Resistor-capacitor analogy and stored energy

The combination rules for resistors and capacitors seem inverted, but they follow a common logic if one looks at which quantity is shared in the connection. Within the same type of connection, the shared quantity is the same (voltage in parallel, charge/current in series); what changes is only whether the equivalent quantity adds directly or by reciprocals.

Connection	Shared quantity	Resistors	Capacitors
Parallel	same $\Delta V$	$\dfrac{1}{R_\text{par}} = \sum \dfrac{1}{R_k}$ (reciprocals)	$C_\text{par} = \sum C_k$ (direct sum)
Series	same current / charge	$R_\text{ser} = \sum R_k$ (direct sum)	$\dfrac{1}{C_\text{ser}} = \sum \dfrac{1}{C_k}$ (reciprocals)

The “simple” rule (direct sum) holds for $R$ in series and for $C$ in parallel. The “reciprocals” rule holds for $R$ in parallel and for $C$ in series. Remembered this way, the apparent inversion becomes a symmetry.

Stored energy

Every capacitor of capacitance $C$ charged to voltage $V$ stores energy in the electric field between the plates: $\ev{U = \tfrac{1}{2}\,C\,V^2 = \frac{Q^2}{2C}}$ Unlike the resistor, which dissipates, the capacitor conserves this energy and can give it back on discharging.

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Topics: Electric circuits Concepts: Capacitance and the capacitor

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