Resistors in parallel

last video we saw what happens when we have resistors in series now let's see what happens when we have resistors in parallel all right let me pick a new color new color will be magenta there's my battery positive negative there is my ideal conducting wire here's my ideal conducting wire but then and this is new it branches off and I have two resistances I have one here that was my printer making a random noise if you heard that I call them printer burps and that's another resistance and let's say that this is this has a resistance r1 this has a resistance r2 and of course the non-intuitive convention is that the current flows from the positive to the negative terminal but we know that the electrons are actually flowing in the other direction and I want to keep saying that because I don't know I think it's so important to just to understand what's actually happening as opposed to the convention well anyway in the previous video we said well when we have when we have devices or components in series that the current through the entire circuit is constant but let's think about what happens here let's think about what happens here so we have these electrons let's think about it from the electron flow the electrons are flowing flowing flowing at a given rate right and then here they have a choice right some of them can take this top path some of them can take this bottom path right so if you think about it the flow of electrons in this branch plus the flow of electrons in that branch have to add up to the flow of the electrons in this branch right and then they're going to meet back up and then the flow of electrons here so if we if we think of it this way and now I'm going to go back to the convention that this is I 1 so you have these electrons flowing at a given rate right this is the current right here they're going to branch off and you know maybe half of them go if the resistances we'll see if the resistances are equal if these are kind of very you know both of these branches that have have an equal amount of capacity in terms of how fast the electrons can flow through if they're equal or since we're going to card in this direction let's talk about positrons or I don't know you know positive charges if the positive charges although as I just want to keep saying that is not the positive charges they're moving as the electrons but if we say that the lack lack of electrons can flow equally easily between boat paths that's if the resistances were the same we could imagine that the current the flow would split itself up and then over here would meet back up and then we would say that the current here would also be i1 but let's figure out what the currents going let's call this current i2 i2 and let's call this current i3 so I think it is reasonable and you could imagine with water pipes or anything that the current going into the branch is equal to the current exiting the branch or you could even think of it that the current entering when the when the current i2 and i1 merge that they combine and they they become current one right I mean to think about it in a given second if this is five coulombs per second I'm just making up numbers and this is six coulombs per second six six I can't write properly six coulombs per second right in a given second right here you're going to have five coulombs per set five coulombs coming from this branch and 6 coulombs coming from this branch so you're going to have eleven coulombs per second coming out once they've merged so this would be 11 coulombs per second so I think hopefully that makes sense to you that they that that this current is equal to the combination of this current and that current now what do we also know we also know that the voltage along this entire ideal across ascent entire ideal wire is constant so the then voltage let me draw that let me do it in another color in blue so for example the voltage anywhere along this blue that I'm filling in is going to be the same because this wire is an ideal conductor and you can almost view this blue part as an extension of the positive terminal of the battery and very similarly the LME see do it in yellow we could draw this wire as an extension of the negative terminal of the battery this is an extension of the negative terminal of the battery right so the voltage difference between here and here so let's call that the total voltage or let's just say that this is called at the voltage write the voltage drift between between that point and that point is the exact same thing as the voltage difference between this point and this point which is the exact same thing as a voltage difference between this point and this point right so what can we can say so what is what is the total current in the system if we just view this as a black box that this is some type of total resistance right well it would be the total current in the system would be the total voltage the voltage divided by let's call this our total resistance right let's say we couldn't see this and we just said oh that's just some total resistance right and that is equal to the current going through r1 this is is this yeah this is i1 this is one right here this is current i1 well what's current i1 what's going to be the voltage across this resistor divided by the resistance right that's what Ohm's law tells us V is equal to IR or another way we could say it is V over R is equal to I right so i1 is equal to the voltage across this resistor well we just said that voltage is the same thing as this voltage right the voltage here is the same thing as a voltage here the voltage here is the same thing as voltage here so this the voltage across that resistor is still V and so the current flowing across that resistor is V over R one and the same logic what that what is i2 i2 is this current what is the voltage across this device well that's just V again right it's the same thing as a voltage across this device so it's V over r2 by Ohm's law well all these V's are the same so we can divide both sides of that equation by V and we get 1 over the total resistance is equal to 1 over r1 plus 1 over r2 and you could make that argument if there was you know if we had an r3 here let's say that you know we had another device and then it was there and as r3 we just you know you could use the exact same argument you would have a plus one over r3 and if you had RN or you know 10 of them you just keep that one over R 4 R 5 etc so let's see if we can use this information we've learned to actually solve a problem and I actually find it it's always easy to solve a problem than to explain the theory behind a problem you'll see that most of these circuit problems it's actually very basic mathematics so let's say I have a 16 volt battery plus minus it's 16 volts and let's say and just to hit the point home that you always don't have to draw circuits the same although it is nice if you're actually drawing complicated circuits I could draw it like this I can draw the circuit like this and let's say that there's a resistor here and then let's say there's a wire and then there's another resistor here and that this decides to do some random loopy thing here and that they connect here and that they come back here this strange thing that I have drawn which you will never see in any textbook because most people are more reasonable than me is the exact same you can almost view it topologically if the the exact same circuit as what I drew in the previous diagram although now I will assign numbers to it let's say that this resistance is 20 ohms 20 ohms and let's say that this resistance is 5 ohms what I want to know is what is the current through the system so or you know first left to figure out what the equivalent resistance is and then we could just use Ohm's law to figure out the current system so we want to know what the current is and we know that the convention is the current flows from the positive terminal to the negative terminal so how do we figure out the equivalent resistance well we know that we just hopefully prove to you that the total resistance is equal to 1 over this resistor plus 1 over this resistor so one over is I won't keep right is equal what's 1 over 20 well actually let's just make a fraction so it's 1 over 20 plus 1 over 5 is 1 over 20 that's for over 20 right so 1 over our total resistance is equal to 5 over 20 which is equal to what 1 over 4 right 5 over 20 so if 1 over r is equal to 1 over 4 r must be equal to 4 so R is equal to 4 ohms so we could redraw this crazy circuit as this I'll try to draw it small down here we could redraw this where this resistance is 4 ohms and this is 16 volts right we could say that this is this whole thing combined is really just a resistor that is 4 ohms well if we have a 16 volt potential difference current is flowing that way even though that's not the electrons are doing and that's what our resistance is 4 ohms what is the current V equals IR Ohm's law the voltage is 16 volts it equals the current times 4 ohms current so current is equal to 16 divided by 4 is equal to 4 amps so let's do something interesting let's figure out what the current is flowing through what's this what's the current i1 and what's this current i2 well we know that the potential difference from here to here is also 16 volts right because this whole thing is is as it is is essentially the same potential and this whole thing is essentially the same potential so you have 16 volts across there 16 volts divided by 20 ohms so let's call this I 1 so i1 is equal to 16 volts divided by 20 ohms which is equal to what 4/5 so it equals 4/5 of an ampere or 0.8 amperes right and similarly what is the amount of current flowing through here I - I'm due this is a different color it's getting confusing I'll do it in the vibrant yellow so the current falling through here once again the potential difference from here that's not different enough the potential difference from here to here is also 16 volts right so the current is going to be i2 is going to be equal to 16 over 5 which is equal to 3 and 1/5 amps so most of the current is actually flowing through this and that makes sense because the resistance is less right so that should hopefully give you a little bit of intuition of what's going on and less current is flowing through here so i1 through the 20 ohm resistor is we can say 0.8 amps it's I 1 and I 2 through the 5 ohm resistor is equal to 3 point 2 amps and it makes sense that when you add these two currents together the current the 3.2 ampere is flowing through here and the point 8 ampere is flowing through here that when they merge they merge and then you have 4 amperes flowing through there anyway hopefully I have given you some intuition on what happens when we put parallel when we put resistors in parallel I will see you in the next video