2. A circuit is constructed as shown in Fig. 1 below. Capacitor C is initially charged up to a voltage Vo and capacitor C2 is fully discharged. At time t

= 0 the switch, S,is closed, allowing a current, 1, to flow around the circuit and through the resistor (a) Apply Kirchoff's voltage law to the circuit to show that the charges Q1(t) andQ2(t) stored on capacitors C1 an C2 respectively are related by the equation: \frac{Q_{1}(t)}{C_{1}}-I(t) R-\frac{Q_{2}(t)}{C_{2}}=0 (b) Using Kirchoff's current law, show that the charges stored on each capacitor are also related by the equation: Q_{1}(t)=Q_{0}-Q_{2}(t) where Qo = C1/Vo is the initial charge stored on capacitor C, at time t = 0. (c) Use both results above to show that we can write a differential equation for the time-dependent charge on capacitor Q2(t) as, R \frac{d Q_{2}}{d t}+\frac{Q_{2}}{C_{s}}-\frac{Q_{0}}{C_{1}}=0 where 1/C, = 1/C, + 1/C2 is the capacitance of both capacitors in series. (d) Finally, show that the differential equation for Q2 given above can be solved using a solution of the form Q2(t)A (1 – e-Be). and in doing so find the value of the constants A and B.[7 marks]