🔗 Two capacitor paradox

🔗 Electronics

The two capacitor paradox or capacitor paradox is a paradox, or counterintuitive thought experiment, in electric circuit theory. The thought experiment is usually described as follows: Two identical capacitors are connected in parallel with an open switch between them. One of the capacitors is charged with a voltage of ${\displaystyle V_{i}}$, the other is uncharged. When the switch is closed, some of the charge ${\displaystyle Q=CV_{i}}$ on the first capacitor flows into the second, reducing the voltage on the first and increasing the voltage on the second. When a steady state is reached and the current goes to zero, the voltage on the two capacitors must be equal since they are connected together. Since they both have the same capacitance ${\displaystyle C}$ the charge will be divided equally between the capacitors so each capacitor will have a charge of ${\displaystyle {Q \over 2}}$ and a voltage of ${\displaystyle V_{f}={Q \over 2C}={V_{i} \over 2}}$. At the beginning of the experiment the total initial energy ${\displaystyle W_{i}}$ in the circuit is the energy stored in the charged capacitor:

${\displaystyle W_{i}={1 \over 2}CV_{i}^{2}}$.

At the end of the experiment the final energy ${\displaystyle W_{f}}$ is equal to the sum of the energy in the two capacitors

${\displaystyle W_{f}={1 \over 2}CV_{f}^{2}+{1 \over 2}CV_{f}^{2}=CV_{f}^{2}=C({V_{i} \over 2})^{2}={1 \over 4}CV_{i}^{2}={1 \over 2}W_{i}}$

Thus the final energy ${\displaystyle W_{f}}$ is equal to half of the initial energy ${\displaystyle W_{i}}$. Where did the other half of the initial energy go?