DC Circuits and Kirchoff's Loop Rules

1 P10-Class 10: OutlineHour 1:DC CircuitsHour 2:Kirchhoff’s Loop Rules2 P10-Last Time:Capacitors & DielectricsP10-Capacitors & DielectricsTo calculate:1) Put on arbitrary ±Q 2) Calculate E3) Calculate ∆VQCV=∆EnergyCapacitance2221212VCVQCQU∆=∆==2332oEEudrdrε==∫∫∫∫∫∫0Sinsidefreeqdκε⋅=∫∫EA􀁇􀁇􀁷DielectricsFilled with Dielectric0CCκ⇒=P10-This Time:DC Circuits5 P10-Examples of Circuits6 P10-Current: Flow Of ChargeavQIt∆=∆Average current Iav: Charge ∆Q flowing across area A in time ∆tInstantaneous current: differential limit of IavdQIdt=Units of Current: Coulombs/second = Ampere7 P10-Direction of The CurrentDirection of current is direction of flow of pos. chargeor, opposite direction of flow of negative charge8 P10-Current Density JJ: current/unit areapoints in direction of currentIJˆAI≡􀁇IˆˆSSIdAd=⋅=⋅∫∫JnJA􀁇􀁇􀁇9 P10-Why Does Current Flow?If an electric field is set up in a conductor, charge will move (making a current in direction of E)Note that when current is flowing, the conductor is not an equipotential surface (and Einside≠0)!10 P10-Microscopic PictureDrift speed is velocity forced by applied electric field in the presence of collisions. It is typically 4x10-5m/sec, or 0.04 mm/second!To go one meter at this speed takes about 10 hours!How Can This Be?11 P10-Conductivity and ResistivityAbility of current to flow depends on density of charges & rate of scatteringTwo quantities summarize this:σ: conductivityρ: resistivity12 P10-Microscopic Ohm’s Laworρσ==EJJE􀁇􀁇􀁇􀁇σρ1≡ρ and σdepend only on the microscopic properties of the material, not on its shape13 P10-Demonstrations:Temperature Effects on ρ14 P10-PRS Questions:Resistance?15 P10-Why Does Current Flow?Instead of thinking of Electric Field, think of potential difference across the conductor 16 P10-Ohm’s LawWhat is relationship between ∆Vand current?bbaaVVVdE∆=−=−⋅=∫Es􀁇􀁇􀁁/EVJVIIRAIJAρρρ∆⎫==⎪⎪⎛⎞⇒∆=≡⎬⎜⎟⎝⎠⎪=⎪⎭􀁁􀁁/EVJIJAρρ∆===􀁁17 P10-Ohm’s LawIRV=∆RAρ=􀁁R has units of Ohms (Ω) = Volts/Amp18 P10-Examples of Circuits19 P10-Symbols for Circuit ElementsBatteryResistorCapacitorSwitch20 P10-Sign Conventions -BatteryMoving from the negative to positive terminal of a battery increasesyour potentialbaVVV∆=−Think:Ski Lift21 P10-Sign Conventions -ResistorMoving across a resistor in the direction of current decreasesyour potentialbaVVV∆=−Think:Ski Slope22 P10-Sign Conventions -CapacitorMoving across a capacitor from the negatively to positively charged plate increasesyour potentialbaVVV∆=−Think:Ski Lodge23 P10-Series vs. ParallelSeries Parallel24 P10-Resistors In SeriesThe same current I must flow through both resistors12VIRIR∆=+12()eqIRRIR=+=12eqRRR=+25 P10-Resistors In ParallelVoltage drop across the resistors must be the same121122eqVVVIRIRIR∆=∆=∆===12111eqRRR=+12III=+12eqVVVRRR∆∆∆=+=26 P10-PRS Questions:Light Bulbs27 P10-Kirchhoff’s Loop Rules28 P10-Kirchhoff’s Rules1.Sum of currents entering any junction in a circuit must equal sum of currents leaving that junction. 123III=+29 P10-Kirchhoff’s Rules2. Sum of potential differences across all elements around any closed circuit loop must be zero. ∫=⋅−=∆PathCloseddV0sE􀁇􀁇P10-Internal ResistanceReal batteries have an internal resistance,r, which is small but non-zerobaVVVIrε∆=−=−Terminal voltage:(Even if you short the leads you don’t get infinite current)31 P10-Steps of Solving Circuit Problem1. Straighten out circuit (make squares) 2. Simplify resistors in series/parallel3. Assign current loops (arbitrary)4. Write loop equations (1 per loop)5. Solve32 P10-Example: Simple CircuitYou can simplify resistors in series (but don’t need to)What is current through the bottom battery?33 P10-Example: Simple CircuitStart at ain both loopsWalk in direction of current()11220IRIIRε−−−−=()210IIRε−−+=120IRεε−−+=1IRε−→=Add these:()21IIRε−=21IIRε→=+20I=We wanted I2:34 P10-Group Problem: CircuitFind meters’ values. All resistors are R, batteries are εHARDEREASIER35 P10-Power36 P10-Electrical PowerPower is change in energy per unit timeSo power to move current through circuit elements:dPUdt=()ddqqVVdtdt=∆=∆VIP∆=37 P10-Power -BatteryMoving from the negative to positive terminal of a battery increasesyour potential. If current flows in that direction the battery suppliespowerIεIVIP=∆=supplied38 P10-Power -ResistorMoving across a resistor in the direction of current decreasesyour potential. Resistors always dissipatepowerRVRIVIP22dissipated∆==∆=39 P10-Power -CapacitorMoving across a capacitor from the positive to negative plate decreasesyour potential. If current flows in that direction the capacitor absorbspower (stores charge)dtdUCQdtdCQdtdQVIP===∆=22absorbed40 P10-Energy Balance0QIRCε−−=Multiplying by I:22212QdQdQIIRIRCdtdtCε⎛⎞=+=+⎜⎟⎝⎠(power delivered by battery) = (power dissipated through resistor)+ (power absorbed by the capacitor) 41 P10-PRS Questions:More Light Bulbs