Question

Consider the following circuit containing a switch S, an ideal battery with e.m.f. ε-20.0 V, resistors R,-: 8.00 Ω, R2-20.0 Ω and R3-30.0 Ω, and capacitors R2-20Ω RI-8Ω R3 -3082 After the switch is closed: (a) Calculate the initial current through resistor Ri. (b) Calculate the power dissipated by resistor R2 just after the switch is closed. (c) Calculate the combined capacitance of Ci and C2 (d) Determine the voltage drop over Ci when it is fully charged and the current has stopped flowing. D2. An inductor L 100 mH and a resistor R 20.02 are connected in series across a i(t)-011-exp(9)] After a long time the current in the circuit is 1.00 mA. () State the general formula for the (ii) Derive a function for the voltage drop over the inductor in the circuit described above. (i) Find the voltage over the inductor immediately after the connection to the battery has been made. (iv) Use Kirchhoff's Voltage Law to derive the differential equation governing the current in this circuit and show that i(0) satisfies this equation (b) Calculate the time it takes for the current to reach half its final value.

I need a full solution for those questions please, although I'm half way through question one and it is quite simple but I would like to double check if Ive got that right. like and comment are rewarded for good answers

Answer 1

R_2 & R_3 are in parallel so, equivalent resistance of R_2 & R_3 is, 1/R_23 = R_2 + R_3/R_2 middot R_3 = 20 + 30/20 times 30 Ohm = 50/600 Ohm R_23 = 12 Ohm The capacitor C_1 & C_2 are in series The equivalent capacitance C_12 = C_1 middot C_2/C_1 + C_2 = 40 times 60/40 + 60 mu F = 2400/100 mu F = 24 muF \r\n The combined system has a circuit diagram It is an CR circuit with total resistance R = R_1 + R_23 = (12 + 8)Ohm = 20 Ohm The differential eqn for the circuit q/c + IR = V q/c + R dq/dt = V dq/dt + 1/Rc q = V/R integrating factor e^integral dt/Rc = e^t/RC d(q middot e^t/RC) = V/R e^t/Rc middot dt q middot e^t/Rc = V/R middot e^t/Rc/1/Rc + (const) 0 = V/R middot Rc + (const) (const) = -vc Boundary condition at t = 0