Chapter 11, Problem 11.1P

Show that E_x8 = Ae^j(k0z+∅) is a solution of the vector Helmholtz equation, Eq. (30), for k₀ =ωâˆ�μ₀ϵ₀ and any ∅ and A.

To determine

To show:

The solution of vector Helmholtz equation is $E_{xs}=A{\text{e}}^{j\left(k_{0}z+\varphi \right)}$.

Explanation of Solution

Given:

   $\begin{array}{l}{E}_{xs}=A{\text{e}}^{j\left(k_{0}z+\varphi \right)}\\ k_0=\omega \sqrt{{\mu }_0{\epsilon }_0}\end{array}$

The vector Helmholtz equation be $\frac{d^2{E}_{xs}}{d{z}^2}=-k_0{}^2{E}_{xs}$.

Substituting $A{\text{e}}^{j\left(k_{0}z+\varphi \right)}$ for $E_{xs}$ in the above equation in left-hand side,

   $\frac{d^2}{d{z}^2}A{\text{e}}^{j{k}_{0}z+\varphi }={\left(j{k}_0\right)}^{2}A{\text{e}}^{j{k}_{0}z+\varphi }=-k_0^2{E}_{xs}$

Hence, $E_{xs}=A{\text{e}}^{j\left(k_{0}z+\varphi \right)}$ is a solution of the vector Helmholtz equation for, $k_0=\omega \sqrt{{\mu }_0{\epsilon }_0}$ and any $\varphi$ and A.