1996E2. Capacitors 1 and 2, of capacitance C1=4μF and C2=12μF, respectively, are connected in a circuit as shown above with a resistor of resistance R=100Ω and two switches. Capacitor 1 is initially charged to a voltage Vo=50V, and capacitor 2 is initially uncharged. Both of the switches S are then closed at time t=0.
a. What are the final charges on the positive plate of each of the capacitors 1 and 2 after equilibrium has been reached?
b. Determine the difference between the initial and the final stored energy of the system after equilibrium has been reached.
c. Write, but do not solve, an equation that, at any time after the switches are closed, relates the charge on capacitor C1, its time derivative (which is the instantaneous current in the circuit), and the parameters Vo,R,C1, and C2.
The current in the resistor is given as a function of time by I=I0e−t/τ, where I0=0.5A and τ=3×10−4s.
d. Determine the rate of energy dissipation in the resistor as an explicit function of time.
e. How much energy is dissipated in the resistor from the instant the switch is closed to when equilibrium is reached?