Homework help please! Consider the following circuit: Its components all have th

Question

Homework help please! Consider the following circuit: Its components all have the values: = 12V R = 22k Ohm C = 1000 mu F L = 300 mu F Assume that when you start all switches are open. Close switches S_1, S_2, and S_4. (a) What is the current through the circuit? Now, open switch S_2 and close S_3. (b) What is the maximum voltage V_0 across the capacitor C? (c) How long does it take to charge the capacitor to 1/2 V_0? What is the energy stored in the capacitor at this time? Open S_3 and S_4 when the voltage across the capacitor is 1/2 V_0. Close S_2 and connect S_5 to the left pole (i.e. the one without the resistor). (d) What is the maximum current I_0 through the inductor L? (e) How long does it take for the current through the inductor to reach 1/2 I_0. What is the energy stored in the inductor at this time? Open S_1 and close S_3 when the current through the inductor is 1/2 I_0. Energy will begin to flow back and forth between the inductor and capacitor in this LC circuit Like energy in a harmonic oscillator. (f) What is the total electrical energy in the system? (g) Into which component will the energy move first? (h) LC circuits like this are used as timers because their frequency of oscillation is very precise. The frequency is given by omega_0 = (LC)^-1. What is the oscillation frequency for the circuit?

Explanation / Answer

4. given, V = 12 V
R = 22 k ohm
C = 1000 micro F
L = 300 micro H

now,
a. when switches S1, S2 and S4 are switched on
Current through the circuit = i
from ohms law
V = iR
i = V/R = 12/22,000 = 5.4545*10^-4 A
b. NOW s2 IS OPENED AND s3 is closed
so Maximum voltage sacross the c apacitor is when current in the circuit is 0
Vo = V = 12 V
c. so capacitor charging equation is given by
q = Qo( 1- e^(-t/RC))
so the capacitor has to be half charged
0.5 = 1 - e^(-t/RC)
e^(-t/RC) = 0.5
e^(t/RC) = 2
t/RC = ln(2)
t = RCln(2) = 22,000*ln(2)/1000 = 22*ln(2) = 15.249 s
d. at the instant when Vo is Vo/2
open S3 and S4
close S2 and connect S5 to the terminal with no resistor
so maximum current through the inductor Io
now, at t = 0, inductor stops the current which is trying to change its value with time
so, with time the inductor provides little or no resistance to flow of current and hence Io = V/R

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