2. Each of the resistors in the diagram is 12 . The resistance of the entire circuit is: A)120  B) 25  C) 48  D) 5.76 

3. RC circuits: Prior to Steady-State +– E C R S1S1 S2S2 Thus far we have been referring to circuits in which the current does not vary in time, i.e., steady-state circuits When we mix capacitors and resistors, the currents can vary with time? Why?! We need to charge the capacitor! A capacitor which is being charged conducts like a wire After charging, the capacitor acts like a broken wire

4. RC circuits: Prior to Steady-State +– E R S1S1 C Recall: the voltage across a capacitor is: V=q/C When the capacitor is fully charged the voltage is  ( e.g. it acts like a broken wire) Prior, the voltage is V, i.e. there is a voltage drop. Apply the loop rule: Close S 1 The result is a differential equation.

5. RC circuits: differential Eqns Differential equation. General Solution: q b and K are determined from boundary conditions and  from the parameters of the differential equation Plausibility argument:

6. RC circuits: Differential Eqns Integrate both sides to solve: K is determined from boundary conditions Plausibility argument: More general equation and solution:

7. RC circuits: Boundary Conditions At t=0, q=0 Charging: As t goes to infinity, q=  C Combining these together and: As an exercise do the same for discharging

8. RC circuits Capacitor/resistor systems charge or discharge over time Charging:  is the time constant, and equals RC. Discharging: Qualitatively: RC controls how long it takes to charge/discharge completely. This depends on how much current can flow (R) and how much charge needs to be stored (C) [As an exercise, show that RC has units of secs]

9. RC circuits: Discharging +– E C R S1S1 S2S2 Circuit with battery, resistor, and capacitor Switch S 1 is closed, then opened At t = 0, switch S 2 is closed What happens? Battery increases voltage on capacitor to  V = E At t=0. Current begins to flow Charge Q = C  V is stored on capacitor –+ What is the current? [exercise for the class]

10. Time Constants Time constants are common in science! Given a time constant, t, how long does one have to wait for something to decay by: .105  .288  .693   2.30   4.60   9.21 

11. Four circuits have the form shown in the diagram. The capacitor is initially uncharged and the switch S is open. The values of the emf, resistance R, and the capacitance C for each of the circuits are circuit 1: 18 V, R = 3, C = 1 µF circuit 2: 18 V, R = 6, C = 9 µF circuit 3: 12 V, R = 1, C = 7 µF circuit 4: 10 V, R = 5, C = 7 µF Which circuit has the largest current right after the switch is closed? Which circuit takes the longest time to charge the capacitor to ½ its final charge? Which circuit takes the least amount of time to charge the capacitor to ½ its final charge?