Conservation of charge requires that equal-magnitude charges be created on the plates of the individual capacitors, since charge is only being separated in these originally neutral devices. The end result is that the combination resembles a single capacitor with an effective plate separation greater than that of the individual capacitors alone.
See Figure 1b. Larger plate separation means smaller capacitance. It is a general feature of series connections of capacitors that the total capacitance is less than any of the individual capacitances. Figure 1. The magnitude of the charge on each plate is Q. Series connections produce a total capacitance that is less than that of any of the individual capacitors.
We can find an expression for the total capacitance by considering the voltage across the individual capacitors shown in Figure 1. Entering the expressions for V 1 , V 2 , and V 3 , we get. Canceling the Q s, we obtain the equation for the total capacitance in series C S to be.
An expression of this form always results in a total capacitance C S that is less than any of the individual capacitances C 1 , C 2 , …, as Example 1 illustrates. Find the total capacitance for three capacitors connected in series, given their individual capacitances are 1. With the given information, the total capacitance can be found using the equation for capacitance in series.
The total series capacitance C s is less than the smallest individual capacitance, as promised. In series connections of capacitors, the sum is less than the parts. In fact, it is less than any individual. Note that it is sometimes possible, and more convenient, to solve an equation like the above by finding the least common denominator, which in this case showing only whole-number calculations is Figure 2a shows a parallel connection of three capacitors with a voltage applied.
Here the total capacitance is easier to find than in the series case. To find the equivalent total capacitance C p , we first note that the voltage across each capacitor is V , the same as that of the source, since they are connected directly to it through a conductor. Conductors are equipotentials, and so the voltage across the capacitors is the same as that across the voltage source.
Thus the capacitors have the same charges on them as they would have if connected individually to the voltage source. Figure 2. Since the capacitors are connected in parallel, they all have the same voltage V across their plates. However, each capacitor in the parallel network may store a different charge. To find the equivalent capacitance of the parallel network, we note that the total charge Q stored by the network is the sum of all the individual charges:. On the left-hand side of this equation, we use the relation , which holds for the entire network.
On the right-hand side of the equation, we use the relations and for the three capacitors in the network. In this way we obtain. This equation, when simplified, is the expression for the equivalent capacitance of the parallel network of three capacitors:.
This expression is easily generalized to any number of capacitors connected in parallel in the network. For capacitors connected in a parallel combination , the equivalent net capacitance is the sum of all individual capacitances in the network,. Equivalent Capacitance of a Parallel Network Find the net capacitance for three capacitors connected in parallel, given their individual capacitances are. Solution Entering the given capacitances into Figure yields. Significance Note that in a parallel network of capacitors, the equivalent capacitance is always larger than any of the individual capacitances in the network.
Capacitor networks are usually some combination of series and parallel connections, as shown in Figure. To find the net capacitance of such combinations, we identify parts that contain only series or only parallel connections, and find their equivalent capacitances. We repeat this process until we can determine the equivalent capacitance of the entire network. The following example illustrates this process. Equivalent Capacitance of a Network Find the total capacitance of the combination of capacitors shown in Figure.
Assume the capacitances are known to three decimal places Round your answer to three decimal places. Strategy We first identify which capacitors are in series and which are in parallel. Capacitors and are in series. Their combination, labeled is in parallel with. Solution Since are in series, their equivalent capacitance is obtained with Figure :.
Capacitance is connected in parallel with the third capacitance , so we use Figure to find the equivalent capacitance C of the entire network:. Network of Capacitors Determine the net capacitance C of the capacitor combination shown in Figure when the capacitances are and.
When a Strategy We first compute the net capacitance of the parallel connection and. Then C is the net capacitance of the series connection and. We use the relation to find the charges , , and , and the voltages , , and , across capacitors 1, 2, and 3, respectively.
Solution The equivalent capacitance for and is. The entire three-capacitor combination is equivalent to two capacitors in series,. Consider the equivalent two-capacitor combination in Figure b. Since the capacitors are in series, they have the same charge,. Also, the capacitors share the Because capacitors 2 and 3 are connected in parallel, they are at the same potential difference:. University Physics Volume 2 8. My highlights. Table of contents. Chapter Review. Electricity and Magnetism.
Answer Key. By the end of this section, you will be able to: Explain how to determine the equivalent capacitance of capacitors in series and in parallel combinations Compute the potential difference across the plates and the charge on the plates for a capacitor in a network and determine the net capacitance of a network of capacitors.
The magnitude of the charge on each plate is Q. Equivalent Capacitance of a Series Network Find the total capacitance for three capacitors connected in series, given their individual capacitances are 1. Strategy Because there are only three capacitors in this network, we can find the equivalent capacitance by using Equation 8.
Solution We enter the given capacitances into Equation 8. Each capacitor is connected directly to the battery. Equivalent Capacitance of a Parallel Network Find the net capacitance for three capacitors connected in parallel, given their individual capacitances are 1.
Solution Entering the given capacitances into Equation 8. Equivalent Capacitance of a Network Find the total capacitance of the combination of capacitors shown in Figure 8.
Round your answer to three decimal places. Strategy We first identify which capacitors are in series and which are in parallel. Capacitors C 1 C 1 and C 2 C 2 are in series. Network of Capacitors Determine the net capacitance C of the capacitor combination shown in Figure 8.
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