6.11 Connection of Resistances- Series and Parallel
Alright, now let’s consider the connections of resistances. Connections of Resistors.
Well, as we have seen in the case of connections of capacitors, resistors also can be connected in two different ways in electric circuits. Namely, parallel and series connections. If you consider first the series connections of resistors, in this case, again like in the case of the capacitors, resistors are connected one after each other like the cufflinks of train cars along the same railroad track. If you consider three resistors with resistances of R1, R2, and R3, we connect them in series through this way and let’s say then, we apply a potential difference across the combination by connecting the ends of the combination to the terminals of the power supply or an electromotive force which is generating ε volts of potential difference.
As soon as we turn the switch on, the current i will emerge from the positive terminal and flow through the combination towards the negative terminal of the power supply. We can easily see that since we have only one available path for the charges to move, then the amount of current passing through each resistor in series combination will be the same. Therefore, the first property of this connection is that i1, will be equal to i2, which is going to be equal to i3, and they all will be equal to i.
Now here’s one point, common mistake, or misconception, sometimes people think that once the current enters to a resistor and when it emerges from the resistor, then somehow it becomes less in comparing to the instant that it was entering into the resistor. Again, that cannot be true because if that were true, it would violate the conservation of charge principle. In other words, if you have 5 charges, let’s say, entering into this resistance R1, then we have to have 5 charges coming out. We cannot have a case that only 3 charges are coming out and the rest is just disappearing.
So, yes, something is getting less as the charges emerge from the resistant medium and that something is not the amount of charge but the energy associated with each one of those charges. In other words, the charges that enter, before they enter into the resistant medium, they have greater potential energy. And as they go through that resistor medium, because of the collisions that they make with the atoms of the medium, they lose some of their potential energy in the form of heat. That’s why when they emerge from the resistor medium they will not have the same electrical potential energy that they possessed before entering into that resistor medium. So you have to make that point clear in your mind.
Well, again, as we recall from the series connections of capacitors, that’s the property, the general property of series connection is that the potential difference across the whole combination. And in this case, that potential difference is going to be equal to the potential difference generated by the power supply, which is ε volts. So across the combination we are going to have ε volts of potential difference and that will be equal to some of the potential differences across each resistor.
In other words, if we just take a volt meter, which is a device that measures the potential difference between two points, and connect the points across the resistor R1, we will read V1 of volts. Then if we do the same thing across R2, it will read V2 of volts. And doing that across R3, we will read V3 of volts. We see that ε, which is the potential difference across the whole combination, is going to be equal to V1 plus V2 plus V3. So that’s the second property of the series connections of the resistors.
Now, as we did in the case in connections of capacitors, again, we will simplify this circuit simply by replacing these three resistors, which are in series with the single resistance, so that it will do the same job that these three resistors are doing in series combination in this circuit. Well, to be able to do the same job, the resistor that will replace these three, whenever this connected to the circuit should draw the same amount of current from the same battery.
Therefore, if we redraw that circuit over here, in a simplified version, we’re going to end up with a simpler circuit like this. And that’s called this resistance as “R equivalent” (Req). And we want this Req to do the same job that these three, R1, R2, and R3, doing in series connection. Means that we wanted to draw the current i from the same battery, or same EMF, or same power supply.
Well, if we recall Ohm’s law, here, then Req is going to be equal to the ratio of the potential difference between the end of this resistor, which is going to be ε volts — that is the voltage generated by the electromotive force — to the amount of currents passing through those points. Therefore, we will have ε over i. Using Ohms law we can express R1, R2, and R3 also, in a similar way. And solving this equation for ε, we will have i times Req.
Therefore, we can say that V1 is going to be equal to i1 times R1. But from property 1, i1 is equal to i, so, this is going to be equal to i times R1. Similarly, V2 will be equal to i2 times R2. Again, i2 is equal to i. So we can write this as i times R2. And finally, we can express V3 as i3 times R3. And that too will be equal to i times R3, because the same current will flow through resistance R3 also, in series connection.
Now, from property 2, ε is equal to V1 plus V2 plus V3. And for ε, we can write it down as, from Ohm’s law, i times Req, which will be equal to i times R1 for V1. i times R2 for V2, and plus i times R3 for V3. Since current i is common in every term, dividing both sides by i, we can eliminate the current and therefore the equivalent resistance of these three resistances, R1, R2, and R3 in series connection, becomes equal to R1 plus R2 plus R3.
Of course we can easily generalize this by saying that for N number of resistances in series connection, equivalent resistance becomes equal to summation, i from 1 to N of Ri. In other words, the equivalent resistance becomes the sum of the resistances of all the resistors in series connection. So, when we connect the resistances in series, then the total resistance of the circuit increases and it becomes greater than the largest resistance in combination.
Moving on, for the second type of connection which is parallel connection of resistances or resistors, and again, similar to the case of capacitors, we connect the resistances such that the potential difference across each resistor or resistance, becomes equal to the other one. In order to do that, we connect them like this, and apply a potential difference across the combination, by connecting them to the terminals of the power supply. Like this.
So, the EMF, electromotive force, generates ε volts of potential difference. Let’s say this is our resistance R1, R2, and R3. If we take our volt meter and connect across from each resistor, we will read the same potential difference across the combination between these two ends, we’re going to have ε volts of potential difference and that will be equal to ε volts across R3, ε volts across R2, and ε volts across R1.
So the first property of the parallel connection will therefore be that the potential difference across each one of the resistances will be equal to one another and they all will be equal to the net potential difference across the whole combination. In this case that will be equal to ε volts.
Of course this is associated with the conservation of energy principle, and the second property will come from the conservation of charge. As soon as we turn the switch on current i is going to emerge from the positive terminal of the power supply and will flow along this path. Whenever it comes to this point, the charge will see three available paths, which we call that point as the junction point. Some fraction of the current will go through this first path along this branch as current i1. Some fraction will go through resistor R2 as i2. And the remaining fraction will go through this branch as i3 through R3.
All these currents will come back and join together again at this junction point and they will join together and they are going to make current i again. Therefore the second property is going to be that the amount of current entering into any one of these junctions will be equal to sum of the current coming out of that junction, i1 plus i2 plus i3. Therefore we can write that over here in a compact form by saying that sum of the currents into a junction point is equal to sum of the currents out of that junction, and this is directly associated with the conservation of charge principle.
Alright, again, let’s do the same process, simplify this circuit such that we can replace all these three resistances in parallel connection with a single resistance such that it will do the same job that these three resistances in parallel connection, doing in this circuit. Therefore, we just take this unit, these three resistors, and replace them with a single one and we will call that single one again Req. And connect the ends of this resistor to the terminals of the same electromotive force, which is generating ε volts.
Here again, as soon as we turn the switch on we want this Req to do the same job that these three are doing therefore it will draw the same current from the same electromotive force, current i, which will flow through this resistance. By applying Ohm’s law, Req is going to be equal to potential across from the resistor, and that is ε volts, in this case, divided by the amount of current passing through the resistor, and that is i.
Well, here, we can write down i as ε over Req. Using a similar type of relationship we can express the current flowing through resistance R1 as the potential difference across R1 divided by the resistance of R1. But since V1 is equal to ε volts, then we can express this as ε over R1. Similarly, i2 is going to be equal to V2 over R2, and that too will be equal to ε over R2 because, again, V2, in parallel connection, will be equal to the potential difference across the whole combination, which is ε volts. And finally, i3 will be V3 over R3, and that will be also equal to ε over R3.
If you go back to the general properties and look at the property 2, from property 2, the current i is equal to i1 plus i2 plus i3. We can write down ε over Req for i, ε over R1 for i1, plus ε over R2 for i2, and plus ε over R3 for i3. Since ε volts is coming in every term, if you divide both sides by ε, we eliminate ε from both sides of the equation and we end up with 1 over Req is equal to 1 over R1 plus 1 over R2 plus 1 over R3.
Again, if we generalize this for N number of resistances, 1 over Req is going to be equal to summation i from 1 to N of 1 over Ri. In other words, inverse of the equivalent resistance will be equal to sum of the inverses of the each resistance in parallel connection. As a result of this, the equivalent resistance will be smaller than the smallest resistance in combination. So, once we connect the resistances in parallel, then, the total resistance of the circuit will decrease.
Now, again, if we go back to our circuit diagram and see the difference between parallel connection and the series connection, and as you can see in the series connection, all the components, all the resistances will have the same current. So if you think of this circuit in terms of the light box, and if we replace R1, R2, and R3 with the light box over here, and if one of the bulbs burns out, then we will have an open circuit. In other words, we will not have a complete path for the charges to move. Therefore, the current will drop to 0 through each one of these bulbs. Therefore, none of the bulbs will emit light.
Let me turn this switch in the on position. So it is the case of for example in Christmas tree lights, for some of them, that if one of the bulbs is burned out, then one has to really look for and find that bulb because none of the bulbs are going to be able to be turned on.
But in the case of parallel connection, every resistance will have their own unique current. Again, if you replace these resistances by lightbulbs, even if one of them burns out, only that one will not emit light, but the other ones will continue to be on — let me turn the switch on — because their current will be independent of the current that the burned one is basically having.
As a result of this, household circuits are always connected in parallel connection, so that if one of the appliances running with the electricity dies out, it will not effect the other ones. In other words, for example, whenever your refrigerator goes bad, your TV will continue to run as a result of this parallel connection. So you can remember these connections by thinking that the independent paths for the current, in the case of parallel connection, the same path for the components in the case of series connection.
Another thing that you should be always careful while you are calculating the equivalent resistance of a parallel connection that, again, we had a similar type of case for the capacitors also but that was for the series connection. You end up with the inverse of the equivalent resistance by applying this expression. So once you calculate the right-hand side, you should just continue and take the inverse of that to be able to get the equivalent resistance of the parallel connection.
Another interesting point over here, the mathematical form of these equations are just the opposite for the connections in comparing to the connections of the capacitors. In other words, in the case of capacitors, when we connected them in parallel, the total capacitance was the sum of the capacitances of each capacitor. Whereas in the case of resistors, we get the inverse of the equivalent resistance as sum of the inverses of each resistance in the combination.