Chapter 28 Direct Current Circuits 1 2 3 4 5 R connections in series and in parallel Define DC direct current AC alternating current Model of a battery Circuits with 2 batteries Kirchhoff s Rules RC circuit Confucius says reviewing helps one learn new knowledge Concepts Charge positive negative conserve induction potential Electric field flux Electrostatic equilibrium no moving charge Current moving charge Capacitance potential over charge Resistance potential over current Resistance and Resistivity conductivity and temperature Laws Coulomb s force and charge Gauss s electric flux and charge Ohm s electric potential and current Circuits and components Symbols Power Capacitor Resistor wire battery C Switch current time potential difference in parallel in series discuss today Resistor connections In series Condition DV2 DV1 I I1 I2 DV DV1 DV2 DV DV1 DV2 DV1 DV2 Req R1 R2 I I I1 I2 In parallel Condition I I1 I2 DV DV1 DV2 1 I I I I I 1 1 1 2 1 2 Req DV DV DV1 DV2 R1 R2 DV1 DV2 Resistor connections In series Q I I1 I2 voltage sharing power sharing DV1 R1 DV2 R2 P1 R1 P2 R2 In parallel Q DV DV1 DV2 current sharing I1 R2 I1R1 I2R2 or I2 R1 power sharing P1 R2 P1R1 P2R2 or P2 R1 DV1 DV2 Resistors connections summary In series Req R1 R2 R3 DV1 R1 DV2 R2 P1 R1 P2 R2 In parallel 1 1 1 1 Req R1 R2 R3 I1R1 I2R2 P1R1 P2R2 Resistors in Series Example Use the active figure to vary the battery voltage and the resistor values Observe the effect on the currents and voltages of the individual resistors PLAY ACTIVE FIGURE Resistors in Parallel Example Use the active figure to vary the battery voltage and the resistor values Observe the effect on the currents and voltages of the individual resistors PLAY ACTIVE FIGURE Combinations of Resistors The 8 0 and 4 0 resistors are in series and can be replaced with their equivalent 12 0 The 6 0 and 3 0 resistors are in parallel and can be replaced with their equivalent 2 0 These equivalent resistances are in series and can be replaced with their equivalent resistance 14 0 More examples Direct Current and Alternating Current When the current direction not magnitude in a circuit does not change with time the current is called direct current Most of the circuits analyzed will be assumed to be in steady state with constant magnitude and direction like the one powered through a battery When the current direction often also the magnitude in a circuit changes with time the current is call alternating current The current from your car s alternator is AC Model of a battery Two parameters electromotive force emf and the internal resistance r are used to model a battery When a battery is connected in a circuit the electric potential measured at its and terminals are called The terminal voltage V with V Ir If the internal resistance is zero the terminal voltage equals the emf The internal resistance r does not change with external load resistance R and this provides the way to measure the internal resistance DV battery DV load PLAY ACTIVE FIGURE Battery power figure The power the battery generates through chemical reactions p I R r I 2 The power the battery delivers to the load hence efficiency battery DV pload DV I R I2 efficiency pload R p R r load The maximum power the battery can deliver to a load R p I We have load R r 2 I and R r From pload R Where the emf is a constant once the battery is given 2 From dpload 1 2R 2 0 2 3 dR R r R r 2 We get R r to be the condition for maximum pload or power delivered to the load Battery power figure One can also obtain this result from the plot of R p load R r 2 2 battery DV Where when R r pload reaches the maximum value The efficiency of the battery at this point is 50 because efficiency pload R p R r load More complicated circuits circuits with 2 batteries Kirchhoff s Rules A typical circuit that goes beyond simplifications with the parallel and series formulas the current in the diagram Kirchhoff s rules can be used to solve problems like this Rule 1 Kirchhoff s Junction Rule Junction Rule from charge conservation The sum of the currents at any junction must equal zero Mathematically I 0 junction The I1 example on the left figure I2 I3 0 Rule 2 Kirchhoff s Loop Rule Choose your loop Loop Rule from energy conservation The sum of the potential differences across all elements around any closed circuit loop must be zero Mathematically DV 0 DV1 DV2 Loop direction Remember two things 1 A battery supplies power Potential rises from the One needs to pay attention the terminal to terminal sign or of these potential changes following the chosen loop 2 Current follows the direction of electric field hence the direction decrease of potential closed loop Kirchhoff s rules Strict steps in solving a problem Step 1 choose and mark the loop Step 2 choose and mark current directions Mark the potential change on resistors Step 3 apply junction rule I2 L1 I1 I1 I2 I3 0 Step 4 apply loop rule L1 2 00I3 12 0 4 00I2 0 L2 8 00 2 00I3 6 00I1 0 Step 5 solve the three equations for the three variables I3 L2 RC Circuits solve with Kirchhoff s rules When a circuit contains a resistor and a capacitor connected in series the circuit is called a RC circuit Current in RC circuit is DC but the current magnitude changes with time There are two cases charging b and discharging c Not a circuit charging Discharging Charging a Capacitor When the switch turns to position a current starts to flow and the capacitor starts to charge Kirchhoff s rule says Loop DVc DVR 0 Re write the equation in terms of the charge q in C and the current I and then only the variable q q q dq RI 0 and then R 0 C C dt Solve for q The current I is t t RC dq q t C 1 e I t e RC dt R Here RC has the unit of time t and is called the time constant Charging a Capacitor graphic presentation The charge on the capacitor varies with time q t C 1 e t RC Q 1 e t RC The current decrease with time t RC I t e R is the time constant RC Discharging a Capacitor When the switch turns to position b after the capacitor is fully charged to Q current starts to flow and the capacitor starts to discharge Kirchhoff s rule says Loop DVc DVR 0 Re write the equation in terms of the charge q in C and the current I and then only the variable q q q dq RI 0 and then R 0 C C dt Solve for q The current I is t t …
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