CP CALC Consider the circuit shown in Fig. P30.71. Switch S is closed at time t = 0,...

CP CALC Consider the circuit shown in Fig. P30.71. Switch S is closed at time t = 0, causing a current i₁ through the inductive branch and a current i₂ through the capacitive branch. The initial charge on the capacitor is zero, and the charge at time t is q₂. (a) Derive expressions for i₁, i₂, and q₂ as functions of time. Express your answers in terms of , L, C, R₁, R₂, and t. For the remainder of the problem let the circuit elements have the following values: = 48 V, L = 8.0 H, C = 20 μF, R₁ = 25 Ω, and R= = 5000 Ω. (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 t₁ (accurate to two significant figures) will the currents i₁ and i₂ be equal? (Hint: You might consider using series expansions for the exponentials.) (e) For the conditions given in part (d), determine i₁. (f) The total current through the battery is i = i₁ + i₂. At what time t₂ (accurate to two significant figures) will i equal one-half of its final value? (Hint: The numerical work is greatly simplified if one makes suitable approximations. A sketch of i₁ and i₂ versus t may help you decide what approximations are valid.)