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

The phasor equations can now be solved for $E$ and $H$ separately, in the same way as the wave equations of Section 5.5 were obtained in the time domain. Consider a homogeneous medium with $μ$ and $ε_{c}$ , containing the impressed sources $J_{i}$ and $ρ_{v}$ .

The equation for the electric field

Take the curl of 1, written as $\nabla \times E = - j ω μ H$ :

\nabla \times \nabla \times E = - j ω μ \nabla \times H .

Insert 2 in the form of Eq. (5.6.13), $\nabla \times H = J_{i} + j ω ε_{c} E$ :

\nabla \times \nabla \times E = - j ω μ J_{i} + ω^{2} μ ε_{c} E .

Use the vector identity $\nabla \times \nabla \times E = \nabla (\nabla \cdot E) - \nabla^{2} E$ (Section 2.2) with $\nabla \cdot E = ρ_{v} / ε_{c}$ (see the box below):

⟹ \frac{1}{ε _{c}} \nabla ρ_{v} - \nabla^{2} E = - j ω μ J_{i} + ω^{2} μ ε_{c} E \nabla^{2} E + ω^{2} μ ε_{c} E = \frac{1}{ε _{c}} \nabla ρ_{v} + j ω μ J_{i} .

The equation for the magnetic field

Take the curl of 2:

\nabla \times \nabla \times H = \nabla \times J_{i} + j ω ε_{c} \nabla \times E = \nabla \times J_{i} + ω^{2} μ ε_{c} H,

and with $\nabla \cdot H = 0$ (from 4), $- \nabla^{2} H = \nabla \times J_{i} + ω^{2} μ ε_{c} H$ .

The Helmholtz equations

In a homogeneous medium with permeability $μ$ and complex permittivity $ε_{c}$ ,

\nabla^{2} E + k^{2} E = \frac{1}{ε _{c}} \nabla ρ_{v} + j ω μ J_{i}, \nabla^{2} H + k^{2} H = - \nabla \times J_{i}, (5.6.15)

with the wave number

k = ω^{2} μ ε (1 - j \frac{σ}{ω ε}) = ω μ ε_{c} = β - j α [m^{- 1}], (5.6.16)

where $\nabla^{2} = Δ$ is the Laplacian. $β$ is the phase constant in $rad/m$ and $α$ the attenuation constant in $Np/m$ . In a source-free region ( $ρ_{v} = 0$ , $J_{i} = 0$ )

\nabla^{2} E + k^{2} E = 0, \nabla^{2} H + k^{2} H = 0, (5.6.17)

and each Cartesian component obeys the scalar equation $\nabla^{2} U + k^{2} U = 0$ .

In plain words

The Helmholtz equation is the wave equation for one frequency. Where the wave equation of Section 5.5 had $- μ ε \partial^{2} / \partial t^{2}$ , the Helmholtz equation has $+ ω^{2} μ ε_{c}$ , a number.

The comparison can be made exact. In a source-free conducting medium the wave equation of Section 5.5 carries the term $σ \partial / \partial t$ ,

\nabla^{2} E - μ ε \frac{\partial ^{2} E}{\partial t ^{2}} - μ σ \frac{\partial E}{\partial t} = 0 phasor \nabla^{2} E + ω^{2} μ ε E - j ω μ σ E = 0,

and $ω^{2} μ ε - j ω μ σ = ω^{2} μ (ε - j σ / ω) = k^{2}$ . The first derivative in time, which made the lossy wave equation awkward, has become the imaginary part of $k^{2}$ . In a lossless medium $k = ω μ ε$ is real, $α = 0$ and $β = ω / v_{p}$ , and in free space

k_{0} = \frac{ω}{c} = \frac{2 π}{λ _{0}} . (5.6.18)

Common mistake · Using the Helmholtz equation alone

A source-free field must satisfy both $\nabla^{2} E + k^{2} E = 0$ and $\nabla \cdot E = 0$ . The Helmholtz equation was derived with the divergence condition already used, and it does not imply it. The field $5 e^{- j 2 π x} a_{x}$ in free space at $300 MHz$ satisfies the Helmholtz equation, but it points along its own direction of variation, has $\nabla \cdot E \neq = 0$ and cannot exist without charge (Exercise 5.6.6).

5.6.7 The Helmholtz equations

The equation for the electric field

The equation for the magnetic field