5.07 Parallel Connection of Capacitors
Before we study the details of how we connect capacitors in a typical electric circuit, let’s introduce some symbols in order to represent some of the typical components for a electric circuit. We’re going to represent a power supply, which can be either a battery or a solar cell or a generator, for example. Which provides the electrical potential energy to the circuit with this symbol. And the large line, or the longer line, represents the positive terminal of the power supply, and the smaller one represents the negative terminal of the power supply.
And we’re going to represent the direction in which the charge is getting electrical potential energy with an arrow, which will always point from negative terminal to the positive terminal of the power supply. And we will denote that by Epsilon. And this is called emf arrow. And we will come back to importance of this direction associated with this arrow later on when we study the Kirchoff Rules. And, of course, we will use straight lines in order to represent the wires in the electric circuit.
And the next component is the one that we have just introduced, is the capacitors. And for that, we will use this symbol to represent a capacitor with a capacitance of C. And we’re going to use this symbol in order to represent a switch in an electric circuit. And also, in similar way, this will represent a two-way switch in the electric circuits.
Okay. When we look at the types of connections of the components of a typical electric circuit, we see two different types of connections. One of them is called parallel connection; the other one is called series connections. In the case of parallel connections, the components are connected to one another such that the potential difference across each component becomes equal to the other ones. And, in the case of series connections, then the components are connected one after each other like the couplings of train cars along the same railroad. And, in this case, the potential difference across the whole combination becomes equal to sum of the potential differences across each of the component in the combination.
And we will study these connections now for the case of capacitors. Let’s start, first, with the parallel connection of the capacitors. In this case, capacitors are connected to one another such that the potential difference across each capacitor within the combination or connection becomes equal to the other one. So capacitors are connected in parallel if the same potential difference is applied to each capacitor.
Let C1, C2, and C3 be 3 capacitors. And we connect these capacitors in parallel this way, in order to apply the same potential difference to each one of them, which is what we call parallel connection. If we connect the ends of the connection to the terminals of the power supply this way, we see the power supply is going to generate V, volts of potential difference. And let’s introduce a switch over here. And when we connect the switch, s, then the positive charges, concentrated at the positive terminal of the power supply, let’s say a battery over here, are continuously repelling one another.
Therefore, as soon as the find this available path, then they will move along this path and they will come to this point at which they will see three possible paths to move. And, let’s say, if q amount of charge is drawn from the positive terminal of this power supply, these positive charges move along through this path, and whenever they come to this point, which they see three available paths, some fraction of them will go through the first path and get collected to the, let’s say, the left plate of the capacitor C1 as charge q1. And then some fraction will go through this available second path and be collected on the, again, the left plate of this capacitor C2 with a value of q2. And then, the remaining of them will go through this available path and get collected through this plate of the capacitor with capacitance of C3 with an amount of, let’s say, q3.
Similarly, the negative charges are repelling one another at the negative terminal of the power supply, and as soon as the switch s is closed, then we will have an available path for them to flow along this path. And then some fraction of them will go through this path and get collected on this terminal of the capacitor C1 as minus q1. Another fraction will go through this path and get collected here as minus q2. And the remaining fraction will go through this path and get collected in the right hand side plate of capacitor C3 as minus q3.
This charging process will continue until we reach high enough charge at the plates of these capacitors such that they will generate strong enough repulsive force to the incoming charges. And at that time, the charging process will stop and the capacitors will be fully charged. Well, since the major property of the parallel connection is such that the potential difference across each component, in this case, across each capacitor, becomes equal to the other one, in parallel connection, we can say that if we call the capacitance, or the potential difference, across C1 as V1 volts, and across C2 as V2 volts, and across C3 as V3 volts. The first property of parallel connection becomes V1 is equal to V2, which is equal to V3. And in this specific circuit, they all will be equal to whatever the potential difference generated by the power supply or by this battery, V volts.
The instrument that measures the potential difference between two points is called voltmeter. And for that, we’re going to use this symbol. Voltmeter: a device that measures the potential difference between two points. And in order to measure the potential difference between two points, we always connect the voltmeter in parallel to those points of interest. In other words, if you want to measure, for example, the potential difference between the terminals of the power supply, we connect our voltmeter to those points of interest in parallel, like this, and therefore, it measures, the potential difference between this end and the other end.
Therefore, if you take a voltmeter, connect to these points, it will give us the potential difference between these two points. We can then take these, the terminals of the voltmeter, and connect across from the combination between these two points, we will still read V volts, whatever the potential difference generated by this battery. And then, if we take the terminals of the voltmeter, connect across the plates of capacitor C1, we will again read the same voltage. And if we do the same thing across C2, again, we will read exactly the same voltage, and across C3 we will read the same potential difference. Indicating that first important property of the parallel combination is that the potential difference across each capacitor in the parallel combination or connection will be equal to one another. Well, this is directly associated with the conservation of energy principle, since we know that the potential is related to the potential energy, which is potential energy per unit charge. And when we connect the capacitors in parallel, in this form, we basically apply the same potential difference across each capacitor.
The second property is going to be associated with the conservation of charge principle. When we turn the switch on, this connection will draw q amount of charge from the power supply, from this battery, which generates V volts of potential difference between its terminals. Then these charges come over here, and see three available paths in this circuit. Then some fraction will go through this first path, some fraction will go through the second path, and the remaining fraction will go through the third path and getting stored on capacitor C1 as q1, on C2 as q2, and on C3 as q3. Conservation of charge principle, therefore, will state that the amount of charge drawn from this power supply is going be equal to sum of the charges stored into the plates of these capacitors. In other words, q1 plus q2 plus q3 will be equal to the total charge drawn from the power supply.
All right. Now, let’s say that we’d like to simplify this circuit simply by replacing all these three capacitors, which are in parallel connection, with a single capacitor. Such that that single capacitor will do the same job that these three are doing in parallel connection. In that case, replacing these three with a single capacitor, our circuit will be simplified into this form.
Here’s the same power supply, generates V volts of potential difference, and here is our switch, s, and here is the capacitor that we are replacing all these three. And let’s call the capacitance of the capacitor as C equivalent. In other words, the equivalent of these three capacitors in parallel form. Well, for this new circuit to do the same job that these three capacitors are doing in parallel connection, when we close the switch s over here, it should withdraw the same amount of charge q from this power supply, from this battery. Therefore, the amount of charge that will be stored in the place of this capacitor is going to be the same amount, q, which was drawn in the previous circuit.
Well, if we recall, let’s say property three over here, if we recall the definition of capacitance, it was equal to the amount of charge, the magnitude of the charge, stored in the place of the capacitor q, divided by the potential difference between the plates of the capacitor. And we can easily see that the potential difference across the plates of this capacitor will be the same potential difference provided by this battery. In other words, if we just hook up, again our voltmeter to the terminals of this battery, we will read V volts, and then if we go, again, and connect our volt meter in parallel, across the plates of this capacitor, we will read exactly the same voltage. So C equivalent, therefore, is going to be equal to q over V.
Well, if we solve for q from this equation, that’s going to be equal to C equivalent times V. Similarly then, we can express the amount of charge stored on C1, which was defined as q1, and that will be equal to a capacitance of the capacitor C1 times the potential difference between the plates of capacitor C1. But, since V1 was equal to V from property one, then we can express this as C1 times V. And similarly, q2 is going to be equal to C2 times V2. And then, again, V2 is equal to V, that will be equal to C2 times V volts. And q3 will be equal to C3 times V3, which will give us, again, since V3 is equal to also V volts, C3 times V.
Now from property two, and that was q is equal to q1 plus q2 plus q3, we can express for q1 in terms of capacitance and the potential difference as C1 times V plus for q2, C2 times V, plus for q3, C3 times V. And similarly, q can be expressed in terms of the equivalent capacitance and the potential difference across one of the plates of that capacitor as C equivalent times V. Since voltage is the same on each term, we can divide both sides by V. And, therefore, we end up with an expression that the C equivalent is equal to C1 plus C2 plus C3.
Now, we can easily generalize this relationship for N number of capacitors in parallel. C equivalent is going to be equal to C1 plus C2 plus C3 plus C sub N. Or, in compact form, we can write this as summation over I from 1 to N of C sub I. This relationship shows us that when we connect capacitors in parallel then the equivalent capacitance of the circuit becomes sum of the capacitances of each individual capacitor in the connection. in other words, the total capacitance of the circuit increases.