Chapter 28Direct Current Circuits1. R connections in series and in parallel2. Define DC (direct current), AC (alternating current)3. Model of a battery4. Circuits with 2+ batteries – Kirchhoff’s Rules5. RC circuit子曰子曰子曰子曰:"温故而知新温故而知新温故而知新温故而知新"Confucius says, reviewing helps one learn new knowledge.Concepts:Charge: positive, negative, conserve, induction.potentialElectric: fieldfluxElectrostatic equilibrium: no moving charge.Current: moving charge Capacitance: potential over chargeResistance: potential over currentResistance 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: wire, battery, C, Switch…Power: current time potential difference:Capacitor: in parallel in seriesResistor: discuss todayResistor connectionsIn series. Condition: = =∆ = ∆ + ∆1 21 2I I IV V V∆ + ∆ ∆ ∆∆≡ = = + ≡ +1 2 1 21 21 2eqV V V VVR R RI I I I∆1V∆2VIn parallel.Condition:= +∆ = ∆ = ∆1 21 2I I IV V V+≡ = = + ≡ +∆ ∆ ∆ ∆1 2 1 21 2 1 21 1 1eqI I I IIR V V V V R R∆1V∆2VResistor connectionsIn series, :voltage sharingpower sharing∆=∆1 12 2V RV R=1 12 2P RP RIn parallel, : current sharingpower sharing= =∵1 2I I I∆1V∆2V∆ = ∆ = ∆∵1 2V V V= =1 21 1 2 22 1, or I RI R I RI R= =1 21 1 2 22 1, or P RPR P RP RResistors connections, summaryIn seriesIn parallel...= + + +1 2 3eqR R R R...= + + +1 2 31 1 1 1eqR R R R∆=∆1 12 2V RV R=1 12 2P RP R=1 1 2 2I R I R=1 1 2 2PR P RResistors in Series – Example Use the active figure to vary the battery voltage and the resistor valuesObserve the effect on the currents and voltages of the individual resistorsPLAYACTIVE FIGUREResistors in Parallel – Example Use the active figure to vary the battery voltage and the resistor valuesObserve the effect on the currents and voltages of the individual resistorsPLAYACTIVE FIGURECombinations ofResistorsThe 8.0-Ω and 4.0-Ωresistors are in series and can be replaced with their equivalent, 12.0 ΩThe 6.0-Ω and 3.0-Ωresistors are in parallel and can be replaced with their equivalent, 2.0 ΩThese equivalent resistances are in series and can be replaced with their equivalent resistance, 14.0 ΩMore examplesDirect Current and Alternating CurrentWhen the current direction (not magnitude) in a circuit does not change with time, the current is called direct currentMost 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 batteryTwo parameters, electromotive force (emf), ε, 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 = ε– IrIf the internal resistance is zero, the terminal voltage equals the emfε.The internal resistance, r, does not change with external load resistance R, and this provides the way to measure the internal resistance.∆Vbatteryload∆VPLAYACTIVE FIGUREBattery power figurebatteryload∆VThe power the battery generates (through chemical reactions):The power the battery delivers to the load, hence efficiency:The maximum power the battery can deliver to a load= ⋅ = + ⋅2( )pε I R r I= ∆ ⋅ = ⋅2loadp V I R I=+efficiency = loadpRp R r= ⋅2loadp R IFromand= + ⋅( )ε R r IWe have=+22( )loadRpεR rWhere the emf is a constant once the battery is given.ε = − = + + 22 31 20( ) ( )loaddpRεdR R r R rFrom We get to be the condition for maximum , or power delivered to the load.=R rloadpBattery power figurebatteryload∆VOne can also obtain this result from the plot of reaches the maximum value=+22( )loadRpεR rWhere whenThe efficiency of the battery at this point is 50% because =R rloadp=+efficiency = loadpRp R rMore complicated circuits, circuits with 2+ batteries: Kirchhoff’s RulesA 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: The example on the left figure:I1- I2 - I3 = 00junctionI=∑Rule 2:Kirchhoff’s Loop RuleChoose your loopLoop Rule, from energy conservation:The sum of the potential differences across all elements around any closed circuit loop must be zeroMathematically: One needs to pay attention the sign (+ or -) of these potential changes, following the chosen loop direction.closedloop0V∆ =∑∆1V∆2VLoop directionRemember two things:1. A battery supplies power. Potential rises from the “–”terminal to “+” terminal.2. Current follows the direction of electric field, hence the decrease of potential.Kirchhoff’s rulesStrict steps in solving a problemStep 1: choose and markthe loop.L1L2Step 2: choose and markcurrent directions. Mark the potential change on resistors.I1I3I2Step 3: apply junction rule:Step 4: apply loop rule:+ − =1 2 30I I I. . .. . .− + =− − − =3 23 1L1: +2 00 12 0 4 00 0L2: 8 00 2 00 6 00 0I II IStep 5: solve the three equations for the three variables.+–––++RC Circuits, solve with Kirchhoff’s rulesWhen 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 circuitchargingDis-chargingCharging a CapacitorWhen the switch turns to position a, current starts to flow and the capacitor starts to charge.Kirchhoff’s rule says:− ∆ − ∆ =0c Rε V VLoop+ –+–Re-write the equation in terms of the charge q in C and the current I, and then only the variable q:− − = − − =0 and then 0q q dqε RI ε RC C dtSolve for q:( )− = − 1tRCq t Cε eThe current I is( )−= =tRCdq εI t edt RHere RC has the unit of time t, and is called the time
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