I am reading Feynman Lecture, Volume I, and I have a question on the linked chapter about interference. In 28-4 he discusses the radiation pattern of two dipole radiators, and discusses how we can arrange them to get directions of maximal radiation.

In his experimental setting two dipoles are set $10\lambda$ apart, and are driven in phase. If we look from the top on them, then we have a maximal radiation on the horizontal line between them (the $E$-$W$-direction). Then he puts another four dipoles uniformly spaced $2\lambda$ between them, and proceeds to explain the resulting radiation pattern.

That is, the outside ones are still $10\lambda$ apart, but between them, say every $2\lambda$, we have put another antenna, and we drive them all in phase. There are now six antennas, and if we looked at the intensity in the $E$–$W$ direction, it would, of course, be much higher with six antennas than with one. The field would be six times and the intensity thirty-six times as great (the square of the field). We get $36$ units of intensity in that direction.

But as I see it, strictly they will not add to give $36$ in that direction. For if we choose some point on this $E$-$W$-direction line, and denote by $\theta_1, \theta_2, \ldots, \theta_6$ the angle this line forms with the line connecting antenna $1,2,\ldots, 6$ and the choosen point. Then by symmetry $\theta_1 = \theta_6, \theta_2 = \theta_3$ and $\theta_4 = \theta_5$. If $r$ is the distance from the midpoint between antenna $1$ and $6$ and the choosen point, then the distances the signals must overcome from each antenna are $$ d_1 = \frac{r}{\cos \theta_1}, \quad d_2 = \frac{r}{\cos \theta_2}, \quad d_3 = \frac{r}{\cos \theta_3} $$ and $d_4 = d_3, d_2 = d_5, d_6 = d_1$. So they generally are not in phase when they arrive at the point, and do not add to give the full intensity of $36$ units? Right? And if so, is there a point where they precisely add?


When we talk of radiation patterns, we're assuming the limit as $r \rightarrow \infty$. After a cursory read of chapters 28 and 29, I didn't find where Feynman explicitly states that, but I'm sure he talks about it somewhere.

So you are correct that the electric fields do not add in phase for a finite $r$, but in the limit of infinite $r$, all of the $\theta_i$s will approach zero, so the fields will sum in phase.


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