Kirchoff's laws / RC Circuits Examples
Example #1
Problem:
Find the currents through all the resistors in the circuit below:
DATA: Va = 12 V, Vb = 12 V, R1 = 10 W, R2 = 15 W, R3 = 20 W
Solution:
Summing the voltages around the left and right loops gives the following two equations:
One can then reduce the problem to "2 equations 2 unknowns" by substituting for i1 and obtaining 2 new equations.
One can add these two equations together to eliminate the the i3 term since all three resistors are the same, which will be now noted by R.
Once i2 is known, Eq. (4) can be used to get i3 and i1 can be found by summing i2 + i3.
i2 = 0.554 amps, i1= .369 amps, i3 = -.185 amps
Example #2
Problem:
Find the charges on all the capacitors in the circuit below:
DATA: Vb = 12 V, C1 = 10 mF, C2 = 15 mF, C3 = 20 mF
Solution:
Summing the voltages around the left and right loops gives the following two equations
where Q3 has been replaced by Q1 - Q2. Dividing Eq. (1) by C3, dividiing Eq. (2) by C1, then adding the equations yields:
which rearranged yields
Once Q2 is known, Eq. (1) can be used to get Q1, and Q3 can be found as the difference Q1 - Q2.
Q2 = 120.0 mC, Q1= 40.0 mC, Q3 = -80 mC
Example #3
The circuit below has been in position a for a long time. At time t = 0 the switch is thrown to position b. DATA: Vb = 12 V, C = 10 mF, R = 20 W
a.) What is the curnent through the resistor just BEFORE the switch is thrown?
I = 0
b.) What is the current through the resistor just AFTER the switch is thrown?
Solution: I = V/R
I = 0.6 amps
c.) What is the charge across the capacitor just BEFORE the switch is thrown?
Solution: Q = CV
Q = 120 mC
d.) What is the charge on the capacitor just AFTER the switch is thrown?
Solution: Charge does not change instantaneously.
Q = 120 mC
e.) What is the charge on the capacitor at at time t = 0.3 msec after the switch is thrown?
Solution: Q = Q0exp(-t/t) , where t = RC = 0.2 msec
Q = 26.8 mC
Example #4
Considering the same circuit, only with the switch thrown from b to a at time t = 0 after having been in position b for a long time. DATA: Vb = 12 V, C = 10 mF, R = 20 W
a.) What is the curnent through the resistor just BEFORE the switch is thrown?
I = 0
b.) What is the current through the resistor just AFTER the switch is thrown?
Solution: I = V/R
I = 0.6 amps
c.) What is the charge across the capacitor just BEFORE the switch is thrown?
Solution: Q = CV
Q = 0
d.) What is the charge on the capacitor just AFTER the switch is thrown?
Solution: Charge does not change instantaneously.
Q = 0
e.) What is the charge on the capacitor at at time t = 0.3 msec after the switch is thrown?
Solution: Q = Q0(1.0 - exp(-t/t)) , where t = RC = 0.2 msec
Q = 93.2 mC