Two satellites at an altitude of 1200 km are separated by 28 km. If they broadcast 3.6-cm microwaves, what minimum receiving-dish diameter is needed to resolve (by Rayleigh’s criterion) the two transmissions?

Video transcript

Hello, fellow physicists today, we're gonna solve the following practice problem together. So first off, let us read the problem and highlight all the key pieces of information that we need to use in order to solve this problem. Two drones are flying approximately 30 kilometers apart from each other at an altitude of approximately 1500 kilometers using relays criterion, determine the minimum size necessary for a receiving antenna to resolve the two transmissions if they transmit radio waves at a wavelength of five centimeters. So that's our end goal is we're trying to figure out what the minimum size is necessary in order to receive two transmissions if they are transmitting radio waves. So I have a wavelength of five centimeters. So we're trying to figure out what the minimum size of this receiving antenna has to be in order to pick up on these signals. So with that in mind, let's read off our multiple choice answers to see what our final answer might be. And let us note that they're all in units of meters. So A is one B is two C S3 and D is five. So first off, we need to recall and use relay's criterion for resolution. So we also need to recall and note that according to relay's criterion, the minimum angular separation theta that can be resolved by a circular aperture is given by the following equation which we should recall and write as theta is equal to 1.22 multiplied by LAMBDA divided by capital D where LAMBDA is the wavelength of the radio waves and capital D is the diameter of the receiving dish. Awesome. So now let's take a moment here to quickly write down all of our given variables. So let's make a little side note in blue. So we're told that our wavelength Lambda is equal to five centimeters, which is also equal to 0.05 m. So I've gone ahead and given you the conversion, but you can use dimensional analysis or you can quickly look it up. But to save time, I've just given you the conversion. And now we also know that the altitude of the drones, which we're gonna call, H for height is equal to 1500 kilometers, which is also equal to 1 million 500,000 meters. Awesome. So moving right along here, the separation between the drones, let's call this lower case D which is the separation distance, which is equal to 30 kilometers is equal to 30 1000 meters. OK. So now our next step is, is we need to find the angular separation theta between the two drones and to do that, we need to recall and use the small angle approximation formula which states that theta is approximately equal to D divided by H Awesome. So now all we have to do is just plug in our known variables to sulfur theta. So we take 30 1000 meters, which is our value for D divided by our value of H which is 1,500,000 meters, which when we plug that into our calculator is equal to 0.02 radiance. So now we need to recall and use Raya's criterion equation and we need to rearrange it to isolate and solve for capital D. So let's call. So we wrote our, that's our very first equation that we wrote down together, which states that theta is equal to 1.22 multiplied by lambda divided by capital D. And to make sure you don't get confused. Let's call this equation one. So we need to rearrange equation one to isolate and solve for capital D where once again, capital D is the diameter of the receiving dish. So when we use a little bit of algebra to rearrange to isolate and solve for D, we will find that capital D is equal to 1.22 multiplied by lambda divided by theta. So now all we have to do is just plug in all of our known variables to solve for capital D, which is our final answer that we're ultimately trying to solve for. So when we do that, we will get 1.22 multiplied by our lambda value, which is equal to 0.05 m divided by our value of theta, which we determine to be 0.02 radiance. So when we plug that into our calculator, we will get when we round to the nearest whole number 3 m and that's it, that's our final answer. So to quickly know here when you initially plug into your calculator, you will get 3.05 meters, but we're going round to the nearest full number. So that means that D is approximately equal to 3 m. So looking at our multiple choice answers, the correct answer has to be the letter C 3 m. Thank you so much for watching. Hopefully, that helped and I can't wait to see you in the next video. Bye.