(II) When two capacitors are connected in parallel and then...

Name: Problem 52, Chapter 24: Physics for Scientists and Engineers with Modern Physics
Uploaded: 2019-07-07T19:42:37-08:00
Duration: 2 min 1 s
Channel: Kai Chen
Description: (II) When two capacitors are connected in parallel and then connected to a battery, the total stored energy is 5.0 times greater than when they are connected in series and then connected to the same battery. What is the ratio of the two capacitances? (Before the battery is connected in each case, the capacitors are fully discharged.)

(II) When two capacitors are connected in parallel and then connected to a battery, the total stored energy is 5.0 times greater than when they are connected in series and then connected to the same battery. What is the ratio of the two capacitances? (Before the battery is connected in each case, the capacitors are fully discharged.)

Key Concepts

Capacitors in Series and Parallel

Understanding how capacitors combine when connected in series versus parallel is essential. In a series connection, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances, leading to a lower net capacitance. In a parallel connection, the equivalent capacitance is simply the sum of the individual capacitances, which maximizes the total capacitance available.

Equivalent Capacitance

Equivalent capacitance is the single capacitance that can replace a network of capacitors without changing the overall behavior of the circuit. For capacitors in series, the effective capacitance is determined by the formula 1/C_eq = 1/C1 + 1/C2, and for capacitors in parallel, it is C_eq = C1 + C2. This concept is crucial in analyzing circuits with different configurations.

Energy Stored in a Capacitor

The energy stored in a capacitor is given by the formula E = 1/2 * C * V^2, where C is the capacitance and V is the voltage across it. When a battery is connected, the voltage is fixed, so the energy storage directly depends on the effective capacitance. Comparing energy storage in different configurations hinges on this relationship.

Energy Ratio and Capacitance Ratio Relationship

To compare the stored energy when the capacitors are connected in parallel versus series, one sets up the ratio of energies using their effective capacitances. Since the energy is proportional to capacitance, the ratio E_parallel/E_series becomes (C1 + C2)^2/(C1 * C2) for identical applied voltage. Setting this ratio equal to a given number, one then derives an equation that relates the two capacitances, leading to the determination of their ratio.

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