Two capacitors are connected to a battery. The battery...

Two capacitors are connected to a battery. The battery voltage is $V=60.0 \mathrm{V}$ , and the capacitances are $C_{1}=2.00 \mu \mathrm{F}$ and $C_{2}=4.00 \mu \mathrm{F}$ . Determine the total energy stored by the two capacitors when they are wired $(\mathrm{a})$ in parallel and $(\mathrm{b})$ in series.

Key Concepts

Capacitors and Energy Storage

Capacitors are devices that store energy electrostatically in an electric field established between two conductors. The energy stored in a capacitor can be quantified by the formula U = ½CV², where U is the energy, C is the capacitance, and V is the voltage applied. This fundamental concept is essential when analyzing how circuits store and manage electrical energy.

Equivalent Capacitance

When multiple capacitors are connected in a circuit, their behavior can be described in terms of an equivalent capacitance. In parallel connections, the equivalent capacitance is the sum of the individual capacitances, whereas in series connections, the reciprocal of the equivalent capacitance is the sum of the reciprocals of the individual capacitances. Understanding how to compute the equivalent capacitance is critical for analyzing voltage distributions and energy storage in complex circuits.

Circuit Configurations - Series versus Parallel

The configuration of capacitors in a circuit—whether in series or parallel—affects how the voltage is distributed across each capacitor and thus influences the total energy stored. In a parallel circuit, all capacitors share the same voltage, resulting in cumulative capacitances, while in a series circuit, the same charge flows through each capacitor with the voltage divided among them. This distinction is vital for determining overall circuit parameters in energy storage applications.

Voltage Sources in Electrical Circuits

A voltage source, such as a battery, provides a constant electromotive force that drives the circuit. The supplied voltage is directly linked to how much energy can be stored in capacitors, as it appears squared in the energy storage formula. Understanding the role of a voltage source is key to analyzing circuits and designing systems that effectively store and utilize electrical energy.

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