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w! 30.71 ... CP CALC Consider the Figure P30.71 circuit shown in Fig. P30.71. Switch S E is closed at time t = 0, causing a cur- rent iſ through the inductive branch and a current in through the capacitive S branch. The initial charge on the capaci- RI L tor is zero, and the charge at time t is 92. mioon (a) Derive expressions for ij, i2, and 92 as functions of time. Express your an- swers in terms of E, L, C, R1, R2, and t. с For the remainder of the problem let the circuit elements have the following val- ues: E = 48 V, L = 8.0 H, C = 20 uF, R, = 25 N, and R2 = 5000 N. Ω (6) What is the initial current through the inductive branch? What is the initial current through the capacitive branch? (c) What are the cur- rents through the inductive and capacitive branches a long time after what time 1, (accurate to two significant figures) will the currents i the switch has been closed? How long is a long time"? Explain. (d) At and iz be equal? (Hint: You might consider using series expansions for e exponentials.) (e) For the conditions given in part (d), determine ij. (f) The total current through the battery is i = i + i. At what time 12 (accurate to two significant figures) will i equal one-half of its final value? (Hint: The numerical work is greatly simplified if one makes approximations. A sketch of i and in versus f may help you = the suitable decide what approximations are valid.)