## Radiation of Two Dipole Antennas

The simplest form of an antenna array are two short dipole antennas. In this problem you will learn how a phase difference between the currents of both antennas affects their combined radiation pattern.

## Problem Statement

A single short dipole antenna with length $$L\ll\lambda$$ shall obey a current$\begin{eqnarray*} I\left(z,t\right)&=&\begin{cases} I_{0}e^{-\mathrm{i}\omega_{0}t}\left(1-2\frac{\left|z\right|}{L}\right) & \left|z\right|\leq L/2\\ 0 & \mathrm{else} \end{cases}\ . \end{eqnarray*}$Calculate the dipole moment $$\mathbf{p}\left(t\right)$$ of such a current and its radiated power.

Now, suppose that two of these antennas are placed along the $$z$$ axis with a separation $$d$$. Assume that there is an intrinsic phase difference between both currents, say $$I_{0,2}=I_{0,1}\exp\left(-\mathrm{i}\Delta\phi_{21}^{0}\right)$$. Then, what is the angular distribution of the radiated power $$dP/d\Omega$$ in this case? Discuss the qualitative behavior of $$dP/d\Omega$$ for $$\Delta\phi_{21}^{0}=0$$ and $$\pi/2$$ with both $$d=\lambda$$ and $$\lambda/2$$.

## Background: Radiation of a Dipole

The time-averaged radiated power of a dipole with dipole moment $$\mathbf{p}$$ oscillating at some angular frequency $$\omega_{0}$$ is given by$\begin{eqnarray*} P&=&\frac{\omega_{0}^{4}}{12\pi\varepsilon_{0}c^{3}}\left\langle \mathbf{p}^{2}\right\rangle \ . \end{eqnarray*}$This relation can be calculated from the time-averaged Poynting vector of such a source in the farfield,$\begin{eqnarray*} \left\langle \mathbf{S}\right\rangle &=&\frac{\omega_{0}^{4}}{32\pi^{2}\varepsilon_{0}c^{3}}\frac{\left\langle \mathbf{p}^{2}\right\rangle \sin^{2}\theta}{r^{2}}\mathbf{e}_{r}\equiv\frac{1}{r^{2}}\frac{dP}{d\Omega}\mathbf{e}_{r}\ , \end{eqnarray*}$where $$\theta\in\left[0,\pi\right]$$ is the angle between $$\mathbf{p}$$ and a vector at the position of (thought) measurement. If $$\mathbf{p}$$ is along the $$z$$ axis, this angle is the usual polar angle in spherical coordinates. The quantity $$dP/d\Omega$$ is usually termed the angular distribution of the radiated power and goes characteristically with $$\sin^{2}\theta$$ for a dipole. Note that as usual $$d\Omega=\sin\theta d\theta d\varphi$$.