25.4:
Equivalent Capacitance
The net capacitance for a capacitor network can be calculated by finding the equivalent capacitances individually for the parallel only and series only combinations.
Consider a capacitor network consisting of a series and parallel combination of four capacitors connected to a battery. What is the charge across each capacitor?
Since capacitors 2 and 3 are connected in parallel, the sum of their capacitances gives the equivalent capacitance.
So, the four-capacitor network reduces to three capacitors connected in series.
Now, the charge on each capacitor has an equal magnitude, while the applied potential difference equals the sum of the voltage across each capacitor.
Since voltage equals the ratio of charge to capacitance, substituting the values of capacitances and applied voltage gives the charge accumulated on each capacitor in series.
The voltage across capacitors 2 and 3 are equal and can be obtained from the calculated charge.
Finally, the product of individual capacitances and voltage values gives the charges on capacitors 2 and 3.
So, the charges on all the capacitors in the network are estimated.
25.4:
Equivalent Capacitance
Multiple capacitors can be connected in a circuit in series or parallel configuration. When the capacitor combination is connected to a battery, the potential drop across each capacitor and the magnitude of charge stored in the individual capacitor depends on the type of the connection. The capacitor combination is replaced by a single equivalent capacitor that stores the same amount of charge as the combination for a given potential difference.
The following strategies are adopted to calculate the net capacitance for a capacitor network:
- The capacitor network is grouped into capacitors connected in series and parallel in the network.
- A series combination's equivalent capacitor has the same charge magnitude as the individual capacitors. The potential difference across the series combination is the sum of potential differences across the individual capacitors. Finally, the sum of reciprocals of individual capacitances gives the reciprocal of the equivalent capacitance in a series circuit.
- The equivalent capacitor for a parallel combination has the same potential difference as the individual capacitor. The sum of the charges of the individual capacitors equals the charge of the equivalent capacitor. The equivalent capacitance equals the sum of all individual capacitances in the network.
- For series combinations, the magnitude of equivalent capacitance is smaller than any of the individual capacitances. On the contrary, the equivalent capacitance value is greater for the capacitors connected in parallel than any of the individual capacitances.
- All the series or parallel groups in the network are replaced with the equivalent capacitance in multiple steps till the net capacitance is obtained.
Suggested Reading
- Young, H. D., and Freedman, R.A. (2012). University Physics with Modern Physics. San Francisco, CA: Pearson. pp. 795.
- OpenStax. (2019). University Physics Vol. 2. [Web version]. pp. 359 Retrieved from https://openstax.org/books/college-physics/pages/19-5-capacitors-and-dielectrics