the radiation pattern of an infinitesimal dipoleIn this problem we first investigate the radiation properties of a very short and thin filament of current. Although this filament of current as an antenna is not physically realizable, as a mathematical model it will help us to understand the general behavior of antennas. Here we also introduce some of the fundamental antenna parameters.

Problem Statement

A very thin and short antenna which works at frequency \(\omega\) has the the following current distribution.

\[\mathbf{J}\left(\mathbf{r},\omega\right)=\mathbf{J_0} \, \delta (\mathbf{r})=J_0 \, \delta (\mathbf{r}) \mathbf{\hat{z}}\]

where \(J_0\) is the amplitude of the current distribution, \(\delta (\mathbf{r})\) is delta function, and \(\mathbf{\hat{z}}\) is the unit vector in z direction of the Cartesian coordinate . Assume that the antenna (current distribution) is placed in a homogeneous environment with permitivity and permeability of \(\varepsilon\) and \(\mu\).

1. Find the generated electric and magnetic fields \(\big(\mathbf{E}\left(\mathbf{r},\omega\right), \mathbf{H}\left(\mathbf{r},\omega\right)\big)\) by the antenna.

2. Sketch the far filed radiation pattern of the antenna in transverse and vertical planes (e.g. x-y and x-z planes).

3. Calculate the half power beamwidth (HW) of the antenna in x-z plane and its directivity.

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