Charge sharing and discharge Joule heating between two capacitors

Problem Statement

A circuit is constructed using a DC power supply with electromotive force $V$V, resistors $\text{R}_1 and \text{R}_2$R1​andR2​ with resistances $R_1 and$R1​andR_2$respectively, capacitors$respectively,capacitors\text{C}_1 and \text{C}_2$with capacitances$withcapacitancesC_1 and $C_2$C2​ respectively, and a selector switch $\text{S}$S.

The selector switch $\text{S}$S has one common terminal and two selector terminals $\text{a} and$aand\text{b}$, and the common terminal can be connected to either terminal$,andthecommonterminalcanbeconnectedtoeitherterminal\text{a}$or terminal$orterminal\text{b}$. The specific connections of the circuit are as follows:

• One plate of capacitor $\text{C}_1$C1​ is connected to the common terminal of switch $\text{S}$S.

• Terminal $a$a of switch $\text{S}$S is connected to one terminal of resistor $\text{R}_1$R1​, and the other terminal of resistor $\text{R}_1$R1​ is connected to the positive terminal of the DC power supply.

• Terminal $b$b of switch $\text{S}$S is connected to one terminal of resistor $\text{R}_2$R2​, and the other terminal of resistor $\text{R}_2$R2​ is connected to one plate of capacitor $\text{C}_2$C2​.

• The negative terminal of the DC power supply, the other plate of capacitor $\text{C}_1$C1​, and the other plate of capacitor $\text{C}_2$C2​ are all connected to a common ground wire.

Initially, switch $\text{S}$S is not connected to anything, and capacitors $\text{C}_1$C1​ and $\text{C}_2$C2​ have no charge stored.

First, switch $\text{S}$S is connected to terminal $\text{a}$a, and a sufficiently long time elapses.

Then, switch $\text{S}$S is removed from terminal $\text{a}$a and switched to terminal $\text{b}$b. Find the algebraic expression, using $C_1$C1​, $C_2$C2​, and $V$V, for the total amount of Joule heat generated in resistor $\text{R}_2$R2​ during the time elapsed between switching to terminal $\text{b}$b and the start of another sufficiently long period of time.

Assume that the resistance of the wires and switches, as well as the self-inductance of the circuit, are all negligible.

Constraints

• $V = 50.0 \text{ V}$V=50.0 V
• $C_1 = 3.00 \text{ F}$C1​=3.00 F
• $C_2 = 2.00 \text{ F}$C2​=2.00 F
• $R_1 = 10.0 \text{ }\Omega$R1​=10.0 Ω
• $R_2 = 15.0 \text{ }\Omega$R2​=15.0 Ω

Input Format

Calculate the total amount of Joule heat $Q_2 \text{ [J]}$Q2​ [J] generated by resistor $\text{R}_2$R2​ within a sufficient amount of time after switching the switch to terminal $\text{b}$b, and input a natural number representing that value.