Consider A' similar sntennas ernitting lincarly polarized...

Consider A' similar sntennas ernitting lincarly polarized electromagnetic radistion of wavelengtb $\lambda$ and velocity c. The antennss are located along the $x$ sxis at a separstion $\lambda$ from each other. An observer is located on the $x$ axis at 8 great dintance from the antennas, BHen a single antenne maliates, the observer measures en intensity (i.e., mesn-square electric-feld amplitude) equal to $I$. (a) If all the antennas are driven in phase by the seme generator of frequency $y=c / \lambda$, what is the total intensity measured by the observer? (b) If the antennss all radiate st the same frequency $v=c / \lambda$ but with completely random phases, what is the mean intensity measured by the observer?

Consider A' similar sntennas ernitting lincarly polarized electromagnetic radistion of wavelengtb $\lambda$ and velocity c. The antennss are located along the $x$ sxis at a separstion $\lambda$ from each other. An observer is located on the $x$ axis at 8 great dintance from the antennas, BHen a single antenne maliates, the observer measures en intensity (i.e., mesn-square electric-feld amplitude) equal to $I$.
(a) If all the antennas are driven in phase by the seme generator of frequency $y=c / \lambda$, what is the total intensity measured by the observer?
(b) If the antennss all radiate st the same frequency $v=c / \lambda$ but with completely random phases, what is the mean intensity measured by the observer?

Key Concepts

Incoherent Addition

Incoherent addition describes the process of summing intensities from sources that have random or varying phase relationships. In this case, because the phases are not correlated, the resultant intensity is simply the sum of the individual intensities, and no constructive interference occurs on average. This distinction is crucial when comparing scenarios with phase-aligned versus random-phase sources.

Phasor Representation

Phasor representation is a method used to express sinusoidal waveforms as complex numbers, encapsulating both magnitude and phase information. This approach allows for the straightforward addition of multiple wave contributions, particularly in the coherent case, by converting time-varying signals into static vectors in the complex plane. It is a fundamental tool in the analysis of wave interference phenomena.

Coherent Superposition

Coherent superposition refers to the addition of waves that have a fixed phase relationship, allowing their amplitudes to add directly. When waves are coherent, constructive interference can occur, leading to an intensity that is proportional to the square of the sum of the individual amplitudes. This concept is key for understanding how multiple sources can combine to produce a significantly amplified resultant wave.

Intensity of Electromagnetic Waves

The intensity of an electromagnetic wave is related to the square of its electric field amplitude. This relationship means that when multiple waves interfere, the total intensity not only depends on the individual amplitudes but also on how those amplitudes combine, whether coherently or incoherently. Understanding this concept is essential in predicting the observed power distribution in antenna arrays and other wave-based systems.

Transcript

00:01 Okay, so the signal will be out of face when the path difference equals an odd number of half wavelengths, so destructive interference, right? and so we can represent the distance from between s1 and s2 is d.

00:17 So d is 175 meters.

00:19 Y is what we don't know.

00:23 It's the length on the y -axis.

00:26 And then lambda, which is wavelength, is just speed of light over frequency.

00:33 So that's 3 times 10 to the 8 meters per second over 6 times 10 to the 6 hertz, 6 megahertz.

00:40 So that's 50 meters for the wavelength.

00:45 And so basically what's left now is to find the left -hand side.

00:51 So m plus 1 half lambda, so that's for destructive interference.

00:56 On the right -hand side is equal to you use the pythagorean theorem and you have square root of y squared plus d squared right because because this is y and this is d and so this is why and this is d and so with the by taking the squares of the of the two and and putting it in a square you have the hypotenuse minus y which is the vertical length and so you can pull y to the other side and square both sides you get y squared plus d squared equal equal equals m uh squared plus two y times m plus half lambda plus m plus m plus and plus half quantity squared, lambda squared.

02:02 So these ys cancel.

02:04 And so you have an expression for y in terms of everything else, which is y equals d squared minus m plus half squared, lambda squared over two lambda times m plus half.

02:22 And so you have, we can plug in values now.

02:27 D is 175 meters...

Please give Ace some feedback