The document discusses Kirchhoff's rules for circuit analysis and RC circuits. Kirchhoff's rules state that the sum of currents entering a junction must be 0, and the sum of potential differences in any closed loop must be 0. RC circuits have time-dependent behavior when charging and discharging based on the time constant, which is the resistance times the capacitance. The document provides examples of applying Kirchhoff's rules to solve for currents in circuits and calculating the time it takes a capacitor to charge to half its final value in an RC circuit.

1. DC Circuits The circuit Resistance in combinations Kirchhoff’s Rules RC transient circuits

2. Work done by a battery on charge Here

3. Real Battery and Single Loop circuits… What’s the current ? Conservation of energy: Kirchoff’s first Law: Sum of voltages in a closed loop is zero. Real Circuit with ammeter and voltmeter

4. Equivalent Resistance Resistors in Series Series requirements Conservation of energy Potential differences add Current is constant Apply Ohm’s Law to each resistor

5. Resistors in parallel Parallel requirements Charge conservation Currents must add Potential difference is same across each resistor Apply Ohm’s Law to each resistor

6. Example 1 What is current through battery? What is current through i 2 ?

7. Kirchhoff’s Rules The algebraic sum of the currents entering a junction is zero. ( Conservation of Charge ) The algebraic sum of the changes in electric potential difference around any closed circuit loop is zero. ( Conservation of Energy ) Signs for Rule 2 The direction of travel when traversing the loop is from a to b.

8. Problem 2 Find the currents in each of the three legs of the circuit, Three unknowns, need three equations. Also since batteries are in there cannot reduce the resistances since none in parallel or series

9. Example: Applying Kirchhoff’s Rules Apply Kirchhoff’s first rule to the three wire junction at the bottom of the diagram Apply Kirchhoff’s second rule to the closed path in red, traversing it clockwise Apply Kirchhoff’s second rule to the closed path in green, traversing it clockwise Note the sign changes for some of the elements

10. Another, example: applying Kirchhoff’s Rules Solve the equations simultaneously for the values if I. If I is negative the current is in the opposite direction

11. RC Circuits and Time dependence Time dependence Recall Lab 7! Resistor slows down the charging of the capacitor Time dependent behavior (transient) 2 cases: switch at “ a” or at “b” Charging discharging

12. In position “a” Charging the Capacitor Use Kirchhoff’s Loop rule Or the voltage across capacitor is …..

13. What’s V R across resistor? Find the current and multiply by R

14. Discharging Position b:

15. Example: RC circuit How long does it take the capacitor to reach ½ its final charge, if the capacitor is uncharged at t = 0? R=10  C=1F

16. 1. Charging the Capacitor Note: is called the time constant What are the units?