Analyzing a resistor circuit with two batteries

hi my name is Willie I'm the Khan Academy electrical engineering content fellow and we got a tweet in from Jane and Jane asked could you take a video could you make a video on solving complex circuit problems in physics for example something something similar to this question so Jane sent us a circuit here here it is over here and it's an interesting looking circuit we're gonna take a shot at solving it here and I'll tell you how I would approach this let's take a quick look at it we have two batteries in it which is a bit unusual there's two resistors here they happen to have the same value 1.4 ohms there's a third resistor connected up and the question we're being asked is to find the what's the current in this resistor right here so that's what we're looking for so the way I approach these kind of questions first I sort of just live with this schematic a little bit I look at it and see if I can imagine how the currents are flowing and and what's going on and that a really good way to do that is to I'm gonna draw this schematic myself I'm just gonna draw it over and let my my own hand kind of learn this circuit as we go so I'll draw the battery the two here's the two batteries and I see that they're connected together okay that's interesting each battery then goes to a resistor this goes to a resistor and those are connected let me check yeah those are connected together and then they go to a third resistor like this and now you notice what I'm doing is I'm taking out all the funny corners and dots and curves and things like that I'm just gonna draw a square schematic that I can understand now I'm gonna label it just to make sure I got everything right so this is 11 volts and I know from the battery symbol that the short the short line is the minus side and the long line is the plus side and same here so these two batteries are hooked up exactly the same way in the same direction this one's also 11 volts that's gonna prove out to be interesting we'll call this one r1 because the problem had I won here we'll call this one r2 and we'll call this one r3 of course and the current that we're looking for is this current right here what's that okay so before I start doing mathematics on this what I'm gonna do is see if I can simplify the diagram and what I'm gonna notice here this is this is kind of a trick that was done here let's look at these batteries these two ends of the batteries are connected together so they are at the same voltage we don't know what that voltage is but we know it's the same and we know they're both 11 volts and that means that this node here this node here and this node here are at exactly the same voltage with respect to this node here this one is 11 volts lower in voltage and this node is 11 volts lower and ultd voltage so what I'm going to imagine in my head if I connected a wire between these two nodes this end would be at 11 volts this end would be at 11 volts so no current would flow through that wire it would have no effect on the circuit it wouldn't Dart there will be no current like diverted from here over to here I'm actually gonna draw that wire in I'm gonna draw that wire in and I didn't change the circuit when I did that the same current goes through this wire as before because zero goes this way the same current goes through this wire because zero current goes this way because I know they're both at 11 at the same potential above here all right so this is a good time to redraw this again if you look here I have two batteries that are hooked up their inputs what they're positive side is hooked up together and their negative side is hooked up together so they're they're actually just acting like one big battery so let me draw that I'm gonna draw the circuit again so it looks like this there's my here's my combined big battery and it goes to we label these again so we don't get mixed up okay this is r1 this is r2 and this is r3 so this circuit looks a little simpler I'm going to look at it again see if I can do any more simplification so what I recognize right here right in this area right here R 1 and R 2 are in parallel they have the same voltage on their terminals that means they're in parallel I know how to simplify parallel resistors we'll just use the the parallel resistor equation did I have in my head and that looks like this let's go to this color here okay so parallel resistors r1 and parallel with R - uh-oh I made up this symbol - two vertical lines that means they're in parallel and the formula for two parallel resistors is r1 times r2 over r1 plus r2 now I'll plug in the values we know that R 1 and R 2 are actually if we look over here at our schematic they're the same value and that has a special thing when did in parallel resistors so it's actually our R over 2 R because those resistors are the same and you can see I can cancel that and I can cancel that and two parallel resistors if the resistors are equal is equal to half the resistance and let's plug in the values 1 point 4 ohms over 2 equals zero point seven ohms that's the equivalent resistance of these two resistors in parallel so this is a good time to redraw this circuit again let's let's do it again here's our battery this time I'm going to draw the equivalent resistance then we have our three we'll put in our three down here nothing's changed there and that goes back to our voltage source again do some labels so this is this is r1 and parallel with our 2 this is our 3 this is the current we want to know right there and this is 11 volts that's the plus side and that's the minus side okay we got a pretty simple circuit here now let's pillo let me fill in the values for our resistor here r1 in parallel with our 2 is 0.7 ohms r3 let's go look that up reminder so it's 2.7 ohms now I can almost do this in my head these are just now two series resistors so Ohm's law equals V equals I R this is the form of it I always memorized if I want to know I I'm solving for a current so I equals V over R we can start filling in values I equals 11 volts over what's the resistance it's the series resistance here so it's the sum of these two resistors which is zero point seven plus two point seven if I work that out equals 11 over 3.4 equals okay let me actually work that out it's 11 volts divided by 3 point 4 ohms and that equals 3 point 2 3 dab dab that's something something something and that will be in amps so this right here is our answer right there so find the current I I equals 3 point 2 amperes and now I'm going to check actually can check to see if that's the answer let's see what let's see what Jane said she actually sent in the answer so thank you James for doing that and there it is so that's it thank you very much for listening