Question

An electric circuit consists of three charged capacitors. The capacitances of the capacitors and the values of their charges are given in the diagram. At some moment, the plates of capacitor X are released, allowing them to move freely toward each other. As a result of inelastic collisions, the plates eventually touch. What amount of heat is released in the system as a result?

• A. $\frac{Q^2}{2C}$
• B. $\frac{2}{3} \frac{Q^2}{C}$
• C. $\frac{Q^2}{C}$
• D. $\frac{2}{5} \frac{Q^2}{C}$

Answer 1

Answer: (B)

$\frac{2}{3}\frac{Q^2}{C}$

Answer 2

Calculate the initial total energy of all three capacitors. When capacitor X’s plates touch, they short-circuit that capacitor and force the remaining capacitors (with $C$ and $C/2$) to redistribute charge in parallel. Using charge conservation on the isolated nodes yields the final common voltage, from which the final energy is computed. The difference $U_i - U_f$ gives the heat released: $\frac{2Q^2}{3C}$.