The Finite Dipole Antenna - small, half-wavelength and larger

One of the most widely used antennas in telecommunication is the dipole antenna. In this problem, we investigate the radiation properties of such an antenna. We first calculate the electromagnetic fields far away from the antenna. Then, as a special cases we discuss half-wavelength, full-wavelength and larger dipole antennas.

As the name suggests, a dipole antenna consist of two terminals (poles), where the radio frequency current is applied by a transmitter or extracted by a receiver. One of the simplest form of dipole antennas consist of two pieces of straight and thin metallic wire, which are oriented on a common axis. Generally, each of these wires may have different lengths depending on the application. Below an image of such an antenna is shown. In one special case the lengths of both wires are equal to one forth of the wavelength of the radiating wave. Therefore, the length of the whole antenna is equal to half of the wavelength. Thus, this antenna is called half-wave dipole antenna.

Problem Statement

A symmetric and very thin dipole antenna which works at frequency $$\omega$$ is placed in a homogeneous environment with permitivity and permeability of $$\varepsilon$$ and $$\mu$$. It can be shown that the antenna has approximately the following current distribution. $\mathbf{J}\left(\mathbf{r},\omega\right) = \left\{ \begin{array}{l l} I_0 \, \sin\left(k(\frac{l}{2}-|z|)\right) \,\frac{\delta(\rho)}{2\pi\rho}\, \mathbf{e}_z & \quad |z|<\frac{l}{2}\\ 0 & \quad |z|\geq\frac{l}{2} \end{array} \right.$ where, $$I_0$$ is the current amplitude at the feed point of the antenna, $$k=\frac{2\pi}{\lambda}$$, $$\lambda$$ is the wavelength of the radiating wave, $$l$$ is the total length of the antenna, $$\delta (\mathbf{\rho})$$ is delta function, $$\rho$$ is the radial distance from the z axis in cylindrical coordinate, and $$\mathbf{e}_z$$ is the unit vector in z direction.

1. Find the far field electric and magnetic fields $$\big(\mathbf{E}\left(\mathbf{r},\omega\right), \mathbf{H}\left(\mathbf{r},\omega\right)\big)$$ generated by the antenna.
2. Approximate your results for $$l\ll\lambda$$ and compare them to the results of "The Infinitesimal Dipole". In this case, calculate the radiation resistance of the antenna.
3. Sketch approximately the far filed radiation pattern in principal planes for
1. Half-wave dipole antenna ($$l=\frac{\lambda}{2}$$)
2. Full-wave dipole antenna ($$l=\lambda$$)
3. $$l=1.4\lambda$$
4. $$l=1.5\lambda$$