Parallel resistors (part 3)

in this video we're going to talk even some more about parallel resistors parallel resistors are resistors that are connected end to end and share the same notes here's r1 and r2 they share the same nodes that one and that one and that means they share the same voltage and we worked out a expression for how to replace that with a single resistor R parallel and we found that a 1 over R parallel is 1 over r1 plus 1 over r2 so in this video I'm going to actually start working with this expression a little bit more and we'll just change it around a little bit to an easier and easy way to remember it and then we're going to do a special case where r1 and r2 are the same value and we're going to see what happens so right now I just want to do a little bit of algebra on that expression 1 over R P is the same as 1 over R 1 plus 1 over r2 what I want to end up with here is an expression that on this side says RP equals something and on this side I just want one expression not not two fractions so we're going to go about that by combining these two fractions first so the least common denominator here l see D equals r1 times r2 and I'm going to convert both these to convert this to that proper denominator I have to multiply it by r2 over r2 so we'll do it we'll do all the steps we'll multiply it by r2 over r2 on this expression we have to multiply it by r1 over r1 and this equals of course one over RP continuing on one over RP equals R 2 over R 1 R 2 plus R 1 over r 1 r 2 and now i can combine them together let's let's move up here one over RP equals will keep everything in numerical order R 1 plus R 2 over R 1 R 2 and now I'm going to take the reciprocal of both sides of the expression so I get an expression in R P R P equals and just flip over this expression R 1 R 2 over R 1 plus R 2 that is a way you can remember how to combine parallel resistors the parallel equivalent resistor RP is the product of the two resistors over the sub so it's a product over the sum that's how I remember it now this arithmetic this expression is exactly the same as the original one that we had these are the same and it's just a question which one do you want to remember which one's easier to remember and which ones easy is to calculate I like to remember this one here let's do a quick example using it I'll move the screen up a little bit let's leave that there so we can see it let's say we have two parallel resistors and we'll say the one the first one is 1000 ohms and the second one is four thousand ohms and the question is what is the parallel combination of those things how can I replace these with one resistor so the same current flows and we'll use this expression here so RP equals a product which is 1000 times 4000 divided by the sum 1000 plus 4000 and that equals oh my goodness there's a lot of zeros here four and six zeros one two three four five six over that's easy five thousand all right and let's say let's take out three zeros out of this one let's knock off three zeros here and three zeros here and I have 4000 divided by five and that equals 800 ohms so that is this and that's how we use this expression not just something to notice here notice that our parallel the equivalent parallel resistor is smaller than both of these and that happens every time that happens every time the parallel resistance is smaller than the smallest resistor so here it was 1,000 it's going to be smaller than that and that's a property of parallel resistors because you have two current paths that allow current to go two different ways the effective resistance is always smaller than the smallest original path because there's a way for current to go around another way so this is now the expression for two parallel resistors and that's a good one to remember now I'm going to show you one special case and we'll do this in this color what if R 1 equals R 2 what is RP and for this special case we use the same expression we say RP we say RP is the product will just use R because R is this it's the same value R times R over hour plus R and that's multiplied and so that is R squared over 2 R and one of these RS cancels so let's cancel let's cancel that are and that's squared and we end up with 0 over 2 so for the special case if R 1 equals R 2 then our P equals R over 2 it's just half and that should make sense if that that if we do something like this if we draw two resistors in parallel like this and we say this is 300 ohms and this is 300 ohms that means the effective effective parallel resistor 150 ohms that's got a very pleasing symmetry to it because these resistors are the same they have the same voltage they're going to have the same current basically twice as much current is going to flow in this circuit as would flow if there was just one of these guys so that's where the divide by two comes from so for two resistors in parallel if the resistors are the same value the effective parallel resistance is just half