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Johns Hopkins University

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Problem 44

A 10 -Hz generator produces a peak emf of 40 $\mathrm{V} .$ (a) Write an expression for the emf as a function of time. Indicate your assumptions. (b) Determine the emf at the following times: $0.025 \mathrm{s}, 0.050 \mathrm{s}, 0.075 \mathrm{s},$ and 0.100 $\mathrm{s} .(\mathrm{c})$ Plot these emf-versus-time data on a graph and connect the points with a smooth curve. What is the shape of the curve?

Answer

a) $(40 \mathrm{V}) \sin (20 \pi t)$

b) graph

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## Discussion

## Video Transcript

here we have a 10 hertz generator. So frequency is 10 hertz that produces a peak IMF of 40 volts. So peaky, MF I'm just gonna say that that's my max e m f produced. So whenever I'm given frequency, I am just going to go ahead and convert that to angular velocity. So I know exactly how fast my generator coil is rotating in the magnetic field. And so if I plug in 10 right here, I get about 40 0 I can't do math. 20 pie radiance per second and part. One of the question is asking you to write an expression for the IMF as a function of time, and then we're going to indicate our assumptions. So in this case, we know that we are changing the angle. So whenever we're doing that are Faraday's law of ah, magnetic induction becomes something like this, and the way to get to this equation involves in calculus. So you're just gonna have to trust that this is the right equation and notice that I wrote Max right here. So whenever you have the peak e m for the maximum yemma, that means you're assuming that whatever this angle is you're looking at the sign of it will give you positive one, because that will be the maximum number that you get with this expression to produce the maximum e m f. So the maximum off then just gives U N B a times omega. And because we know from the problem, Max is about 40. So this gives us a way to find a value for this portion of the equation. And now, if we are to start to write a general equation for Yemma, even when it is not at the maximum value So we have an equation for the M F as a function of time, then we can just use 40 to represent the first few constants as so. In this case, E M F is equal to 40 times the sign of omega times time. But of course, earlier we found an expression for Omega s well, and that is 20 pie radiance for a second. So I'm just going to put 20 pie times time inside the sign. So this is an expression or a function of Yemma with respect to time, right here and now let's move on to, uh, talking about the assumptions within this equation, so notice we are using a constant for thes values. And so these are the assumptions. We are assuming that we have constant magnetic field so constant Byfield and we're also assuming that Theo area of the generator coil is not changing. What else are reassuming were also assuming that the oil is turning at a constant rate. So oil is rotating at constant angular velocity, which is omega. So these are all the assumptions that we have to have in order for this function to work. And now let's go on to let her be part b of the problem is asking us to determine the IMF in the following times. So we're just gonna plug this into the equation that we just found in part A. So, um, F is equal to 40 times sign of 20 pie times. The first time that we see is 0.25 seconds. Now because we are putting in 20 pie for Omega and because omega is measured in radiance per second, we're going to use ah, the sign and are in our calculator in radiance mode instead of degree mode, and I find that the IMF is about 40 bolts. So that's our max here. Now, if I continue to do the same process for all the other times listed, I find that at 0.5 seconds Yemma zero. And to save time, I'm just gonna go ahead and tell you that at 0.75 seconds, E m. F. Is naked of 40 and at 0.100 seconds, E M f is about zero again, and Part C simply asks us to plot this CMF versus time graph. So I am going to go ahead and draw a straight line that was not very straight. Let's try again, all right, and we are going to put in the values. So this is going to be the white access is going to be Iemma on the X axis is going to be time yeh Memphis Measuring bolts. Time is measured in seconds and you start at zero right here, and the maximum value we need to plot is positive. 40 on the minimum value we need to plot is negative 40 and our time intervals can be split into force. So let's just go with one to 34 This is zero. This is 0.25 0.5 0.75 and 0.10 on. And lastly Ah, we find all the values at thes time intervals. And so the 1st 1 is 40. This is zero. This is negative 40 and that's back at zero. So let's try our best to connect the dots predicting the shape of the graph. So hopefully you can kind of predict that by the time you get to hear it's gonna be zero right? And so what type of shape is this? This turns out to be signing so little. So the's shape of the graph is a sine wave, and that's how you answer this question.

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(FIGURE CANNOT COPY)

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(FIGURE CANNOT COPY)