DC Circuits Series and parallel rules for resistors Kirchhoff’s circuit rules

“DC” Circuits “Direct Current or DC”: current always flows in one direction. For circuits containing only resistors and emf’s the current is always constant in time. Circuits containing other elements such as capacitors and inductors as well as resistors will have currents that change with time. Alternating current or AC is current that reverses direction many times (eg: 60 Hz current in Canada) and will not be treated in this course

Resistors in Series R1R1 R2R2 R3R3 I A B We want to replace this combination by a single resistor with resistance R eff AB R eff I

Same current through all resistors Voltages add: V eff = V 1 + V 2 + V 3 + … IR eff = IR 1 + IR 2 + IR 3 + … (same current through all) So, Easy way to remember: think of the length of a string: L tot = L 1 + L 2 + L 3 where L 1 etc are the segments

Resistors in Parallel R1R1 R2R2 R3R3 I I1I1 I2I2 I3I3 I We want to replace these resistors by a single resistance R eff : A A B B R eff I

Same voltage across each resistor Currents add: I eff = I 1 + I 2 + I 3 +

Example 1 11 V R1R1 R2R2 R3R3 R4R4 R5R5 R6R6 Find: Effective resistance across the battery All resistors = 1 Ω

Quiz Find the effective resistance of a network of identical resistors: A) 11/6 R B) 6 R C) 6/11 R D) 1/6 R RR RR R R

Example 2 Find a) the current in each resistor b) the power dissipated by each resistor c) the equivalent resistance of the three resistors 3 Ω6Ω9Ω 18 V a b

Example 3 A regular “40 watt” bulb and a “60 watt” bulb are connected in SERIES across 120 V. What power does each bulb give? (Assume that the resistances don’t change with temperature—these are special bulbs.)

Kirchhoff’s Circuit Rules Junction Rule: total current in = total current out at each junction (from conservation of charge). Loop Rule: Sum of potential differences around any closed loop is zero (from conservation of energy). Charge q moves through circuit changing its potential energy qV but eventually there is no overall change.

Junction Rule: conservation of charge. I1I1 I2I2 I 1 -I 2 I 1 = I 2 + I 3 Sum of currents entering a junction equals the sum of currents leaving the junction Alternately: I1I1 I2I2 I3I3 I 1 = I 2 + (I 1 -I 2 )

R I loop going from left to right ΔV = -IR -Q+Q C -+ ΔV = ΔV = Q/C Loop Rule: conservation of energy. Follow a test charge q around a loop:  around any loop in circuit. loop

Example 5 A dead battery is charged by connecting it to the live battery of another car with jumper cables. Determine the current in the starter and in the dead battery.

Solution

Example In the circuit shown, currents of 2.50 A and 1.60 A flow in the branches indicated in the diagram. Apply Kirchhoff’s circuit rules and calculate the potential difference Ve  Vb between points b and e in the diagram.  R 3.0  2.5 A 4.0  2.0  6.0  1.6 A a b c d e f 2.5 A I2I2 I1I1 I3I3