RF/Microwave Engineering and Applications in Energy Systems. Abdullah Eroglu. Читать онлайн. Hotlib. HOTLIB.NET

RF/Microwave Engineering and Applications in Energy Systems

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.

Figure 1.18 Illustration of Stokes' theorem.

Schematic illustration of geometry of Example 1.5.

Figure 1.19 Geometry of Example 1.5.

which leads to

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

for Example 1.5.

1.4 Maxwell's Equations

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]

(1.88) nabla times upper E overbar equals minus StartFraction partial-differential upper B overbar Over partial-differential t EndFraction minus ModifyingAbove upper M With bar left-parenthesis Faraday prime normal s l a w right-parenthesis

(1.89) nabla times upper H overbar equals StartFraction partial-differential upper D overbar Over partial-differential t EndFraction plus ModifyingAbove upper J With bar left-parenthesis Ampere prime normal s l a w right-parenthesis

(1.90) nabla dot upper B overbar equals 0 left-parenthesis Gauss prime normal s l a w for magnetic field right-parenthesis

(1.91) nabla dot upper D overbar equals rho left-parenthesis Gauss prime normal s l a w for electric field right-parenthesis

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).

The continuity equation is derived by taking the divergence of Eq. (1.89) and using Eq. (1.91) as

(1.92) nabla dot left-parenthesis nabla times upper H overbar right-parenthesis equals nabla dot left-parenthesis StartFraction partial-differential upper D overbar Over partial-differential t EndFraction plus upper J overbar right-parenthesis

Since nabla dot left-parenthesis nabla times upper H overbar right-parenthesis equals 0 , then

(1.93) nabla dot upper J overbar equals minus StartFraction partial-differential rho Over partial-differential t EndFraction

Equation (1.93) also represents the fundamental law of physics which is known as conservation of an electric charge.

In an isotropic medium, the material properties do not depend on the direction of the field vectors. In other words, the electric field vector is in parallel with the electric flux density and the magnetic field vector is in parallel with the magnetic flux density.

(1.94) upper D overbar equals epsilon upper E overbar

(1.95)

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RF/Microwave Engineering and Applications in Energy Systems

1.4 Maxwell's Equations 1.4.1 Differential Forms of Maxwell's Equations

1.4 Maxwell's Equations

1.4.1 Differential Forms of Maxwell's Equations