Problem

Using the exact exponential treatment, find how m…

01:18

Problem 71 Hard Difficulty

Figure 21.58 shows how a bleeder resistor is used to discharge a capacitor after an electronic device is shut off, allowing a person to work on the electronics with less risk of shock. (a) What is the time constant? (b) How long will it take to reduce the voltage on the capacitor to 0.250% (5% of 5%) of its full value once discharge begins? (c) If the capacitor is charged to a voltage $V_{0}$ through a $100-\Omega$ resistance, calculate the time it takes to rise to 0.865$V_{0}$ (This is about two time constants.

Answer

a) $\tau=20 \mathrm{s}$
b) $t=120 \mathrm{s}$
c) $t=0.016 \mathrm{s}$

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Physics 102 Electricity and Magnetism

College Physics for AP® Courses

Chapter 21

Circuits, Bioelectricity, and DC Instruments

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Video Transcript

So in the first part, we're supposed to find the time Constant time Constant of RC circuit is simply multiplication off resistance on the capacitance resistance given in there circuit is to 15 alone. And that means that it is to 50 times 10 powerfully over. And the capacitance is 18. Microphone, it's 80 times 10 power. Negative. Six. This is equal to 20 seconds sometime. Constant here is 20 seconds. Uh, for the part B, we are asked, How long will it take to reduce the voltage capacitor? 2.25%. So wall teach as a function of time. This be not or the initial brutish times 10 power Negative. TB, I tell. This is the discharging equation. So initial the final voltage is supposed to be 0.25% off the initial voltage, which will be 0.25 divided by 100. Because this is percent. We not is equal to me. Not exponential. Negative TV, I tell. So this will be 0.25 is equal to exponential Negativity by top implies negativity divided by child is equal to natural law off 0.25 So listen that natural log off you upon 00 to 5 is 5.99 on this is then and this is negative. So it means that the T is equal to 5.99 times top on Valley of Chow was 20 seconds. So this is 5.99 times 20. So this is equal to 1 20 seconds in the last part. The questions is if the capacitor is charged to the voltage V not through 100 organ resistance, what is the time it takes to rise to 0.865 mean we know. So in this case, we are going to use the charge in equation. So free of T is equal. Be off T is equal to be not one minus explanation. Negative. He divided by top meet easy quitter as the questions is 0.865 0.865 We not this is equal to be not one minus exponential negative team by tower. As you can see, those were not concert on both sides and we are left with exponential negative t divided by two not equals one minus zero point 865 So that teeth point 13 fight on. Now let's take the natural knock on both sides to get negative two divided by child. So this is equal to point natural love off 0.135 This is too so negative too. So t is equal to negative. Also positive. Two times tell the child here is bit different. This is capacitance times the resistance through which it was charged. So two times two capacitance is still the same 80 times 10,082 6 times the resistance is 100 open. So that makes it times 100. So let's do that. This is equal to no zero point 016 seconds. So it takes the it take 0.16 seconds to judge to this potential on the capacitor. Thank you.

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