RF/Microwave Engineering and Applications in Energy Systems. Abdullah Eroglu
Чтение книги онлайн.

Читать онлайн книгу RF/Microwave Engineering and Applications in Energy Systems - Abdullah Eroglu страница 28

СКАЧАТЬ href="#ulink_3f1abf4d-261e-5c3e-88fb-61321dc23627">Figure 1.18 [1]. The net nonzero contribution occurs due to contours with having a side on the open boundary L. Then, the total result of adding the contributions for all the contours can be represented as given in (1.87) by Stokes' theorem.

      Solution

      We first calculate the left‐hand side of Eq. (1.87) as

nabla times upper T overbar equals left-parenthesis StartFraction partial-differential Over partial-differential x EndFraction ModifyingAbove x With ampersand c period circ semicolon plus StartFraction partial-differential Over partial-differential y EndFraction ModifyingAbove y With ampersand c period circ semicolon plus StartFraction partial-differential Over partial-differential z EndFraction ModifyingAbove z With ampersand c period circ semicolon right-parenthesis times left-parenthesis 4 y squared ModifyingAbove y With ampersand c period circ semicolon plus 2 italic y z ModifyingAbove z With ampersand c period circ semicolon right-parenthesis Schematic illustration of Stokes' theorem. Schematic illustration of geometry of Example 1.5. nabla times upper T overbar equals ModifyingAbove x With ampersand c period circ semicolon StartFraction partial-differential Over partial-differential y EndFraction left-parenthesis 4 italic y z squared right-parenthesis equals ModifyingAbove x With ampersand c period circ semicolon 2 z

      The differential surface area is found from

normal d s overbar equals ModifyingAbove x With ampersand c period circ semicolon italic dydz

      Then

integral Underscript upper S Endscripts left-parenthesis nabla times upper T overbar right-parenthesis dot normal d s overbar equals integral Subscript z equals 0 Superscript 1 Baseline integral Subscript y equals 0 Superscript 1 Baseline 2 italic zdydz equals 1

      Now, we calculate the right‐hand side of the equation for each segment as

      1 x = 0, z = 0, ,

      2 x = 0, y = 1, ,

      3 x = 0, z = 1, ,

      4 x = 0, y = 0, ,

      Hence

contour-integral Underscript upper C Endscripts upper T overbar period normal d l overbar equals four thirds plus 1 minus four thirds plus 0 equals 1

      As a result, it is confirmed that

integral Underscript upper S Endscripts left-parenthesis nabla times upper T overbar right-parenthesis dot normal d s overbar equals contour-integral Underscript upper C Endscripts upper T overbar dot normal d l overbar equals 1

      1.4.1 Differential Forms of Maxwell's Equations

      Maxwell's equations in differential forms in the presence of an impressed magnetic current density upper M overbar and an electric current density upper J overbar can be written as [2]

      where upper E overbar is electrical field intensity vector in volts/meter (V/m), upper H overbar is magnetic field intensity vector in amperes/meter (A/m), upper J overbar is electric current density in amperes/meter2 (A/m2), upper M overbar is magnetic current density in amperes/meter2 (A/m2), upper B overbar is magnetic flux density in webers/meter2 (W/m2), upper D overbar is electric flux density in coulombs/meter2 (C/m2), and ρ is electric charge density in coulombs/meter3 (C/m3).