Crkt 1 More Circuits In an electrical circuit circuit elements such as resistors and batteries can be connected together in series or in parallel Resistors in series are connected like links in a chain resistors in parallel are side by side like so in series R1 R2 Rtot R3 R1 R2 in parallel Rtot R3 Series R tot R1 R 2 R 3 Parallel R tot 1 1 1 1 R1 R2 R3 Rtot R1 R2 R3 Rtot R1 R2 R3 Resistors in series act like a single large resistor Resistors in parallel act like a single small resistor Proof Resistors in Series Itot I1 I2 Vtot V1 V2 R tot Vtot V1 V2 V1 V2 R1 R 2 I tot I tot I1 I2 Resistors in Parallel Itot I1 I2 Vtot V1 V2 I tot I I2 I1 I 1 1 1 1 2 Vtot Vtot V1 V2 R tot R1 R 2 Last update 9 28 2009 Dubson Phys1120 Notes University of Colorado Crkt 2 Examples of parallel resistors 100 1 Two 100 resistors in parallel 50 100 R tot 1 1 100 50 2 1 1 2 100 100 100 2 10 in parallel with 0 wire R tot 1 1 1 0 10 1 0 0 0 R2 Last update 9 28 2009 Dubson Phys1120 Notes University of Colorado Crkt 3 Key points The current is the same for resistors in series Current is not used up I I R1 R2 Think of the water pipe analogy two gravel plugs in series same flow same gal min through both plugs assuming no leaks or bubble in the pipe gravel plug gravel plug Adding another resistor in series always increases the total resistance The voltage difference across each resistor is the same for resistors in parallel Rbig bigger plug less flow Ismall Rsmall lo P hi P Ibig V same across both R s smaller plug more flow pressure same across both plugs Both resistors in parallel have the same V V I big R small Ismall R big Adding another R in parallel always decreases the total resistance Like adding another pipe along side the original pipe allows more flow smaller total resistance Last update 9 28 2009 Dubson Phys1120 Notes University of Colorado Crkt 4 Kirchhoff s two rules for analyzing circuits Kirchhoff is really spelled that way 2 h s 2 f s Kirchhoff s Current Rule also called the Junction Rule The total current into any junction total current out junction place where 3 or more wires meet I2 I1 I1 I1 I 2 I3 I3 junction junction This is also called Conservation of Current In steady state the charge is not building up anywhere it is just flowing along at a steady rate So the current into any portion of the circuit must equal the current coming out of that portion otherwise charge would be building up in that part of the circuit Kirchhoff s Voltage Rule also called the Loop Rule The sum of the voltage rises around any complete loop in a circuit sum of the voltage drops around the same loop Voltage rises and drops must sum to zero since we must return to the same voltage after one complete loop hi P R1 hi V med P med V V I R2 Pump lo V lo pressure V N rise IN R1 I R 2 N fall fall I R1 R 2 I V R1 R 2 Remember voltage is a kind of electrical pressure or electrical height If you go around a complete circuit and return to the same place you are back at the same pressure or height So rises must equal drops Last update 9 28 2009 Dubson Phys1120 Notes University of Colorado Crkt 5 Ammeters and Voltmeters I I A An ammeter measures the current through itself A voltmeter measures the voltage difference between its terminals V To measure the current through a resistor R must place the ammeter in series with R To measure the voltage across R must place voltmeter in parallel with R A I V R An ideal ammeter has zero internal resistance rinternal 0 so current I is not affected An ideal voltmeter has rinternal so no current flows through currents and voltages in rest of circuit are not affected i 0 rint ideal ammeter rint 0 Last update 9 28 2009 A V I ideal voltmeter rint Dubson Phys1120 Notes University of Colorado Crkt 6 Circuits with multiple loops and batteries Have a circuit with known V s R2 10 R1 10 and known R s Seek the I s I1 I3 V1 10 V R3 30 I2 V2 10 V Procedure I Guess direction of I through each R Draw I arrows label each I1 thru R1 etc Directions of currents not always obvious so just guess If you guess wrong value of current I will come out with a negative value II K s Current Law gives 1 or more equations eq n 1 I1 I 2 I3 3 unknowns I1 I2 I3 will need 3 eq ns to solve III K s Voltage Law gives an equation for each complete loop in the circuit 3 loops in this circuit Only need 2 more equations so only 2 of the 3 loop equations are needed Loop 1 V1 I1 R1 I3 R 3 eq n 2 Loop 2 V2 I 2 R 2 I3 R 3 eq n 3 V1 Loop 3 N rise I1 R1 drop 1 3 V2 0 N I2 R 2 rise 2 Moving CW around loop 3 drop Don t need the equation from loop 3 because already have 3 equations Remember In a resistor if we move in the direction of the current V drops V is negative hi V lo V if we move in the direction opposite current V rises V is positive In a battery if move from to terminal V rises lo V hi V if move from to terminal V drops Last update 9 28 2009 Dubson Phys1120 Notes University of Colorado Crkt 7 We now have 3 equations in 3 unknowns I1 I2 I3 1 I1 I 2 I3 2 V1 I1 R1 I3 R 3 3 V2 I 2 R 2 I3 R 3 The physics part of this problem is over now we have a messy algebra problem How do we solve Eqn 1 says we can substitute I1 I2 for I3 Eliminate I3 in equations 2 3 V1 I1 R1 I1 I 2 R 3 V2 I 2 R 2 I1 I 2 R 3 Rearrange A V1 I1 R1 R 3 I 2 R 3 B V2 I1 R 3 I 2 R 2 R 3 Now have 2 equations A B in 2 unknowns I1 I2 Now combine these to eliminate either I1 or I2 For instance can solve eqn A for I1 I1 V1 I 2 R 3 R1 R 3 eqn C V I 2 R 3 R 3 I2 R 2 R 3 Then plug this into eqn B V2 1 R1 R 3 Can solve this for I2 messy Then plug into C to get …
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