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Interpreting expressions with multiple variables: Resistors

Video transcript

- [Lecturer] We're told an electronic circuit has two resistors with resistances R one and R two connected in parallel. The circuit's total resistance, R sub t or Rt is given by this formula. Suppose we increase the value R one while keeping R two constant. Does the value of R sub t increase, decrease, or stay the same? So pause the video and see if you can answer this question. All right, now let's work through this together, and some of you might be familiar with the idea of an electronic circuit and resistors, and what they represent, but you really don't need to understand that in order to understand what's going on in this expression. There's some quantity, R sub t, that's equal to 1 over, and then in the denominator we have 1 over R one plus 1 over R two. So if we increase the value of R one So if we increase the value of R one while keeping R two constant, what happens? So this is going to increase, and R two is going to be constant. So one way to think about it, we have two variables here, especially in this denominator, but really, in this entire expression, but if R two is going to be constant, we really just have to focus our analysis on R one. If R two is constant, that means it's just a number. It could be two, it could be five, it could be pi. Whatever, but that is not going to change as we increase the value of R one. So let's think about what's happening here. If R sub 1 increases, If R sub 1 increases, then what does that do to 1 over R one? Well, if you increase the denominator, then you're going to decrease the reciprocal of that. So that means that this whole thing right over here is going to decrease. Now, if 1 over R one is decreasing, if 1 over R one is decreasing, what's going to happen to 1 over R one plus 1 over R two? what's going to happen to 1 over R one plus 1 over R two? Will this entire expression increase or decrease? Well, this part is staying constant. R two is constant. So 1 over R two is constant. Just imagine, R two could be two or three. So this should just be 1/2 or 1/3 or whatever it is, while over here, this part of the expression is going down. So if you're taking the sum of two things, one part's going down, the other part's constant, then that means this whole thing is going to be going down. So the entire denominator of this entire thing is going down. Now, if the entire denominator is going down, if 1 over R one plus 1 over R two, if this whole thing is going down, what's going to happen to the reciprocal of that? 1 over 1 over R one plus 1 over R two. Well, if something is going down, the reciprocal of that is going to go up. If you get smaller and smaller denominators, 1 over that is going to be a larger and larger value. So the value or Rt increases if R one increases and R two is constant. if R one increases and R two is constant. And for those of you who know about resistance, which is really how well a current can flow through a circuit, that will also make intuitive sense, but you don't need to understand resistance to analyze this mathematically.