Two charged capacitors connected together

Problem

Synthesis: two charged capacitors connected together. Two isolated capacitors: $C_1 = 10\;\mu\text{F}$ charged to $V_1 = 200\;\text{V}$ and $C_2 = 30\;\mu\text{F}$ charged to $V_2 = 100\;\text{V}$, with the same polarity. They are connected in parallel (positive plate to positive plate). (a) Total charge before connection; (b) common voltage after equilibrium; (c) energy before and after. Explain why energy is not conserved: where does it go?

Solution

(a) Total charge. The initial charges add up (same polarity): $Q_1 = C_1 V_1 = 2\;\text{mC}, \qquad Q_2 = C_2 V_2 = 3\;\text{mC},$ $\ev{Q_\text{tot} = Q_1 + Q_2 = 5\;\text{mC}}.$ The total charge is conserved: it can neither disappear nor be created by the connection.

(b) Common voltage. In parallel the two capacitors share the same voltage; the equivalent capacitance is $C_\text{eq} = C_1 + C_2 = 40\;\mu\text{F}$, so $V_\text{eq} = \frac{Q_\text{tot}}{C_\text{eq}} = \frac{5\cdot 10^{-3}}{40\cdot 10^{-6}} = \ev{V_\text{eq} = 125\;\text{V}}.$

(c) Energy before and after. $U_\text{pre} = \tfrac{1}{2}C_1 V_1^2 + \tfrac{1}{2}C_2 V_2^2 = 0{,}20 + 0{,}15 = \ev{U_\text{pre} = 0{,}35\;\text{J}},$ $U_\text{post} = \tfrac{1}{2}C_\text{eq} V_\text{eq}^2 = \tfrac{1}{2}\cdot 40\cdot 10^{-6}\cdot 125^2 \approx \ev{U_\text{post} \approx 0{,}31\;\text{J}}.$

Why energy is not conserved. About $0{,}04$ J ($\sim 11\%$) is missing. Charge is always conserved, but energy is not: during the connection transient the two voltages differ, so a current flows through the connecting wires (which have some resistance, however small). This current dissipates energy through the Joule effect in the wires, and partly radiates it away as electromagnetic waves. This is a general result: connecting two capacitors at different voltages always entails an energy loss, no matter how small the resistance.

Links

Topics: Campo elettrico e potenziale Concepts: Capacità e condensatore · Densità di energia del campo Skills: Conservazione dell’energia Methods: Semplificazione serie-parallelo Objects: Condensatore piano