Problems

P1-1 Prove that (1.76) and (1.107) are equivalent.

P1-2 With reference to Fig. 1.4, consider an electric current element at the origin of coordinates, oriented along the z-axis, and having length $\Delta l << \lambda_0$. Let $I$ be the phasor representing the current. Calculate the field components at a point of spherical coordinates $(r, \theta, \phi)$ located anywhere in the free space surrounding the current element. Prove that:

Figure 1.4: Geometry for Problem P1-1.

$$\left\{ \begin{aligned}[l]E_{\phi} &= H_r=H_{\theta}=0 \\ E_r &=... ..._0 r}}{r} \left( jk_0 + \frac{1}{r} \right) \sin \theta . \end{aligned} \right.$$

Observe that the radiated field, which involves terms whose amplitude is proportional to $r^{-1}$, has only $E_{\theta}$ and $H_{\phi}$ components related by $E_{\theta}=Z_0H_{\phi}$

P1-3 In phasor domain, a special class of Hertz vectors, especially useful when spherical coordinates are employed, is given by

$$\underline{\Pi}_e=u \underline{r} \space , \underline{\Pi}_m=v \underline{r} \space ,$$ (1.144)

where $\underline{r}=r \hat{r}$ is the oriented distance from a fixed origin, and the scalar functions of position $u$ and $v$ are called Debye potentials. Prove that, in a source free region of space, $u$ and $v$ satisfy:

$$\left( \nabla^2 + k_0^2 \right) \left\{ \begin{array}{l} u \\ v \end{array} \right. = 0 \space .$$ (1.145)

It was proven by Wilcox (1957) that every electromagnetic field in a source free region of space between two concentric spheres can be represented in that region by two Debye potentials, $u$ and $v$. Prove that the field components in the concentric region are:

$$\left\{ \begin{aligned}E_r&=\left( \frac{\partial ^2}{\partial r^... ...\varphi}(rv)- j k_0Y_0 \frac{\partial u}{\partial \theta} \end{aligned} \right.$$