in terms of H i i Jy AdV i VxH -AdV 18 Jv Jv where the volume V is just the volume occupied by the current. Larger volumes including all space can be used in 18 for the region outside the current has J 0 so that no additional contributions arise. Using the vector identity V AxH H VxA -A VxH H-B A- VxH 19 we rewrite 18 as W H H-B-V- AxH dV 20 Jv The second term on the right-hand side can be converted to a surface integral using the divergence theorem I V- AxH dV AxH -dS 21 Jv Js It now becomes convenient to let the volume extend over all space so that the surface is at infinity. If the current distribution does not extend to infinity the vector potential dies off at least as 1 r and the magnetic field as 1 r. Then even though the area increases as r2 the surface integral in 21 decreases at least as 1 r and thus is zero when S is at infinity. Then 20 becomes simply PV H-BdV i i nH2dN i i dV 22 Jv Jv Jv F- where the volume V now extends over all space. The magnetic energy density is thus 1 5 2 iv H-B mH2 - 23 2 z 456 Electromagnetic Induction These results are only true for linear materials where p does not depend on the magnetic field although it can depend on position. For a single coil the total energy in 22 must be identical to 13 which gives us an alternate method to calculating the self-inductance from the magnetic Held. 6-5-5 The Coaxial Cable a External Inductance A typical cable geometry consists of two perfectly conducting cylindrical shells of radii a and b and length I as shown in Figure 6-31. An imposed current I flows axially as a .