Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Let all sources vary as $cos ω t$ . In a linear medium all fields then vary at the same frequency, and every field and source can be replaced by its phasor. By the rule of Eq. (5.6.9) each time derivative becomes a factor $j ω$ , and Maxwell's equations, Eq. (5.1.1) in Section 5.1, turn into equations for complex functions of position only.

Maxwell's equations in time-harmonic (phasor) form

\overset{◯}{1} \nabla \times E + \frac{\partial B}{\partial t} = 0 ⟶ \nabla \times E + j ω B = 0, \overset{◯}{2} \nabla \times H - \frac{\partial D}{\partial t} = J ⟶ \nabla \times H - j ω D = J, \overset{◯}{3} \nabla \cdot D = ρ_{v} ⟶ \nabla \cdot D = ρ_{v}, \overset{◯}{4} \nabla \cdot B = 0 ⟶ \nabla \cdot B = 0. (5.6.10)

On the right, $E$ , $H$ , $D$ , $B$ , $J$ and $ρ_{v}$ stand for phasors, complex functions of $r$ only. The continuity equation of Section 5.4 becomes $\nabla \cdot J = - j ω ρ_{v}$ .

In plain words

The time is gone. There are as many equations as before, but one variable fewer: every $\partial / \partial t$ has become a multiplication by $j ω$ , and the wave problem has become a problem in space alone.

In a linear, isotropic and homogeneous medium the constitutive relations, Eq. (5.1.2) in Section 5.1, hold for the phasors as well, $D = ε E$ and $B = μ H$ . The current density splits into the conduction current, given by Ohm's law, and an impressed current $J_{i}$ , the source current fed in from outside (the current on an antenna, for instance):

J = σ E + J_{i} . (5.6.11)

For $ω \neq = 0$ the two divergence equations are not independent. Take the divergence of 1: $j ω \nabla \cdot B = 0$ , which is 4. Take the divergence of 2 and use continuity: $- j ω \nabla \cdot D = \nabla \cdot J = - j ω ρ_{v}$ , which is 3. This is the phasor version of the argument of Section 5.1, and in time-harmonic problems it means that the two curl equations contain everything.

5.6.5 Maxwell's equations in time-harmonic form