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Ch 28: Resistors & DC CircuitsWorksheetSee all chapters
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Intro to Current
Resistors and Ohm's Law
Power in Circuits
Microscopic View of Current
Combining Resistors in Series & Parallel
Kirchhoff's Junction Rule
Solving Resistor Circuits
Kirchhoff's Loop Rule

Concept #1: Solving Resistor Circuits


Hey guys. So, in this video we're going to talk about how to solve more complicated circuit problems, let's check it out. Alright, so in some circuit problems you'll be asked to find not just the equivalent resistance but also the current i and the voltage V of different resistors. Alright, so let's see how we do that, but first, let me remind you that resistors can be connected in series or in parallel and if they're in series, this is the equation used to find the equivalent resistance, this is the equation for equivalent resistance in parallel. Remember, that in parallel there are also two shortcut equations you can use, two shortcuts, so this is old news, what's new here is this stuff here, right? And, when you have resistors in series they will share current with each other, they will share current with each other. So, for example, if I have a 1 ohm resistor 2 and 3, 2 ohms, 3 ohms and if I know that this current here is 2 amps then I know, if I'm given the current 2 amps then I know that this here has to be 2 amps and then I know that this here also has to be 2 amps, because they're in series, the current is flowing this way, it has nowhere to go so it has to just flow with a constant 2 amps. So, they share current with each other, cool? Now, you need to learn that but you also have to remember that they share current with the equivalent resistor as well So, they share current with the equivalent resistor, what does that mean? Well, if I combine these I just add them. So, I have one, two and three, this is 6 ohm resistor, but this 2 here is not only the same for these guys but it's also the same here. So, I'm going to know that this has to be a 2 amp current going through this resistor. So, not only is the current the same with each other when you're combining in series but also with the combined merged resistor, okay? Very important.

One way to think about, this is that these guys combine to form the equivalent resistor and the equivalent resistor will inherit, inherit, the current, because it's coming from a series combination, in parallel it's not current it's the voltage that will be the same, okay? So, the voltage will be the same with each other and also with the equivalent resistor. So, let's say, this is a 9 ohm resistor, 9 ohm resistor, 9 ohm resistor, you may remember from the short cut that if I have three 9 ohms the equivalent resistance will just be 9 Ohm's divided by 3 because are three of them. So, equal resistance is 3, that's old news, but let's say, I know that this is, let's say, I know that this is 5 volts, Well, if this guy is 5 volts it must follow that this guy right here is also 5 volts and this guy here is also 5 volts because if they are in parallel, which they are, they have to share the same voltage with each other and not only with each other but also with the merged equivalent resistor, so the voltage here would also be 5 volts, you can think that, when you combine them in parallel, the equivalent resistor will inherit, inherit the voltage, okay? These points are super important for you to remember, please do not forget. Now, that's important for you to solve the problems. Now, here's the three steps that you're going to use in solving all of these problems, we're going to be very methodical about this, very systematic, cool? The first one is you're going to get, you're going to get some sort of weird resistor network here, like this, and you're going to collapse, combine, merge down to one equivalent resistor and we're going to do this using the techniques we already talked about so this is old news, okay? This is old, you know how to do this, then you're going to find voltage and current on the equivalent resistor, once you get down to one resistor you're going to be able to find the voltage and the current on it, and by the way, you're going to do this using Ohm's law V equals i, R which combines current and voltage with resistance, okay? This is old because we've seen V equals i, R and it's also really easy, it's a very quick step. Now, the hard part is step three where we're going to work backwards noting voltage and current on each resistor. Now, don't worry about what that means yet, I'm going to show you and the best way to show you is just by doing an example, okay? So, let's walk through this simple example here and we're going to use these steps to find the current and voltage across each one of these resistors, okay? So, let's draw small here, because we're going to need a lot of space, so the first thing we're going to do. Remember, from the rules up there is that we're going to get everything down to one resistor and you know how to do this the first place I'm going to start is by combining these two in parallel, okay? I'm going to combine these two in parallel. So, I'm going to draw the circuit here, in black and then, circuit here in black and leave some space there, this is the 6 ohm resistor and this is the 10 volt battery right here, and this is the combination of the, combination of the 2 with the 1, I'm going to do one of these and I'm going to write here that this is a parallel merge, okay? I got these two, when I compress them is a parallel merge into this single resistor here, let's, if we want we can calculate that, let's do that real quick, just to get it out of the way. So, I'm going to say that the equivalent resistor, let's call this, let's call this R1, which is a new resistor here. So, we're going to say that R1, I'm going to use the shortcut equation because I have two resistors in parallel. So, I can use a shortcut equation, right? And, I'm going to have, it's a 2 an a 1. So, I have 2, 1, 2, 1, 2 times 1 is 2, 1 plus 2 is 3. So, 2 over 3 which is 0.67. So, this resistor here has a resistance of 6.7 ohms, 6.7 ohms, cool? Now, I have two resistors, I'm going to get down to one. So, I still have to merge these other two right here, and hopefully you can clearly see here that they are in series, okay? I'm going to get a single series resistor here, that's hooked up to the 10 volt battery, can you see that 10 volt? Yes, you can, and I have to combine those two there. So, this was us combining these two resistors. Now, let's combine these two resistors over here and I'm going to call this resistance R2 and R2 just 6.7 plus 6, I'm sorry, I wrote 6.7, I hope it caught that, it's 0.67, hopefully didn't freak out, 0.67, okay? I'm just going to add the 2. So, it's going to be 6.67, right? Once you add them you get 6.67 ohms, this one is ohms as well, cool? So, this resistance right here, is 6.67. I've completed step one which is getting all the way down to one resistor, this merge here, by the way, was a series merge, okay? Once you do that, step two is, write V equals i, R to find other things about that resistor that you may not know, okay? So, now we're going to write V equals i, R and let's see, for this resistor here, we know the resistance, we know the voltage is a 10 because if this is a 10 then this has to be a 10, right? The voltage of the battery there. So, I can find the i, and by the way, this is almost always what's going to happen, you're going to know the v and you will just have found out the equivalent resistance. So, you're going to be able to find the current coming out of the battery, this is the current drawn by the resistors of the battery, cool? So, i equals V over R, the voltage is 10 volts and the resistance will be 6.67 ohms and if you do this you get 1.5 amps. So, your current is 1.5 amps, what this means that 1.5 is the amount of current that comes out of the battery or that is drawn out of the battery by the resistor. So, I'm going to write you that this is a current 1.5 amps, by the way, this current keeps going and it's the current that goes through this resistor right here. So, I now, know the resistance of, I know the resistance of this guy, I know that this guy has a voltage of 10 volts, it has to be the same, right? When you get down to one resistor that resistor has the same voltage as the battery and I know that this guy also has a current of 1.5 amps, let me write it like this, let me just write 10 volts and 1.5 amps, cool? I'm done with that resistor. So, now what you're going to do is, you're going to work backwards, meaning you've compressed and now you're going to work backwards and go back to this circuit here and try to find as many as much information as you can about every single one of these resistors, and the way we're going to do that is by remembering how we merge that in the first place. So, I went, I got to this circuit by going by doing a series merge. So, when I go back I have to remember this was in the series merge because this resistor here, this green resistor, came from these two resistors and remember, when you merge two resistors into an equivalent resistor and they were in series they share the same current. So, what that means is that this current right here, 1.5 is going to be the current of these two guys here, because that same current goes through both of them, okay? So, the current of the 6 is 1.5 amps, and the current of the 0,67 seven is also 1.5 amps, okay? Now, remember you can use Ohm's law, v equals i, R, anytime you know two out of these three you can find a third one. So, if we look at this resistor right here, this first one, okay? I know the resistance is 6, I know the current is 1.5. So, we're able to find its voltage, okay? So, let's just plug in the numbers, so the current is 1.5, the resistance is a 6 and if you multiply, this is a 9 its volt, voltage, that's the voltage, so the answer is in the units of volts, 9 volts. So, I'm going to write here, that this is 9 volts, okay? Now, this is a little messy, you want to make sure that you are as organized as possible, right? There's a lot of numbers, it's easy to get lost, you want to make sure that you're organized as possible, and as you're working your way back you want to make sure that you're getting all the information, you should know the voltage, the current and the resistor, for the resistance, for every one of these elements. Now, what about this red 1 over here, okay? Again, we're going to write V equals i, R. Notice that, I know that the resistance is 0.67. So, I have that, I know that current is 1.5. So, I have that. So, I'm able to find, again, the voltage, okay? The current is 1.5 the resistance is 0.6t7 and if you multiply this you get 1, 1 volt, okay? So, I know that this guy is 1 volt, the voltage across this resistor here is 1 volt, let me highlight all the information for that resistor. So, it's all together without that 10 volts, that's the battery, okay? This is all the information on the second, on the resistor on the Left. So, now we're going to go one step back, okay? We're going to go another step back here, and we're going to go to this original drawing, and notice that this 6 is the same as this 6 over here, they didn't change. So, I can just transfer the information I can say, okay, this 6 has a current of 1.5 amps and it has a voltage of 9 volts, so I'm done here, this is basically a big puzzle where you're trying to find out all the pieces of the puzzle, what about here? I know that that these guys, resistances, but I don't know i and V? Well, remember, where do they come from, right? So, or, what do they merge into? These two guys merged in parallel into this one. Resistors in parallel share the same voltage therefore these two guys are going to have the same voltage as each other but they're also going to have the same voltage as this one that they merged into and that voltage, we just found, was 1 volts, okay? So, this is going to be 1 volt and then this is going to be 1 volt, okay? Again, whenever you have two of these things you can use Ohm's law to find the third. So, let's write Ohm's law two more times here, V equals i, R, in this case I know the resistance for both of them and I've just got the voltage for both of them. So, we're going to be able to solve for the current because current is going to be V over R, the voltage, let's do the top one first, top, the voltage for the top one is 1 and the resistance is 2, so the current is 0.5 amps. So, I now, know that this is 0.5 amps of current, I'm done with this resistor because I know all three properties and the bottom one is going to have a current, I can find the current as well. So, if I write V equals i, R, once again, I know the resistance and I know the voltage of this bottom resistor here. So, i is going to be V over R and voltage is 1, I'm just looking here, right? Notice, how if you have all the numbers in the picture it's easier, it's more organized, and you can just kind of follow what's going on there and then the resistance is 1 over 1 is just 1. So, 1.0 amps. So, this has a current of 1 amp or one amp, and this is all the information for that resistor, okay? So, we now know the current voltage, and the resistances were given. So, now we know everything for all of these resistors. So, if you want to line it up you can sort of, you can make a table and say resistance voltage and current and the 2 ohm resistor here has a voltage of 1 volt, and a current of 0.5, the 1 ohm resistor here has a current of, voltage of 1, and a current of 1, and the 6 over here, the 6 ohm resistor has a voltage of 9 volts and a current of 1.5 amps, we're essentially done, I just want to show you one last thing. Notice, I have this 1.5 here and that's the current going this way, 1.5. Notice that the current here is 0.5. So, this 1.5 amps splits this way as a 0.5 amps and notice that here it splits down as a 1 amp and this is consistent with Kirchhoff's Junction rule or current law that says that the current, the current in equal to current out, 1.5 went in, 1.5 has to come out, the reason I'm telling you this is, because once you knew this and once you knew one of these guys you didn't really even have to calculate this using v equals i, R you could have just known okay? Well, 0.5 right here, I got one left. So, this current here, has to be 1. So, these are just some of the things that help you move a little bit faster, so this is long and hairy the first time, we're going to do quite a few of these examples and practice problems. So, you can nail this but you absolutely have to know how to do this, I highly recommend that you're very organized and systematic so that you don't get lost, cool? Let's do some more.

Practice: What is current and voltage across each resistor below?

Example #1: Find Current of One Resistor


Hey guys. So, this example I want to show you how some of these problems are actually deceivingly simple, they look complicated but they're very straightforward, this is an example of them. So, in this question instead of asking for everything, we're being asked for just one thing, which is what is the current through the swinging ohm resistor which is this guy here. So, that's all we need to know, let's call this the current through the 4 ohm, what is it okay? Well, remember, the equation that connects all these variables is Ohm's law V equals i, R. So, you can find i as long as you have V and R if you solve for i, u get V over R, I know the resistance is 3 ohms. So, really all I need is the voltage. Now, you may also remember that the voltage is the same in parallel and it's not just the same between resistors it's the same for any circuit, any circuit element, so the voltage across these two points is the same through that branch is the same as the voltage across these two points and it's the same for this voltage and it's the same for the voltage between these two resistors. So, because the voltage through this battery is 5, I can say that the voltage on the 3 ohms has to be 5 as well because they're parallel. So, that's it, this voltage is 5 volts, I can divide the two and the answer is going to be 1.67, 1.67 amps and we're done, super straightforward, but you have to be able to recognize certain properties, okay? So, some of these questions are going to be very long and some of them are going to be tricky and simple. So, you gotta be ready for both, let's keep going.

Practice: What is the voltage of the battery below?