Two radio antennas A and B radiate in phase. Antenna B is 120 m to the right of antenna A. Consider point Q along the extension of the line connecting the antennas, a horizontal distance of 40 m to the right of antenna B. The frequency, and hence the wavelength, of the emitted waves can be varied. (a) What is the longest wavelength for which there will be destructive interference at point Q?

Video transcript

Hey everyone in this problem. Digital television broadcast towers F and G are 257 m apart. Tower F is located left of tower G. Okay. Their antennas ready in phase at the same frequency, a tv antenna is located at Point H, which is one kilometer to the right of antenna ji on the axis connecting the two antennas, Tv networks broadcast at various frequencies that are called channels, assuming a channel can have any frequency. And similarly wavelength were asked to determine the longest wavelength of a channel that will cause destructive interference at point H. Alright, so let's draw this out. Okay, we have two towers and we're just gonna draw them as points. Okay, Just to make the diagram simpler. Okay, we have tower F located to the left of tower G. Okay, so this one on the left is going to be F. The one on the right is going to be G. The distance between the two is m. Now we have a tv antenna that is located at POINT H. And that is one kilometer to the right of antenna gee. Okay, so over here this is going to be point H. This is where we have our tv antenna and it is one kilometer to the right of G along that same axis. So they kind of make a straight line and just note this is not drawn to scale. Alright, so we have our diagram here. Now we're asked to find the longest wavelength that causes destructive interference and when we're thinking about destructive interference, what we want to think about is the path difference. Okay, that's really important and the path difference is going to be the difference in the distance that one wave has to travel to get to h from the other wave. Okay. So let's first figure out the distance that the wave has to travel from F to H and then the distance that it has to travel from G to H. Okay. And the difference between those is going to give us our path difference. So we're gonna say that D. F. It's gonna be the distance from broadcast tower F to T V antenna H. And let's just convert this one kilometer. Okay, Everything else is a meter. So we're going to convert one kilometer first, before we do this, this is going to be one kilometer times m per kilometer. Okay, So when we're converting from kilometer two m, we multiply by 1000 those units of kilometer cancel. And we're left with 1000 m. All right, so the difference from F two H. Well, these all lie along the same axis. Okay. And so we can just add the distances between F and G. And then the distance between G and H. So this is going to be 257 m plus 1000 m. Okay, Which gives us a distance of 1257 m. Alright, we're gonna do the same for point G. Okay, Or broadcast tower G. Okay. So D. G. Is going to be the distance from broadcast tower G to T. V antenna H. And this we know is a 1000 meters. Alright, so we have our two distances now. We want to calculate that path difference and recall that the path difference, we're gonna call it D. It's just gonna be the absolute value of the difference between these two distances. Okay, D. F minus D. G. You could write it the other way around if you want A D. G minus D. F. We have the absolute value. All we care about is the magnitude of the difference. Okay, so either way you write it as long as you have those absolute values, you're okay, So this is going to be 1,257 m -1000 m. Okay, we can drop the absolute values because we wrote the bigger one first. So this is going to give us a positive value. We get 257 m for our path difference. Alright, so now we have our path difference and we want to find the longest wavelength for destructive interference. Okay, so what's the relationship between path difference and destructive interference? Well, let's write it out. Okay, recall that destructive interference. Okay, it's gonna occur when the path difference. D Okay. Is a half multiple of the wavelength. Okay, so when we have The path difference equal to λ over two three lambda over two dot that. Okay. In general we can write this as D. Is equal to N. Lambda over two. Where lambda is equal to 1357 dot dot dot K. So lambda is odd. All right. Why do we want this? Okay, why does destructive interference occur here? Well, if the path difference is half a wavelength or an intruder multiple of half a wavelength different. Okay, that means that when the two waves arrive at point H. Okay, they're gonna be out of face and if they're out of face, those waves are gonna kind of cancel each other out. Okay. And that is what destructive interferences. Okay? So because the difference in the distance that they travel is a multiple of half a wavelength. Okay, they're out of phase when they arrive and we get destructive interference. All right. So we want to find lambda and we want to find when lambda is the longest. Okay, So when lambda is the biggest, So let's isolate for lambda and we're gonna call it lambda subscript N. Because it's going to depend on that value of N. Okay, So we get lambda, N. Is equal to two D. Divided by N. Now we want to find the largest Lamba. Okay, the largest wave length are the longest wavelength. Okay? It's gonna occur when N. Is its smallest. Okay, we're dividing by N. So the smaller we can make end the larger this whole thing is going to be Okay, what's the smallest value of N. Well that's gonna be an is equal to one. Okay, so lambda will call it lambda max. For the longest wavelength or the largest wavelength. It's going to be equal to lambda one. It's going to be equal to two times our path difference. 257 m, divided by the end value, which is one Which gives us a lambda of 514 m. Alright, So if we go back up to our answer choices, we see that we have answer choice. A. Okay. The longest wavelength that's gonna cause destructive interference at point H. Is going to be 514 m. Thanks everyone for watching. I hope this video helped see you in the next one.