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Resistors in Parallel
PhD., University of Maryland
Jonathan is a published author and recently completed a book on physics and applied mathematics.
Resistors in parallel consist of two separate independent circuits so that when the current reaches a resistor, the current can choose which to go through. Most homes are wired in parallel so that unlike resistors in series not all appliances in our homes have to be turned on for a single one to work. Adding resistors in parallel reduces the overall resistance. The smallest current will go through the largest resistor. To write all the resistors as a single resistor use the equation 1/(effective resistance) = 1/(resistor 1) + 1/(resistor 2). The effective resistance is smaller than either of the two individual resistors.
So let's talk about resistor networks that are connected in parallel. So first off what does parallel mean? Parallel means there's a choice, it means that there's a branching off in thee circuit and if the current can choose to go one direction or another direction. So in parallel a resistor network looks like this. The current comes in and then it has a choice, it's going between the same 2 points but there's 2 different roads that it can take. Alright so basically parallel networks consist of 2 separate circuits that are independent of one another. Alright so in parallel potential difference is the same, so the potential difference between these 2 points is the same because they're the same 2 points. They were just 2 different branches that went through. The current is going to add, so what we would say here is that i1 plus i2 is equal to the total current i. i1 plus i2 alright now as always what we'd like to do is consider this parallel network as a single effective resistor. So I want the potential difference to be the same obviously the current going in here into my effective resistor is going to be i and so what we're going to do is we're going to write the current as delta v over r the potential difference divided by the resistance wiht of course our obligatory minus sign. So it'll be minus potential difference over r parallel equals minus potential difference over r1 minus potential difference over r2.
Now the potential difference is the same so that means that that's going to cancel out along with the minus sign and that gives us a formula for adding resistors in parallel. 1 over r parallel equals 1 over r1 plus 1 over r2. So adding resistors in parallel is more complicated than adding them in series. When I add them in series I just add, when I add them in parallel what we say is that the reciprocals add. So 1 over the effective resistance equals 1 over r1 plus 1 over r2. Now very, very, tempting but absolutely incorrect to try to flip this up-side-down and say that just means r parallel equals r1 plus r2 that is not true. Now one thing that we can think about parallel circuits as is kind of like you're trying to get from point a to point b. If there's only 1 road then it's going to be annoying, anybody who wants to go from point a to point b got to take that road.
If I add in parallel it's like constructing a new road that also goes from point a to point b and so that's actually going to reduce the traffic. So that's an important thing to remember about parallel and it will immediately tell us that you can't just add. When I add resistors in parallel I reduce the overall resistance alright now the way that this parallel network is going to work I've got 5 amps coming in and I've got 2 resistors the 15 ohm and the 10 ohm. The potential difference is the same, you can kind of think of these as point a point b 2 different roads, think of the resistors as representing traffic lights. The larger the resistance the more traffic lights along this path so the less cars are going to go there. So the smallest current will go through the largest resistor in a parallel combination. Alright so how are we going to determine how much current goes through the 15 and how much current goes through the 10? It's actually really easy, what we're going to do first is we're going to add these 2 resistors in parallel. Now remember adding in parallel is a little bit tricky, so we'll say 1 over r parallel equals 1 over 15 plus 1 over 10.
Alright now so like middle school we got to cross multiply. So it'll be 10 plus 15 over 150 alright. 10 plus 15 is 25 over 150 and if we do this carefully we can cancel. So this gives us one sixth. 1 over r parallel is 1 over 6 so that means r parallel equals 6 okay. Notice that our effective resistance is smaller than either of the 2 resistances that we added together smaller than either of them okay. And in act this number will always lie between this one divided by 2 and this one divided by 2, 5, 7 and a half, 6 alright it'll always work that way. Alright so now that we've got the effective resistance how are we going to determine the current through each of these guys? Well we're going to use the idea that the potential difference has to be the same in parallel always, always, always. So what is the potential difference? Well the potential difference got to be 5 times 6 because this network is really just this 6 ohms, 5 amps going through. Well 5 times 6 is 30 so that means the potential difference got to be 30.
Alright in order to make a potential difference of 30 volts across a 15 ohms resistor, 2 amps because 2 times 15 is 30 what about down here I want 30 volts across it so how many amps do I need? 3 amps and of course this follows our earlier decision that the current add 5 amps goes in 2 and 1 branch, 3 in the other. Alright one other way to think about parallel circuits is houses, that's the way that houses are wired because I don't want it to be necessary for me to turn on the TV in order to get the dish washer to work right, so we wire those 2 circuits separately. Every single outlet in your house is a separate parallel branch of the same circuit and so that's how we get housing to work. Alright those are parallel resistor circuits.
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