6.12 Kirchoff’s Rules
Alright, we have seen that we can use Ohm’s law in order to analyze simple electric circuits provided that it has only one electromotive source. If we have more than one electromotive in a given electric circuit, with multi-loops, then we cannot apply Ohm’s law directly. In order to analyze those type of circuits, we’re going to apply a specific set of rules which are called Kirchoff’s rules.
Multi-loop circuits, and let’s introduced Kirchoff’s rules in order to analyze these type of circuits. The first rule simply states that algebraic sum of the changes in potential from the complete traversal of a closed loop is equal to 0.
And the second rule is a familiar one. We have seen this earlier, which we call it, “junction rule”. It simply states that the sum of the currents into a junction is equal to the sum of the currents out of that junction. These rules are also known in short form, as the first one, “potential rule”, and the second one is known as “junction rule”.
Okay. First, let’s try to understand the first rule, through a simple mechanical example. Let’s say potential rule. Let’s assume that we’re hiking along a mountain. We start at a certain point — let’s say initial point — and we climb up for a while, and then walk at a level, and then climb up again, through these paths, and then turn, climb up again, walk at a level, and then go down along this path. And then go down again, and let’s say turn, and walk down again. And finally, go back to the point that we have started.
Now, along this closed loop of path, it portrays the changes in gravitational potential energy that we will experience as we hike along this path. Since our final point is the same point that we started with, the change in gravitational potential energy, the total change in our gravitational potential energy, will add up to 0. In other words, we can easily say that ΔU over here, the gravitational potential energy, U total is going to be equal to 0, because we end up at the same point that we started.
Well, if we just trace the gravitational potential energy change that we experience, as we climb up from initial point to this point, since our height is going to increase relative to the level that we started, our potential energy will increase, let’s say, by ΔU1. As we walk level relative to our original height, let’s say, then we will not experience any change in gravitational potential energy. Then from this point and that point again, the height relative to the original level will increase so that the potential energy will increase, again, let’s say, +ΔU2 times.
Again, turning back to the mountain, and then walking up, is going to cause an increase in our potential energy by ΔU3. Here again, as we walk level, relative to the original level, or parallel to the original level, we will not experience any change in altitude. Therefore, the gravitational potential energy will remain the same over here.
Now, we’re going to go downhill. Therefore, the height will get smaller, so the potential energy will decrease by Δ, let’s say U4. And then finally, going downhill through this path will cause a decrease by ΔU5 in our gravitational potential energy, Δ-U6, through this interval, and finally, -ΔU7 through this interval. Then we end up at the point that we started with.
Since, again, at the final destination point, since it is the same point that we started when we looked at our total change in our gravitational potential energy, we will see that that will be equal to 0 because the gravitational potential energy for the initial and the final case will be the same and their difference will give us just simply 0. So, we can therefore, write down that ΔU total, the total change in gravitational potential energy, which is going to be equal to 0, and that will be equal to algebraic sum of all these changes in gravitational potential energy. In other words, plus ΔU1 plus ΔU2, and then plus ΔU3, and minus ΔU4, minus ΔU5, minus ΔU6, and finally, minus ΔU7. So when we add all these changes algebraically, we’re going to end up with 0.
Now, moving from here to our closed circuit loop, instead of looking at the changes in electrical potential energy in this case, we will just trace the changes in electric potential as we traverse or as we cross every component along that loop. To be able to do that, we will follow some certain rules. Before we introduce those rules for each component, let’s review the junction rule over here also.
That rule simply states that if we have bunch of branches entering into a junction point and a bunch of branches coming out of that junction point, and if we look at the currents flowing through these branches as i1, i2 and i3 entering into this point, which we called a junction point, and if the currents i4, i5, i6, and i7 is coming out of from that junction point, this rule simply states i1, plus i2, plus i3 are the sum of the currents entering into this junction point, is equal to sum of the currents coming out of that junction, and those are i4, plus i5, plus i6, and plus i7. Of course, this is directly associated with the conservation of charge principle, and as you can easily see that the potential rule, which we use as mechanical example over here, is directly associated with the conservation of charge principle.
Now, let’s look at some certain rules for the change in potential as we cross different types of components. First, we’re going to start with the electromotive force, EMF. Now, let’s say this is our electromotive force. And as you recall, the short side represents the negative terminal, the long one represents the positive terminal. And we have introduced that the EMF arrow gives us the direction in which the charges are gaining electrical potential energy, and that always points from negative terminal to positive terminal. So here’s the direction of the EMF arrow.
I’m going to represent the direction in which that I will be crossing the component with dashed lines, in purple color. So, if I cross an electromotive force in the circuit, in the direction of EMF arrow, then the potential will change. ΔV is going to be equal to +ε volts. In other words, as I go from negative to positive terminal, with a charge, I will see that that charge will gain potential of +ε volts. We will go from low potential region to high potential region.
The second rule is just the opposite of this. And if I have the same electromotive force, an EMF arrow is always pointing from negative to positive. And if I trace this component, in opposite direction to the EMF arrow, like this — so therefore, let me just make a note over here, the purple line, dash line represents direction to traverse — then the potential will change by –ε volts. In other words, as we go from positive to negative, we will go from high potential region to low potential region and the difference is ε volts. So we will experience –ε volts of decrease in potential.
The third one is related to the capacitors, which is very similar to the electromotive force. Let’s assume that we charge a capacitor such that this plate is charged positively. The other plate is charged negatively, to some q coulombs. Now, obviously, if we cross this capacitor from negative to positive, then we will go from low potential to high potential region. And as you recall, C, by definition, capacitance was equal to amount of charge, magnitude of the charge stored on the capacitor, divided by the potential difference between the plates of the capacitor. Therefore, the potential difference is going to be equal to q over C. So, the potential difference between the plates of the capacitor, V, is q over C.
Now, if we go from negative to positive plate, then the potential is going to increase by plus q over C times. On the other hand, if we just cross the same unit, the same capacitor, with its negatively charged plate here, and positively charged plate here, in this case, from positive to negative plate, then we’re going to experience a change in potential which will be equal to minus q over C. In other words, as we go from positive to negative plate, we will experience a decrease in potential, which will be equal to as much as the potential difference between the plates of this capacitor.
Alright, the fifth rule is associated with the resistance. Let’s assume that we have a resistor with a resistance of R, and a current is flowing through this resistor from left to right. Well, if we cross this resistor in the same direction with the direction of flow of current, in that case, as we recall from the Ohm’s law, the resistance is equal to potential difference between the ends of the resistor divided by the amount of current passing through the resistor, V over i, then the potential difference between the ends of the resistor is going to be equal to i times R.
Now, if we cross this resistor in the same direction with the flow of current, then we will experience that the potential will decrease by –i times R. Why is it going to decrease? Because if we just sit on a charge in this current and move with that charge, just right before the resistance, before it enters the resistance or resistant medium, it has a certain amount of electrical potential energy.
Now, as it goes through this medium, it will make many collisions. Therefore, some of its electrical potential energy will be converted into heat. And whenever it emerges from the resistors, it will not have the same electrical potential energy that it had before it entered the resistant medium. So it will end up with lesser electrical potential energy. And since potential is electrical potential energy per unit charge, therefore, we’re going to have low potential once the current emerges in comparing to the end that the current enters into this resistor.
Going, therefore, along the current flow direction, we will experience a decrease in potential, and that’s why we have a negative sign over here. You can also remember this case by comparing to a mechanical analogy such that if you walk downstream along a river or a creek, in other words, if you walk in the same direction with the current of water flow, and then the gravitational potential energy will decrease, because water always flows in downhill direction. There’s no way that it will flow in uphill unless it is pumped. So walking downhill means that your height, relative to some origin is going to get smaller and smaller. It means that you will experience a decrease in your gravitational potential energy.
Of course, the opposite of this is that crossing the resistor in opposite direction, to the direction of flow of current. In this case, we’re going to go from low potential region to high potential region. Then we will experience an increase in potential, and that is going to be equal to plus i times the resistance R. Why? Again, this is equivalent to walking uphill, or in opposite direction to the flow of current and creek. And if we’re walking uphill in the opposite direction to the flow of water, then our gravitational potential energy will increase.
In a similar way, if we cross the resistor, in opposite direction to the flow of current, then we’re going to end up increase in potential, and that will be equal to the amount of current passing through the resistor times the resistance of the resistor directly from the Ohm’s law. Later on, we’re going to introduce the related rules when we introduce more components for the typical simple electric circuits.