Problem

When a certain resistor is connected to an ac ge…

03:34

Problem 79 Hard Difficulty

Trapped on a deserted island, you salvage some copper wire and a generator to build a makeshift radio transmitter. (a) If you wrap the wire into an 8 -turn coil of radius $3.8 \mathrm{~cm}$ and length $25 \mathrm{~cm},$ what is its inductance? (b) What capacitance do you need to make an RLC circuit with the coil that oscillates at the emergency beacon frequency of $406 \mathrm{MHz}$ ? (c) If you fashion the parallel-plate capacitor out of two aluminum squares, $5.0 \mathrm{~cm}$ on a side, what should be the separation distance between the two plates?

Answer

a) 1.50$\mu \mathrm{T}$
b) 0.11 $\mathrm{pF}$
c) 0.21 $\mathrm{m}$

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Physics 102 Electricity and Magnetism

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Video Transcript

the first item. We have to remember that they looked as if a solenoid is given by music through times and squared times. The area divided by the length ones off the solo night and is the number off loops off that store tonight, which is he goes to eight a is the area off the solar night and the area off the solenoid is these area, which is the area of a circle. So we have museums times and squared times by times R squared for the area, divided by the length off the solar night. Then we can plug in the even values off the problem to get mu zero times eight squared times by times 3.8 squared times 10 to the miners for because we have a centimeter here and we are squaring it. So we have 10 to the minus three times 10 to the minus two, which results in 10 to the minus former, and this is all divided by 25 times 10 to the minus troop because there is a ST year in the length of the coil, then within supposedto also the value off the magnetic constant, which is four pi times 10 to the minus seven and then we get these times eight squared times pi times 3.8 squared times 10 to the finals for divided by 25 times, 10 to the minus two. And this is approximately 1.5 me Crow Harris in the second. Bite them. We want our circle to isolate the frequency off. 406 Maga hurts, and all we can do is include a capacitor in our circuit. Then what you have to do is considered that these is the resonant frequency, the circuit. And then we calculate the capacitance that is necessary to achieve that. So it goes as follows. 406 times. 10 to the six. We changed the resonant frequency and we have a mega here. So we have 10 to the sixth Izzy course. True. One divided by to play times the square it off. L time see, But these we consider the value that we have just calculated for the duck tends to get 406 times 10 to the sixth is equals two to pry times the square it off 1.5. Thanks. 10 with my new six times the capacitance under one now with so for the capacitance and goes as follows four or six times, 10 to the sixth times two pi times the square root off 1.5 times 10 to the minus six. Is it close to one divided by the square root of the capacitance, Then this square root of the capacitance is equals. Two, 406 I'm staying to the sixth times to buy times discredited off 1.5 times, 10 to the minus six under one. And finally we get the following result May capacitance is it goes to 406 times 10 to the sixth times to buy squared times 1.5 times 10 to the minus six under one and these is approximately 0.1. Be cool for it. On the last item, we are building a part of plate capacitor using a square off Dominion, actually, two squares off aluminium, and each of them has five centimeters. We didn't have to complete what is the neccessary distance to get the capacitance off 0.1 pickle for its I wouldn't have to remember that the capacitance off a parallel plate capacitor is he goes to the electric constant times, the area off the capacitor, divided by the distance between the plates. Then we can sew for distance. To get that distance is it goes to the electric constant times, the area divided by the capacitance. No, we can plug in the given values off our problem to get absolutely root times. The area is, of course, to five times 10 to the minus two, which is five centimeters squared. So it's the Argonaut office queer, and these is divided by 0.1 time. Stand to the miners 14. Now we have to remember that the electric constant is approximately eight 0.85 times 10 to the minus 12. Then we get these times five times 10 to the minus. Two squared, divided by 0.1 time Stan to the miners 13. And these is approximately zero and two meters

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