The tuning circuit in an FM radio receiver is a series RLC circui...

Question

The tuning circuit in an FM radio receiver is a series RLC circuit with a 0.200 μH inductor.b. FM radio stations are assigned frequencies every 0.2 MHz, but two nearby stations cannot use adjacent frequencies. What is the maximum resistance the tuning circuit can have if the peak current at a frequency of 103.9 MHz, the closest frequency that can be used by a nearby station, is to be no more than 0.10% of the peak current at 104.3 MHz? The radio is still tuned to 104.3 MHz, and you can assume the two stations have equal strength.

Accepted Answer

Hello, fellow physicists today, we are 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, an AM radio receiver designed to detect oscillating signals is based on an RLC circuit with an inductance of 320 nano henries. This receiver is tuned to detect signals at 780 kg Hertz. However, a nearby generator is producing signals at 750 kertz two generators have a frequency separation of 30 kilohertz, which is the minimum permissible separation for adjacent AM signal generators to effectively combat the undesired 750 kilohertz signal. The peak current at this frequency should be 0.08% of the peak current at 780 kilohertz determine the maximum resistance of the RLC circuit when decoding the 780 kilohertz signal assuming both generators radiate equal power. So that's our end goals. We're trying to figure out what the maximum resistance of this particular RLC circuit will be when decoding a 780 kilohertz signal. And we also need to assume that both generators are radiating equal power in this case. Awesome. So we're also given some multiple choice answers. Let's read them off to see what our final answer might be. A is 320 ohms. B is 1.0 kg ohms C is 0.12 ohms. And finally, for D it's 0.10 mily ohms. OK. So first off, let us recall and use the equation for the total impedance of an RLC circuit. So the total impedance represented by Z is equal to the square root of R squared where R is the resistance plus X subscript L XL which represents the inductive reactants minus X subscript C where XC represents the capacitive reactants squared. Awesome. Let us also take a moment to note that in for this problem, we're gonna denote subscript zero to mean resonant. So anything that's subscript zero will mean resonant moving forward. So the current in the tuning circuit at 780 kilohertz. So let's make a note here. 780 Kah Hertz. So the current in the turning circuit at 100 and 80 kilohertz is I zero, I substantive zero, which is the resonant current is equal to the resonant EMF divided by the resonant impedance. Awesome. And the enf the EMF or the resident EMF I should refer say is epsilon subscript zero. OK. So the circuit oscillates at a resonance of let's use a semi colon here to separate our equation. So the circuit oscillates at a resonance of Z zero equals R. Therefore, we can write that I zero equals epsilon zero divided by R. And let's also note that the unwanted current can be expressed as epsilon zero divided by Z. And we need to note that Z cannot equal R because 750 khz is not a resonant frequency. Thus, we can go on to write that epsilon zero divided by Z is equal to. And I'll take a moment here to pause. So we need to take our 0.08% and we need to turn that into a decimal. So we need to take 0.08 and divide it by 100 to turn it into a decimal. So when we do that, we'll get 0.000 eight, multiplied by epsilon zero all divided by R. OK. So now that we have our equation here, we need to uh take this equation and we're gonna be rearranging it and simplifying it in order to isolate and solve for R. So to start off, let's start by dividing epsilon zero to the other side. And when we do that, we will get one divided by the square root of R squared plus XL minus XC squared is equal to 0.000 eight divided by R. Now we need to take the reciprocal and square. And when we do that. We will get that R squared plus XL minus XC squared is equal to R squared divided by 0.0008 squared is equal to 1250 squared multiplied by R squared. OK. Moving right along we're still in the simplifying process. So our next step is to write that XL minus XC squared is equal to 1250 squared, multiplied by R squared minus R squared is equal to 1250 squared minus one multiplied by R square. And we need to note that as 1250 squared minus one approaches zero, we can write that XL minus XC squared is equal to 1250 squared multiplied by R squared. And when we square root both sides, we will get that the absolute value of XL minus XC is equal to 1250 multiplied by R. And now it's really simple. We can get R by itself and we can write that R is equal to the absolute value of XL minus XC, all divided by 1250. But we can also write that R is equal to the absolute value of XL minus XC divided by 1250. We can also say that this is equal to the absolute value of Omega multiplied by L minus one divided by Omega multiplied by C all divided by 1250. OK. So before we move any further, let's take a second to discuss what these different variables mean. You're probably wondering what in the world Omega is and what it represents. So let's take a moment here to talk about our different variables here and what units they will have. So let's start with Omega. So Omega in this case represents the angular velocity. And let's recall that the angular velocity Omega is equal to two multiplied by pi multiplied by the frequency. So let's solve for what our angular velocity in this case, for this problem will be. So Omega equals two multiplied by pi multiplied by the frequency which we are given 750 kilohertz. But we need to convert Kertz to Hertz. So if we take 750 we multiply it by 10 to the power of three, we could convert kilohertz to Hertz. So our angular velocity in this case will be 1.5 multiplied by pi multiplied by 10 to the power of six radiance per second, which radiance per second is the standard units for angular velocity. So that's good. And L in this case, or I should say capital L represents the inductance. And in this case, it's given to us as 320 nano Henry. But let's convert Nano Henry to Henry. So we just multiply it by 10 to the power of negative nine. And finally, what does C mean? So C represents the capacitance. And in this case, we do not know what the capacitance is, it wasn't provided to us in the problem itself. So we need to take a moment here and quickly solve for the capacitance. So let's note and recall that the resonant angular velocity is equal to two multiplied by pi multiplied by the A like by the resonant frequency, which is also equal to one divided by the square root of L multiplied by C. So now we can rearrange this equation to solve for C. So let's do that. So we'll get that C is equal to one divided by four, multiplied by pi squared multiplied by the resonant frequency squared multiplied by the inductance L. OK. So now we can plug in our known variables and solve for the capacitance. So let's do that. So one divided by or multiplied by pi squared multiplied by the resonant frequency which is 780 kertz. But let's convert Kah Hertz to Hertz. So it's gonna be multiplied by 10 to the power of three Hertz. And then for the or don't forget that it's squared. And then we need to multiply it by the inductance which is 320 multiplied by 10 to the power of negative nine henries. Awesome. Awesome. So now when we take that and we plug it into a calculator, we will get 1.30 multiplied by 10 to the power of negative seven ferrets. So when we take one divided by four, multiplied by pi squared multiplied by our resonant frequency squared multiplied by the inductance, we will get 1.30 multiplied by 10 to the power of negative seven fair adds. So finally, at this stage, we can solve for the maximum resistance, which is our final answer and our end goal what we're trying to solve for. So as we know, we know that the resistance is equal to the absolute value of the angular velocity multiplied by the inductance minus one divided by the angular velocity multiplied by the capacitance all divided by 1200 and 50. So now we can plug in our known variables and solve for the resistance. OK. So let's do that. So the absolute value of our angular velocity, which we found to be 1.5 pi multiplied by 10 to the power of six radians per second multiplied by the inductance, which is 320 multiplied by 10 to the power of negative nine henries minus K minus one divided by 1.5 pi multiplied by 10 to the power of six radiance per second multiplied by our capacitance, which we discovered to be one point 30 multiplied by 10 to the power of negative seven. They are ads all divided by, don't create our absolute value sign all divided by 1250. So once we plug in that big old long equation indoor calculator, we will find that the resistance is equal to 9.95 multiplied by 10 to the power of negative five ohms, which is roughly or approximately equal, I should say, which is approximately equal to 1.0 multiplied by 10 to the power of negative four arms which is equal to 0.10 Milly alms. And that is our final answer. Hooray, we did it. So I've gone ahead and done the conversion for you, but you can perform dimensional analysis to get this conversion done or you could just quickly look it up, but I just went ahead and gave it to you this time around. So let's go look at our multiple choice answers to see which of them, which of them are or which of the answers is our final answer. So the correct answer and for this problem has to be the letter D 0.10 Milli Ohms. Thank you so much for watching. Hopefully that helped and I can't wait to see you in the next video. Bye.

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