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

it can be seen that the time derivative of a harmonic function means multiplying the same function by jω in the frequency domain. This can be shown as

      (1.121a)

      (1.121b)

      Example 1.6 Maxwell's Equations

      Derive the phasor representation of Maxwell's equations in free space with no source.

      Solution

      We begin with the equation given in (1.88) as

(1.122)

Using the relation given in (1.122), this can be represented as

      or

(1.123)

From Eq. (1.123), we can then express the phasor representation of the Maxwell's equation as

      or

      (1.124)

This can be applied to obtain the phasor form of all of the Maxwell's equations. The phasor form of the Maxwell's equations for (1.88)–(1.91) are

      (1.125)

References

      1 Zahn, M. (1987). Electromagnetic Field Theory: A Problem Solving Approach. Krieger Pub Co.

      2 Eroglu, A. (2010). Wave Propagation and Radiation in Gyrotropic and Anisotropic Media. Springer.

Problems

      Problem 1.1

      If K(1,2,0), L(2,5,0), and M(0,4,7) are given, calculate

      1 KL × KM

      2 the angle between KL and KM

      Problem 1.2

      Find vector AB in the Cartesian coordinate system if points A(2m,π,0) and B(2m,3π/2,0) are given in a cylindrical coordinate system.

      Problem 1.3

      If and , then find unit vector nor mal to these two vectors.

      Problem 1.4

      If a plane is defined by 3x + 4y + 5z = 10, find the unit vector normal to this plane.

      Problem 1.5

      If two arbitrary vectors are given as and , how can you prove they are perpendicular?

      Problem 1.6

      Calculate the gradients of the following functions.

      1 f = x3y2z

      2  (cylindrical coordinate)

      3 

      Problem 1.7

      Calculate the divergence of the following vectors

      1 

      2 Figure 1.20 Geometry for Problem 1.9.Figure 1.21 Geometry for Problem 1.10.

      3 

      Problem 1.8

      Find the curl of the following vectors (a) K = 3xy²z⁻¹ (b) L