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

In (1.117), is defined as the phasor form of the function g ( r , t ) and expressed as

(1.118)

This can be applied for vectorial function as

(1.119)

The representation of the time harmonic functions in phasor form provides several advantages. They convert the time domain differential equations to frequency domain algebraic equations. This can be better understood by studying the derivative property as follows. Let's take derivative function g ( r , t ) with respect to time as

(1.120)

As a result of (1.120), it can be seen that the time derivative of a harmonic function means multiplying the same function by j ω in the frequency domain. This can be shown as

(1.121a)

(1.121b)

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

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)

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.

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

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

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

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

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

Calculate the gradients of the following functions.

1 f = x3y2z

2 (cylindrical coordinate)

3

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

Find the curl of the following vectors (a) K = 3xy 2 z −1 (b) L= (sin 2 ϕ ) r − r 2 z cos ϕ (c) M= R 2sin θ + r (cos ϕ ) −1

A= y 2 + x 2 . Verify Stokes' theorem for the geometry given in Figure 1.20.

Prove divergence theorem using the function

and a cube situated at the origin as in Figure 1.21.