Resistors in series and in parallel (review) R1R1 R2R2 V1V1 V2V2 V I R1R1 I1I1 R2R2 I2I2 II R1R1 R2R2 r R1R1 R2R2 Example r 1

10. Kirchhoff’s rules 1)Kirchhoff's first rule (Junction or current rule) Follows from the conservation of charges I1I1 I2I2 I3I3 Example: I 1 =I 2 +I 3 2

2) Kirchhoff’s second rule (Loop or voltage rule) Follows from the conservation of energy R1R1 R2R2 R3R3 I V 3) Using Kirchhoff’s rules For N junctions write N-1 equations Write equations only for independent loops 3

Example 1: Determine the currents through the elements of this circuit. R1 R1 ε2ε2 R2 R2 R3 R3 I1I1 I2I2 I3I3 ε1ε1 We have 3 different currents We need 3 equations We have 2 junctions (N=2)We can use N -1=1 equation We have 2 independent loopsWe can use 2 equations 4

Junction equation: (Both junction give the same equation) Loop equations: If we add two equations above, we obtain the equation for the third (big, combined) loop. We do not need this equation – it is dependent from two previous Let us use numbers: I 3 flows opposite to our assumption Example 1 (continued) An extra loop equation: We have three equations for three unknowns, we can solve this system of equations. 5

Example 2: R1R1 R2R2 I1I1 I2I2 I3I3 Example 3: In the circuit below, the switch is initially open. When the switch is closed, the current through the bottom resistor: R R 9V A. IncreasesB. DecreasesC. Stays the same 6

11. Microscopic picture L vdvd + For electrons in copper : drift speed ~ m/s random-motion speed ~ 10 6 m/s A n – number of electric carriers per unit volume A – area of cross-section AL - volume 7

Example: A copper wire 2.0 mm in diameter, caries 1.0 A. What is the drift speed of electrons? Assume one electron per atom is free to move. 8