Lecture 7 Circuits Chp. 28PowerPoint PresentationDirect Current CircuitsExample (Single Loop Circuit)Slide 5Example with numbersExample with numbers (continued)Multiloop CircuitsMethod of determinants for solving simultaneous equationsAnother exampleRules for solving multiloop circuitsHow does a capacitor behave in a circuit with a resistor?Discharging a capacitor through a resistorSlide 14What is the current?How the charge on a capacitor varies with time as it is being chargedSlide 17OhmmeterAmmeterVoltmeterWarm up set 7Lecture 7 Circuits Chp. 28•Cartoon -Kirchoff’s Laws•Opening Demo- transmission lines•Physlet •Topics–Direct Current Circuits–Kirchoff’s Two Rules–Analysis of Circuits Examples–Ammeter and voltmeter–RC circuits•Demos–Ohms Law–Power loss in transmission lines–Resistivity of a pencil–Blowing a fuse• Warm-up problemsTransmission line demoDirect Current Circuits1. The sum of the potential charges around a closed loop is zero. This follows from energy conservation and the fact that the electric field is a conservative force.2. The sum of currents into any junction of a closed circuit must equal the sum of currents out of the junction. This follows from charge conservation.Example (Single Loop Circuit)No junction so we don’t need that rule.How do we apply Kirchoff’s rule?Must assume the direction of the current – assume clockwise.Choose a starting point and apply Ohm’s Law as you go around the circuit.a. Potential across resistors is negativeb. Sign of E for a battery depends on assumed current flowc. If you guessed wrong on the sign, your answer will be negativeStart in the upper left hand corner.213212111322210rrRRREEiirEiRirEiRiR++++−==−+−−−−−2132121rrRRREEi++++−=Put in numbers. VEVErrRRR5101102121321==Ω==Ω===Suppose:32511101010510=++++−=iampVEVE10521==Suppose:32532105 −=−=iampWe get a minus sign. It means our assumed direction of current must be reversed.Note that we could have simply added all resistors and get the Req. and added the EMFs to get the Eeq. And simply divided.325)(32)(5R .e.=Ω==VEiqeqampSign of EMFBattery 1 current flows from - to + in battery +E1Battery 2 current flows from + to - in battery -E2In 1 the electrical potential energy increasesIn 2 the electrical potential energy decreasesExample with numbersQuick solution:AEIRVVVVEqeqiiii1610R16102412.e.6131==Ω==+−=∑∑==Question: What is the current in the circuit?Write down Kirchoff’s loop equation.Loop equationAssume current flow is clockwise.Do the batteries first – Then the current.AampsVii625.0625.016100)311551()2412(==Ω==+++++−+−+Example with numbers (continued)Question: What are the terminal voltages of each battery?12V:2V:4V:€ V =ε− ir =12V − 0.625A • 1Ω =11.375VV =ε− ir = 2V − 0.625A • 1Ω =1.375VV =ε− ir = 4V + 0.625A • 1Ω = 4.625VMultiloop CircuitsKirchoff’s Rules1. in any loop2. at any junction0=∑iiV∑∑=outiniiFind i, i1, and i2Rule 1 – Apply to 2 loops (2 inner loops)a.b.Rule 2a.045203412121=+−−=−−iiii21iii +=We now have 3 equations with 3 unknowns.0245037120)(34122121211=−+−=−−=+−−iiiiiii0612150614242121=−+−=−−iiiiAiAiAii0.25.05.1263902639211=====−multiply by 2multiply by 3subtract themFind the Joule heating in each resistor P=i2R.Is the 5V battery being charged?Method of determinants for solving simultaneous equations524012043021121=−+−=+−−=−−iiiiiiiFor example solve for iAi 2265261282420482400431112450412110==+++−=−+−−−−−+−−−−=You try it for i1 and i2.See Appendix in your book on how to use Cramer’s Rule.Another exampleFind all the currents including directions.211211358023380234480iiiiiiiVVV−−=−−−=−−−++=012120010166024611212=+−=−+−=++−iiiii0)1(2426 =++− AiLoop 1 Loop 2Ai 11=AiAi212==multiply by 2i = i1+ i2Loop 1Loop 2i iiii1i2i2Rules for solving multiloop circuits1. Replace series resistors or batteries with their equivalent values.2. Choose a direction for i in each loop and label diagram.3. Write the junction rule equation for each junction.4. Apply the loop rule n times for n interior loops.5. Solve the equations for the unknowns. Use Cramer’s Rule if necessary.6. Check your results by evaluating potential differences.How does a capacitor behave in a circuit with a resistor?Charge capacitor with 9V battery with switch open, then remove battery.Now close the switch. What happens?Discharging a capacitor through a resistorV(t)Potential across capacitor = V = just before you throw switch at time t = 0.Potential across Resistor = iR at t > 0.RCQiRiCQoooo=⇒=CQoWhat is the current I at time t?€ i(t) =Q(t)RCTime constant = RCWhat is the current?€ Q = Q0e−tRC€ i =dQdt= −Q0RCe−tRC= −V0Re−tRCIgnore - signRCHow the charge on a capacitor varies with time as it is being chargedOhmmeterAmmeterVoltmeterWarm up set 7Warm up set 7 Due 8:00 am Tuesday 1. HRW6 28.TB.05. [119859] In the context of the loop and junctions rules for electrical circuits a junction is: where a wire is connected to a battery where three or more wires are joined where a wire is bent where a wire is connected to a resistor where only two wires are joined2. HRW6 28.TB.18. [119872] Two wires made of the same material have the same length but different diameter. They are connected in parallel to a battery. The quantity that is NOT the same for the wires is: the electric field the electron drift velocity the current the current density the end-to-end potential difference3. HRW6 28.TB.26. [119880] The emf of a battery is equal to its terminal potential difference: only when there is no current in the battery only when a large current is in the battery under all conditions under no conditions only when the battery is being