Phasors

If all field quantities vary sinusoidally with time, with angular frequency $\omega_0$, the electric field (1.31) may be written as:

$$\underline{\mathcal{ E}}(\underline{r}, t)= \frac{1}{2} \left[ \u... ...hrm{Re} \left\{ \underline{E}(\underline{r}) e^{j \omega_0 t} \right\} \space ,$$ (1.50)

where the asterisk means the complex conjugate, and

$$\underline{E}(\underline{r})=\underline{E}_0 e^{j \varphi( \underline{r} ) }$$ (1.51)

should be compared with the field $\underline{\tilde E}$ of (1.33) in the frequency domain. While $\underline{E}$ is measured in $\mathrm{V/m}$, $\underline{\tilde E}$ is measured in $\mathrm{V \, s /m}$ as seen from (1.23). The electric field $\underline{\mathcal{E}}$ is obtained from $\underline{E}$ by multiplying $\underline{E}$ times the time-dependence factor $\exp(j \omega_0 t)$ and taking the real part of the product, and from $\underline{\tilde E}$ via the inverse transform (1.24).

When all field quantities are written as phasors, Maxwell's equations assume the form (1.25-1.30) and are thus indistinguishable from the equations in the frequency domain. Basically, phasor quantities and frequency-domain quantities lead to the same analytical derivations. Unless indicated otherwise, in the following we will use phasors and indicate the angular frequency with $\omega$ instead of $\omega_0$. Then Maxwell's equations and the continuity equations in the phasor domain are:

$$\nabla \times \underline{H} =\underline{J}_e + j \omega \underline{D} \space ,$$ (1.52)

$$\nabla \times \underline{E} =-\underline{J}_m - j \omega \underline{B} \space ,$$ (1.53)

$$\nabla \cdot \underline{D} = \rho_e \space ,$$ (1.54)

$$\nabla \cdot \underline{B} = \rho_m \space ,$$ (1.55)

$$\nabla \cdot \underline{J}_e + j \omega \rho_e =0 \space ,$$ (1.56)

$$\nabla \cdot \underline{J}_m + j \omega \rho_m = 0 \space .$$ (1.57)

The scalar phasors $\rho_e$ and $\rho_m$ are complex scalar quantities. The vector phasors $\underline{E}$, $\underline{H}$, $\underline{D}$, $\underline{B}$, $\underline{J}_e$, and $\underline{J}_m$ may be seen as vectors whose components are complex scalar quantities or, equivalently, as complex vectors whose real and imaginary parts are vectors in the ordinary sense.