Abdullah Eroglu - RF/Microwave Engineering and Applications in Energy Systems

α in (1.79)is the angle between the xy plane and surface 4. Substituting (1.79)into (1.78)gives

(1.80)

Then, from (1.69)

(1.81)

In (1.81), the first three terms are zero, hence (1.81)reduces to

or

(1.82)

From (1.76)

(1.83)

Hence, (1.82)can be written as

(1.84)

(1.84)leads to

(1.85)

The left‐hand side of the equation now can be found as

(1.86)

Stokes' theorem states that the line integral of a vector around a closed contour is equal to the surface integral of the curl of a vector over an open surface. It is expressed by

(1.87)

This can be illustrated in Figure 1.18. The small interior contours have adjacent sides that are in opposite directions yielding no net line integral contribution, as shown in 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.

Please verify Stokes' theorem for the geometry given in Figure 1.19for the vector

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

Figure 1.18 Illustration of Stokes' theorem.

Figure 1.19 Geometry of Example 1.5.

which leads to

The differential surface area is found from

Then

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

As a result, it is confirmed that

for Example 1.5.

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

(1.88)

(1.89)

(1.90)

(1.91)

where is electrical field intensity vector in volts/meter (V/m), is magnetic field intensity vector in amperes/meter (A/m), is electric current density in amperes/meter 2(A/m 2), is magnetic current density in amperes/meter 2(A/m 2), is magnetic flux density in webers/meter 2(W/m 2), is electric flux density in coulombs/meter 2(C/m 2), and ρ is electric charge density in coulombs/meter 3(C/m 3).

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