A 6.0 pF capacitor is needed to construct a circuit. The...

A $6.0 \mathrm{pF}$ capacitor is needed to construct a circuit. The only capacitors available are rated as $9.0 \mathrm{pF}$. How can a combination of three 9.0 pF capacitors be assembled so that the equivalent capacitance of the combination is $6.0 \mathrm{pF} ?$

Key Concepts

Series Combination

When capacitors are connected end-to-end in a series arrangement, the charge on each capacitor is the same and the equivalent capacitance of the network is always less than the smallest individual capacitance. The formula for two capacitors in series is given by 1/C_eq = 1/C_1 + 1/C_2, and it can be extended for more than two capacitors. This concept is essential when designing circuits that require a specific capacitance that is lower than the available capacitance values.

Parallel Combination

In a parallel arrangement, the capacitors all share the same voltage across them and their equivalent capacitance is the sum of the individual capacitances (C_eq = C_1 + C_2 + ...). This property allows us to increase the total capacitance available in a circuit. Understanding how to combine capacitors in parallel is crucial when the design requires a capacitance that is higher than the capacitance of any individual capacitor available.

Combining Series and Parallel Arrangements

Often, achieving a desired capacitance value that isn't directly available from individual components necessitates using both series and parallel connections in one network. By strategically configuring capacitors in parallel to increase capacitance and then arranging the resulting network in series to reduce it, one can realize non-standard equivalent capacitance values. This approach is particularly useful in fine-tuning circuits when only a limited set of component values are on hand.

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