Consider the circuit shown in . Switch is closed at time , causing a current through the inductive

Consider the circuit shown in . Switch is closed at time , causing a current through the inductive branch and a current through the capacitive branch. The initial charge on the capacitor is zero, and the charge at time t is . (a) Derive expressions for and as functions of time. Express your answers in terms of and . For the remainder of the problem let the circuit elements have the following values: and . (b) What is the initial current through the inductive branch? What is the initial current through the capacitive branch? (c) What are the currents through the inductive and capacitive branches a long time after the switch has been closed? How long is a "long time"? Explain. (d) At what time (accurate to two significant figures) will the currents and be equal? ( You might consider using series expansions for the exponentials.) (e) For the conditions given in part (d), determine . (f) The total current through the battery is . At what time (accurate to two significant figures) will i equal one-half of its final value? ( The numerical work is greatly simplified if one makes suitable approximations. A sketch of and versus t may help you decide what approximations are valid.)