Page 61 - Maxwell House
P. 61
BASIC EQUATIONS OF MACROSCOPIC ELECTRODYNAMICS 41
1
2
2
2
〈 () 〉 = |()| cos ()
2 � (1.84)
1
2
2
2
〈 () 〉 = |()| sin ()
2
If so, the total average energy of the monochromatic signal is
1
〈() 〉 = 〈 () 〉 + 〈 () 〉 = |()| (1.85)
2
2
2
2
2
or using complex notifications
1
∗
2
〈() 〉 = () ∙ ()
2 � (1.86)
2
|()| = �2〈() 〉
Here () = () + () and () = () − (), the complex conjugative
∗
of ().
1.8.2 Maxwell’s Equations in Phasor Form
According to (1.78), the derivative of () is
() ∞
= ∫ [()] exp() (1.87)
−∞
Therefore, the time differentiation of the signal () has effect of multiplying its spectrum ()
by the factor . By means of this result, we may apply the Fourier transform (1.77) to
Maxwell’s equations in Table 1.7 and obtain Maxwell’s equations shown in Table 1.9 for
monochromatic EM field in the spectral complex number form = (, ), = (, ),
= (, ), = (, ), and so on. It should be noticed that all these quantities, in
general, have in-phase and quadrature components.
Table 1.9
Integral Form Differential Form
1 − � ∘ = � ∘ + + − x = + +
2 � ∘ = � ∘ + + x = + +
3 � ∘ = + ∘ = +
4 � ∘ = + ∘ = +
5 � ∘ = − ∘ = −
6 � ∘ = ∘ =
0
7 = � ∘ =
