Physics 2102 Jonathan Dowling Physics 2102 Lecture 12 DC circuits RC circuits How to Solve Multi Loop Circuits Step I Simplify Compile Circuits Resistors Key formula V iR In series same current Req Rj In parallel same voltage 1 Req 1 Rj Capacitors Q CV same charge 1 Ceq 1 Cj same voltage Ceq Cj Step II Apply Loop Rule Around every loop add E if you cross a battery from minus to plus E if plus to minus and iR for each resistor Then sum to Zero E1 E2 iR1 iR2 0 R1 E1 R2 Conservation of ENERGY E2 Step II Apply Junction Rule At every junction sum the ingoing currents and outgoing currents and set them equal i1 i2 i3 i1 i2 i3 Conservation of CHARGE Step III Equations to Unknowns Continue Steps I III until you have as many equations as unknowns Given E1 E2 i R1 R2 E1 E2 i1R1 i2 R2 0 and i i 1 i2 Example Find the equivalent resistance between points a F and H and b F and G Hint For each pair of points imagine that a battery is connected across the pair Compile R s in Series Compile equivalent R s in Parallel Example Assume the batteries are ideal and have emf E1 8V E2 5V E3 4V and R1 140 R2 75 and R3 2 What is the current in each branch What is the power delivered by each battery Which point is at a higher potential a or b Apply loop rule three times and junction rule twice Example What s the current through resistor R1 What s the current through resistor R2 What s the current through each battery Apply loop rule three times and junction rule twice Non Ideal Batteries You have two ideal identical batteries and a resistor Do you connect the batteries in series or in parallel to get maximum current through R Does the answer change if you have non ideal but still identical batteries Apply loop and junction rules until you have current in R More Light Bulbs If all batteries are ideal and all batteries and light bulbs are identical in which arrangements will the light bulbs as bright as the one in circuit X Does the answer change if batteries are not ideal Calculate i and V across each bulb P iV brightness or Calculate each i with R s the same P i2R RC Circuits Charging a Capacitor In these circuits current will change for a while and then stay constant We want to solve for current as a function of time i t The charge on the capacitor will also be a function of time q t The voltage across the resistor and the capacitor also change with time To charge the capacitor close the switch on a E VR t VC t 0 E i t R q t C 0 E dq t dt R q t C 0 A differential equation for q t The solution is q t CE 1 e t RC And then i t dq dt E R e t RC i t E R Time constant RC RC Circuits Discharging a Capacitor Assume the switch has been closed on a for a long time the capacitor will be charged with Q CE Then close the switch on b charges find their way across the circuit establishing a current VR VC 0 i t R q t C 0 dq dt R q t C 0 Solution q t q0e t RC CEe t RC i t dq dt q0 RC e t RC E R e t RC C i t E R Example The three circuits below are connected to the same ideal battery with emf E All resistors have resistance R and all capacitors have capacitance C Which capacitor takes the longest in getting charged Which capacitor ends up with the largest charge What s the final current delivered by each battery What happens when we disconnect the battery Compile R s into into Req Then apply charging formula with ReqC Example In the figure E 1 kV C 10 F R1 R2 R3 1 M With C completely uncharged switch S is suddenly closed at t 0 What s the current through each resistor at t 0 What s the current through each resistor after a long time How long is a long time Compile R1 R2 and R3 into Req Then apply discharging formula with ReqC
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