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RC Circuits Physics Problems, Time Constant Explained, Capacitor Charging and Discharging
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00:00:00.030 in this video we're going to talk about 00:00:03.13900:00:03.149 capacitors particularly when they're 00:00:07.46000:00:07.470 charging and discharging so let's say if 00:00:10.33900:00:10.349 we have an RC circuit that is a circuit 00:00:13.24900:00:13.259 containing a capacitor with a resistor 00:00:15.16900:00:15.179 and we're going to have an open switch 00:00:17.68900:00:17.699 initially let's say that the voltage of 00:00:21.38000:00:21.390 the battery is 12 volts this is the 00:00:27.76900:00:27.779 resistor R and here we have the 00:00:30.65000:00:30.660 capacitor represented by the symbol C 00:00:34.36000:00:34.370 now let's switch is closed charge will 00:00:39.47000:00:39.480 begin to flow current will flow from the 00:00:44.09000:00:44.100 positive terminal of the battery to 00:00:48.86000:00:48.870 charge the capacitor the longer side is 00:00:51.41000:00:51.420 the positive terminal now keep in mind 00:00:53.66000:00:53.670 no electrons may flow in the opposite 00:00:56.06000:00:56.070 direction electrons flow from the 00:00:58.27900:00:58.289 negative terminal and it answers the 00:01:01.18900:01:01.199 positive terminal now initially when a 00:01:06.10900:01:06.119 switch is open the voltage of the 00:01:08.60000:01:08.610 capacitor will be zero as soon as the 00:01:13.64000:01:13.650 switch is closed when T is zero the 00:01:17.24000:01:17.250 voltage will still be zero but the 00:01:19.16000:01:19.170 voltage across the resistor initially 00:01:21.52000:01:21.530 will equal the voltage of value which is 00:01:24.23000:01:24.240 12 now after some significant time has 00:01:27.77000:01:27.780 passed the capacitor will be fully 00:01:30.23000:01:30.240 charged let's call it V final it's going 00:01:33.95000:01:33.960 to have a final voltage of 12 when the 00:01:36.89000:01:36.900 capacitor is fully charged the voltage 00:01:39.67900:01:39.689 across the resistance is going to be 00:01:41.31900:01:41.329 zero the reason mean is once you muscle 00:01:44.78000:01:44.790 capacity fully charged no current will 00:01:47.33000:01:47.340 flow in the circuit and if there's no 00:01:49.31000:01:49.320 current flowing to the resistor the 00:01:50.99000:01:51.000 voltage across that resistor will be 00:01:52.99900:01:53.009 zero so initially when a capacitor has a 00:01:58.58000:01:58.590 voltage of zero the voltage as a 00:02:00.49900:02:00.509 resistor will be 12 because it has to 00:02:02.81000:02:02.820 add up to the battery voltage when the 00:02:05.20900:02:05.219 capacitor is fully charged to 12 volts 00:02:07.13000:02:07.140 the voltage across the resistor will be 00:02:09.46900:02:09.479 zero because I have absolute there's no 00:02:11.90000:02:11.910 current flowing to the circuit 00:02:13.43000:02:13.440 and keep in mind no current actually 00:02:16.55000:02:16.560 passes through the capacitor the current 00:02:19.88000:02:19.890 that flows through the circuit is simply 00:02:22.79000:02:22.800 used to pump charge from one plate of 00:02:25.61000:02:25.620 the capacitor to the other plate the 00:02:29.54000:02:29.550 equation that describes a capacitor 00:02:31.58000:02:31.590 charging is this equation that's equal 00:02:34.49000:02:34.500 to the Engelmann of the battery which in 00:02:37.13000:02:37.140 this problem is 12 times 1 minus e is 00:02:41.78000:02:41.790 basically the inverse of the natural log 00:02:43.73000:02:43.740 function minus the time divided by RC 00:02:49.21000:02:49.220 now the graph looks something like this 00:02:59.16000:02:59.170 this is the EMF of the battery which is 00:03:01.08000:03:01.090 12 volts and the capacitor will charge 00:03:05.43000:03:05.440 progressively towards 12 volts but it 00:03:11.72900:03:11.739 starts from zero and so depending on the 00:03:15.75000:03:15.760 resistance and the values of the 00:03:17.22000:03:17.230 pathogens it will affect the shape of 00:03:19.97900:03:19.989 this graph but for the most part that's 00:03:22.41000:03:22.420 how it's going to look like it's going 00:03:24.42000:03:24.430 to increase towards 12 at which point is 00:03:26.85000:03:26.860 just going to be a horizontal line so 00:03:32.00900:03:32.019 this is the voltage of the capacitor and 00:03:33.63000:03:33.640 on the x-axis you have the time 00:03:37.28000:03:37.290 now there's something called tau which 00:03:39.93000:03:39.940 is the time constant and that's equal to 00:03:42.69000:03:42.700 the product of the resistance and the 00:03:45.09000:03:45.100 capacitance it's our scene and so it has 00:03:47.67000:03:47.680 its own times farad's now let's talk 00:03:51.90000:03:51.910 about the time constant let's make a 00:03:54.42000:03:54.430 table 00:04:08.47000:04:08.480 this is going to be the voltage at any 00:04:11.08000:04:11.090 time team that is the voltage of the 00:04:13.42000:04:13.430 capacitor so how much will the capacitor 00:04:18.09900:04:18.109 be charged at the runtime constant one 00:04:23.32000:04:23.330 time constant is basically it's equal to 00:04:29.40000:04:29.410 run our scene two time constants it's 00:04:33.30000:04:33.310 just two RC so what you need to do is 00:04:38.65000:04:38.660 replace T with one RC in the equation 00:04:44.89000:04:44.900 and RC will cancel so basically you just 00:04:49.33000:04:49.340 got a plug in run e to the minus one if 00:04:52.51000:04:52.520 you type that in you should get point 00:04:55.00000:04:55.010 six three two which is sixty three point 00:04:58.45000:04:58.460 two percent now the maximum voltage of 00:05:02.40900:05:02.419 the capacitor is the voltage of the 00:05:04.54000:05:04.550 battery which is twelve sixty three 00:05:06.67000:05:06.680 point two percent of twelve is seven 00:05:09.43000:05:09.440 point five eight volts so after run time 00:05:12.85000:05:12.860 constant the voltage of the capacitor 00:05:15.04000:05:15.050 will be seven point five eight volts the 00:05:18.45000:05:18.460 63.2% of its maximum now what about 00:05:22.00000:05:22.010 after two time constants well if we type 00:05:25.00000:05:25.010 in 1 minus e to negative two you should 00:05:28.51000:05:28.520 see 2865 or 86.5% 86.5% of twelve is ten 00:05:36.64000:05:36.650 point three eight so after two time 00:05:39.27900:05:39.289 constants the voltage of the capacitor 00:05:40.77000:05:40.780 will be ten point three eight now one 00:05:44.65000:05:44.660 minus e to the minus three will give you 00:05:48.19000:05:48.200 point nine five one ninety five percent 00:05:51.90000:05:51.910 which correlates to a voltage of eleven 00:05:55.06000:05:55.070 point four volts after fourth time 00:05:59.05000:05:59.060 constant the capacitor will be ninety 00:06:01.54000:06:01.550 eight point two percent charged 00:06:07.07000:06:07.080 after five constants it's going to be 00:06:11.74000:06:11.750 99.3% charge it's going to have a 00:06:14.77900:06:14.789 voltage of eleven point nine zero so it 00:06:19.49000:06:19.500 takes about 5 time constants for a 00:06:22.96900:06:22.979 capacitor to be fully charged 00:06:25.39000:06:25.400 it's about 99 percent charge at that 00:06:28.12900:06:28.139 point now what about this charge in a 00:06:30.89000:06:30.900 capacitor 00:06:31.62900:06:31.639 let's say if the capacitor is fully 00:06:33.77000:06:33.780 charged and we connected across the 00:06:36.92000:06:36.930 resistor let's say it now has a voltage 00:06:40.52000:06:40.530 of 12 volts what type of graphs will we 00:06:43.55000:06:43.560 have when a capacitor is discharging the 00:06:49.52000:06:49.530 graph will look like this 00:06:50.95900:06:50.969 it's going to start from its initial 00:06:52.99900:06:53.009 value of 12 and it's going to 00:06:55.27900:06:55.289 progressively decrease to 0 the equation 00:07:00.70900:07:00.719 that describes this graph is this one Z 00:07:04.10000:07:04.110 is equal to the initial that's the 00:07:06.70900:07:06.719 original voltage of the capacitor from T 00:07:08.92900:07:08.939 is zero at 12 volts times e raised to 00:07:12.74000:07:12.750 the negative T over RC now let's make 00:07:19.07000:07:19.080 another table 00:07:27.48900:07:27.499 vfc is the voltage of the capacitor and 00:07:30.93900:07:30.949 on the x-axis we have time now after one 00:07:36.49900:07:36.509 time constant 00:07:39.07000:07:39.080 how much will percentage of the charge 00:07:42.46900:07:42.479 does the capacitor still have so that is 00:07:46.99900:07:47.009 this equation if we replace T with 00:07:49.64000:07:49.650 negative R seen this is going to be e to 00:07:56.36000:07:56.370 negative 1 if you type in each negative 00:07:58.24900:07:58.259 one you're going to get 0.368 what's 00:08:01.93900:08:01.949 going to be 36.8% charged the voltage is 00:08:06.30900:08:06.319 now four point four two volts that's 00:08:10.70000:08:10.710 thirty six point eight a set of twelve 00:08:13.39000:08:13.400 after two time constants it's going to 00:08:17.80900:08:17.819 be thirteen point five percent then the 00:08:22.21900:08:22.229 other table after two time constants it 00:08:24.40900:08:24.419 was eighty-six corn-factor sir these two 00:08:26.92900:08:26.939 numbers have to add up to 100 00:08:28.86900:08:28.879 so the voltage at this point is going to 00:08:31.21900:08:31.229 be one point six two volts 00:08:35.50000:08:35.510 after three time constants it's going to 00:08:38.93000:08:38.940 be five percent charge 95 percent 00:08:42.94900:08:42.959 discharged 00:08:44.41000:08:44.420 so it's voltage is now going to be point 00:08:48.25900:08:48.269 6 volts after four time constants it's 00:08:54.43900:08:54.449 going to have one point eight percent of 00:08:56.26900:08:56.279 its initial charge so the voltage is 00:08:58.85000:08:58.860 going to be point twenty two so it takes 00:09:02.93000:09:02.940 about 5 time constants for the capacitor 00:09:06.25900:09:06.269 to be 99 percent discharge where it's 00:09:09.94900:09:09.959 going to have 27 percent remaining of 00:09:13.06900:09:13.079 the original truck and so the voltage is 00:09:15.47000:09:15.480 going to be point zero eight so at that 00:09:17.62900:09:17.639 point you can say that the capacitor for 00:09:19.87900:09:19.889 the most part is discharged now let's 00:09:23.26900:09:23.279 work on an example problem so let's say 00:09:26.48000:09:26.490 we have an RC circuit and the capacitor 00:09:29.60000:09:29.610 is attached to the battery by means of a 00:09:32.26900:09:32.279 resistor and initially at T equals zero 00:09:35.84000:09:35.850 let's say the voltage of the capacitor 00:09:37.04000:09:37.050 is there 00:09:39.45000:09:39.460 and the capacitor is going to have a 00:09:41.85000:09:41.860 value of 500 micro frats and let's say 00:09:48.26900:09:48.279 that the resistor it's a 1 kilo ohm 00:09:51.75000:09:51.760 resistor and the voltage of the battery 00:09:56.01000:09:56.020 is going to be 20 volts with this 00:10:01.23000:10:01.240 information how long will it take for 00:10:04.82900:10:04.839 the capacitor to reach 90% of its 00:10:09.32900:10:09.339 maximum charge so what is 90% of 20 00:10:15.78000:10:15.790 volts 20 times 90% is 18 volts so we 00:10:24.51000:10:24.520 want to know how long will it take for 00:10:27.09000:10:27.100 the capacitor to be 90% charge or to 00:10:30.81000:10:30.820 reach a voltage of 18 volts 00:10:35.12000:10:35.130 since the capacitor is charging we can 00:10:38.19000:10:38.200 use this equation 00:10:45.71000:10:45.720 z is the voltage of the capacitor which 00:10:49.40000:10:49.410 is 18 volts the EMF of the battery is 20 00:10:53.71000:10:53.720 and let's find out the value of RC which 00:10:59.15000:10:59.160 is one time constant R is 1 kilo ohm 00:11:03.47000:11:03.480 which is a thousand ohms C is 500 micro 00:11:08.72000:11:08.730 farad which is 500 times 10 to minus six 00:11:12.92000:11:12.930 farad now if we multiply these two a 00:11:18.10000:11:18.110 thousand times 500 times 10 to the minus 00:11:21.56000:11:21.570 6 and this is equal to 0.5 0.5 is the 00:11:27.86000:11:27.870 value of one time constant so a time 00:11:31.31000:11:31.320 constant is 0.5 seconds in this 00:11:33.77000:11:33.780 particular problem because sometimes you 00:11:36.86000:11:36.870 need to know what the time constants are 00:11:39.73000:11:39.740 so let's replace RC with point five and 00:11:43.63000:11:43.640 now our goal is to solve for team 00:11:52.63000:11:52.640 so the first thing we need to do is 00:11:54.77000:11:54.780 divide both sides by 18 I mean rather by 00:11:58.34000:11:58.350 20 18 divided by 20 is 0.9 so point 9 is 00:12:04.25000:12:04.260 equal to 1 minus e raised to the 00:12:07.01000:12:07.020 negative T divided by point 5 next 00:12:10.12000:12:10.130 subtract both sides by 1 point 9 minus 1 00:12:14.33000:12:14.340 is negative point 1 so we have negative 00:12:17.66000:12:17.670 point 1 minus e raised to the negative T 00:12:20.77000:12:20.780 divided by point 5 next multiply both 00:12:24.35000:12:24.360 sides by negative 1 this will change 00:12:26.60000:12:26.610 negative signs into a positive sign 00:12:29.98000:12:29.990 after that let's take the natural log of 00:12:32.99000:12:33.000 both sides so on the Left we have the 00:12:38.75000:12:38.760 natural log of point 1 and on the right 00:12:42.11000:12:42.120 we have the natural log of e to the 00:12:45.05000:12:45.060 negative T divided by point 5 now a 00:12:49.07000:12:49.080 property of logs allows you to take the 00:12:51.86000:12:51.870 exponent and move it to the front so 00:12:54.97000:12:54.980 therefore what we now have is the 00:12:58.37000:12:58.380 natural log of point 1 is equal to 00:13:01.67000:13:01.680 negative team divided by 0.5 times ln e 00:13:06.79000:13:06.800 ln e is equal to 1 now let's just get 00:13:15.20000:13:15.210 rid of this stuff 00:13:22.25000:13:22.260 now let's multiply both sides by 00:13:26.29000:13:26.300 negative 0.5 this will cancel the point 00:13:30.95000:13:30.960 5 on the right as well as the negative 00:13:33.11000:13:33.120 sign so T is equal to negative 0.5 times 00:13:40.25000:13:40.260 the natural log of 0.1 so if we type 00:13:43.67000:13:43.680 this in this will give us 1 point 1 5 00:13:52.16000:13:52.170 seconds so that's the time it takes for 00:13:56.72000:13:56.73000:13:59.93000:13:59.940 maximum value to reach a voltage of 18 00:14:02.72000:14:02.730 volts now is there an equation that can 00:14:06.05000:14:06.060 help us get this answer a lot faster if 00:14:09.10000:14:09.110 you rearrange the equation you can use 00:14:12.23000:14:12.240 this formula T is equal to negative RC 00:14:16.15000:14:16.160 times the natural log of 1 minus V 00:14:25.46000:14:25.470 divided by E where Z is the voltage of 00:14:29.30000:14:29.310 the capacitor at any time T and this 00:14:33.92000:14:33.930 epsilon symbol is basically the voltage 00:14:37.31000:14:37.320 of the battery now the way you should 00:14:42.29000:14:42.300 type it in should be like this R make 00:14:44.66000:14:44.670 sure you use a thousand don't use one 00:14:46.40000:14:46.410 kilo C is going to be 500 times 10 to 00:14:50.36000:14:50.370 the minus 6 and then multiply that by 00:14:52.73000:14:52.740 the natural log of 1 minus 18 volts 00:14:57.26000:14:57.270 divided by 20 if you type this in 00:15:00.65000:15:00.660 exactly the way you see it and this is a 00:15:03.02000:15:03.030 multiplication sign this will give you 00:15:05.78000:15:05.790 one point one five seconds so we have 00:15:10.94000:15:10.950 the time and we also have the time 00:15:14.96000:15:14.970 constant as you said before one time 00:15:18.80000:15:18.810 constant is equal to RC which is point 00:15:23.03000:15:23.040 five seconds so using this information 00:15:26.53000:15:26.540 how many time constants does it take for 00:15:31.16000:15:31.170 the capacitor to be 90% charge 00:15:34.26900:15:34.279 how can you figure this out to find the 00:15:38.96000:15:38.970 number of time constants convert it 00:15:42.37000:15:42.380 start with the time that you have which 00:15:45.29000:15:45.300 is 1.15 seconds over 1 and we know that 00:15:49.81900:15:49.829 one time constant is equivalent to point 00:15:53.84000:15:53.850 five seconds so notice that the unit 00:15:57.41000:15:57.420 seconds will cancel giving us the number 00:15:59.93000:15:59.940 of time constants so anytime you have to 00:16:03.11000:16:03.120 find the number of time constants which 00:16:04.61000:16:04.620 is we'll call it n it's equal to the 00:16:07.22000:16:07.230 time divided by tau that's how you can 00:16:12.86000:16:12.870 find it so it's 1.15 divided by point 00:16:16.16000:16:16.170 five and so it takes about two point 00:16:22.55000:16:22.560 three time constants for the capacitor 00:16:25.34000:16:25.350 to be 90% charged now does this answer 00:16:28.34000:16:28.350 make sense 00:16:31.75000:16:31.760 consider the first table that we went 00:16:34.34000:16:34.350 over early in this video we said that it 00:16:37.63900:16:37.649 takes two time constants for the 00:16:40.97000:16:40.980 capacitor to be 86.5% charged and that 00:16:47.24000:16:47.250 it takes three time constants for the 00:16:51.23000:16:51.240 capacitor to be 95% charge 00:16:55.36000:16:55.370 so therefore 90% is between 86.5% and 90 00:17:01.72000:17:01.730 which means that it should take 00:17:04.64000:17:04.650 somewhere between two and three time 00:17:06.47000:17:06.480 constants to be 90% charged and 2.3 is 00:17:11.36000:17:11.370 between 2 and 3 so that answer does 00:17:14.05900:17:14.069 indeed make sense and so now you know 00:17:17.17900:17:17.189 how to find the number of time constants 00:17:19.49000:17:19.500 it takes to charge or even discharge the 00:17:22.54900:17:22.559 capacitor to a certain level and so 00:17:26.09000:17:26.100 we're going to stop here so that's it 00:17:28.18900:17:28.199 for this video thanks for watching and 00:17:30.28900:17:30.299 have a great thing
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