2. The 'R+LC' circuit (25 pts) The 'R+LC' circuit shown in figure 1 is an RLC circuit with a resistor followed by a capacitor and inductor in a parallel configuration. Consider R, L, C arbitrary positive constants. R C L Figure 1: 'R+LC' circuit (a) Find the differential equation that describes the transient current in the in- ductor. Show all your work (i. e. all the Kirchhoff rules used in the derivation, any relationship among the variables used to get the final result, etc.) (12 pts) (b) What combination of R, L, C will generate i) overdamped, ii) critically damped and iii) underdamped behavior? (3 pts) Let's now consider the case when the transient behavior generates (non-zero) underdamped oscillations. (c) Find the time dependent expression of the current in the inductor. The initial conditions are: IL(t = 0) = 0 and VI(t = 0) = Vo. Recommendation: check units by using the units of RC, R/L, LC given in class. (5 pts) (d) An LC circuit (i.e. with R = 0) shows SHO at an angular frequency, called the resonant frequency, LRes = 1/VLC. Find the ratio OR+LC/res, where OR+LC is the angular frequency of the underdamped oscillations of the 'R+LC' circuit. Hint: the result must be dimensionless. Use the units of RC, R/L provided in class to check it out. (2 pts) (e) Suppose we are given some values for R, L, C. If we're able to change L, keeping R, C constant, should we increase or decrease L in order to get the underdamped oscillations closer to the LC resonant oscillations? What if we could change the value of C, keeping R, L constant? And what if we could change R, keeping L, C constant? (3 pts)