Analyzing Electromagnetic Radiation Intensity: N Antennas with Random Phases

[buttersrocks]

1. Consider N similar antennas emitting linearly polarized electromagnetic radiation of wavelength $$\lambda$$ and velocity c. The antennas are located along the x-axis at a separation $$\lambda$$ form each other. An observer is located on the x-axis at a great distance from the antennas. When a single antenna radiates, the observer measures an intensity (i.e., mean-square electric-field amplitude) equal to I.

(a) If all the antennas are driven in phase by the same generator of frequency $$\nu = c / \lambda$$, what is the total intensity measured by the observer?

(b) If the antennas all radiate at the same frequency $$\nu=c / \lambda$$ but with completely random phases, what is the mean intensity measured by the observer? (Hint: Represent the amplitudes by vectors, and deduce the observed intensity from the resultant amplitude.)

Homework Equations

None in particular3. The Attempt at a Solution
I know the answer to part (a) should be I(n^2). My problem is when I get to part b. Using E(X) to mean the expectation value of X, I get the following:


The first sum on the left isn't displaying like it should. The entire Sum should be squared, not just the Ai. That's only true for the lefthand sides.

$$E\left[{\left(\sum_i{A_i}\right)}^2\right]=E\left[\sum_i{A_i^2}+\sum_{i \neq j}{A_i A_j}\right]$$
$$E\left[(\sum_i{A_i})^2\right]=\sum_i{E\left[A_i^2\right]}+\sum_{i \neq j}E\left[A_i A_j\right]$$

At this point I know that the leftmost sum is nI and that the rightmost sum is functions of two independent random variables multiplied together, which is just the expected values of each multiplied together and therefore 0. So, I get nI as my final answer. This just doesn't sit right though because it seems to me that if the phase is entirely random, as n->infinity we should see a flat E-field (it would be equally likely for any particular A to be in phase as out of phase with any other and since they would be in equal numbers, they should all cancel out) and therefore the expected intensity should be 0. What am I doing wrong, or, is my intuition leading me astray?

 

[buttersrocks]

solution

Hi,
Figured since nobody replied either A, nobody cares or B, nobody knows the answer. I wrote a lil computer code to do a bit of an experiment. Turns out that my nI answer is correct. I interpret it as follows:

The nI is the result of the intensity due to each individual antenna as if the others were not even present. Because we expect the interferences send the e-field to zero, there is no collective contribution. The way I'm thinking about it, from the perspective of each antenna the E field remains unperturbed by the other N-1 antennas as N-> infinity, since for each constructive contribution there is an equal destructive contribution, therefore negating any change made. I guess, in a manner of speaking, what really matters is the phase differences to one another, which are equally likely to be constructive and destructive.