V =! " Ir. Resistors in series! Ch 28-DC Circuits! EMF & Terminal Voltage!

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1 Ch 28-DC Circuits! Resistors in series! One of the bits of nastiness about DC circuits is that they can be disguised to look like something they are not. Look at the circuit on the left. Its general form looks like a parallel combination. Think about the characteristics of a parallel combination, though, (each element connected to its neighbor on either side of the element) and you see it clearly isn t (in fact, each element is connected to each neighbor at only one place, and there are no nodes junctions between any of the elements these are the signs of a series combination which is is). The moral of the story is this: if a circuit looks complicated, you many be able to redraw it (see circuit to right) in a way that makes it not so obscure..) 3.) EMF & Terminal Voltage! 9. V 8.7 V 8.7 V Resistors in series! What is the current flow in this circuit?! What is the voltage drop across each resistor?!! 3! 6! (Look at series movie in Richard s stuff ) V =! " Ir volts Terminal Open circuit internal! voltage voltage (emf) resistance" 2.) 4.)

2 Resistors in series! What is the current flow in this model?! The euivalent resistance of a group of resistors is the single resistor that will do the same thing in the circuit as does the combination. In the case of a series combination of resistors, it s the single resistor that, when put across the circuit s battery, will draw the same current as the combination. We know that:! 3! 6! volts Resistors in parallel! R euivalent = R + R 2 + R (for resistors in series) so we can write: 5.) 7.) Resistors in series! What is the current flow in this model?! The euivalent resistance of a group of resistors is the single resistor that will do the same thing in the circuit as does the combination. In the case of a series combination of resistors, it s the single resistor that, when put across the circuit s battery, will draw the same current as the combination. We know that: R euivalent = R + R 2 + R (for resistors in series)! 3! 6! volts Resistors in parallel! What is the voltage drop over each resistor in this model?! What is the current through each resistor?!! 8! 6! (Look at parallel movie in Richard s stuff ) so we can write: What is the voltage drop across each resistor?! 6.) 24 volts R euivalent = R + R 2 + R (for resistors in parallel) 8.)

3 Example! 3! 6! 9! 8V a. What is the euivalent resistance?" R euiv =3! +6! +9! =8!! b. Find the current flowing throughout the circuit." I=V/R=8V/8! =.A! c. What is the potential at every point in the circuit." V 3 =IR=(A)(3!)=3V; V 6 =6V; V 9 =9V! d. What is the Power delivered by the battery?" P=IV=(.A)(8V)=8W! e. What is the Power consumed by each resistor?" P 3 =IV=(.A)(3V)=3W; or use P 3 =I 2 R=(A)(3!)=3W; or use P 3 =V 2 /R=(3V) 2 /3A=3W; P 6 =6W; P 9 =9W! 9.) Example 3! 6! 2! 36V 8! a. What is the euivalent resistance?" b. Find the current flowing throughout the circuit." c. What is the potential at every point in the circuit." d. What is the Power delivered by the battery?" e. What is the Power consumed by each resistor?".) Example 2! 24V 6! 4! 2! a. What is the euivalent resistance?" /R euiv =/4! +/6! +/2!; R euiv =2!! b. Find the current flowing throughout the circuit." V drop identical for each resistor=24v! c. What is the potential at every point in the circuit." I from battery = V/R euiv = 24V/2! =2A; I 4 =V/R= (24V)/(4!)=6A; I 6 =4A; I 2 =2A! d. What is the Power delivered by the battery?" Total power P = (I) V = (2 amps)(24 volts) = 288 watts.! e. What is the Power consumed by each resistor?" P = (I)^2 R yields power values of 48 watts for the 2 ohm resistor, 96 watts for the 6 ohm resistor and 44 watts for the 6 ohm resistor for a total of (tada) 288 watts.!.) Kirchhoff s Rules! 2.)

4 Kirchhoff s Rule! The Junction Rule - Total current into a node = total current out of a node! I I 2 I 3 I I 2 I 3 I I 2 I 3 Using Kirchhoff s!. On circuit diagram, identify and label a current and direction of flow for each separate branch of the circuit.! 2. Write out a series of Node euations using Kirchhoff s Junction Rule.(Make sure you don t duplicate any euations!)! 3. Write out a series of Loop euations using Kirchhoff s Loop Rule. (The total number of Junction and Loop euations = the number of unknown currents you re solving for.)! 4. Solve these simultaneous euations.! 3.) 5.) Kirchhoff s Rule 2! The Loop Rule - The sum of the potential changes around a closed loop must eual zero.! Example 4! 8 Ω 2 Ω 5 V 5 Ω a. What is the current flow, everywhere?" b. What are the voltages, everywhere?" c. What is the Power consumed/ produced, everywhere?" 5 Ω V 4.) 6.)

5 Example 4! 2Ω I + I3 = I2!5I3! 8I2! 5! 5I3 = 5 + 8I2 + 2I = I 5V I2 8Ω 5 Ω Apply Kirchoff s rules to circuit with currents (arbitrarily chosen here) to get:" 5Ω V I3 Six Flags/Magic Mountain! When: Monday, April 5, 2! Who: All able-bodied senior physics students! How much: Don't worry about it. (Book bill)! Leaving: In AM. Come to school and meet in 23.! Returning: That depends. Let's talk about it.! Assignment: One "applied physics" problem, assigned at beginning of April! Write-up: A "nice" one, word processed, 3-7 pages, with diagrams, illustrations, data tables, graphs, calculations, blurbs, explanations, sources of error, as reuired.! 7.) Example 4! I! I2 + I3 = I! 8I2! 2I3 =!5 2I + 8I2 + I3 =!5 2!! 8! ) Six Flags/Magic Mountain!!5!5 a. Math -> Matrix -> Edit -> A (for name of matrix)" b. 3 [Enter] 4 [Enter]" c. Enter coefficients and values into Matrix" d. Do rref A (reduced row echelon form)" e. Interpret answers from matrix" "! % $ ' $! 8! 2! 5 ' $#2 8! 5 '& " $ $ $#! % '! ' '& 8.) 2.)

6 Six Flags/Magic Mountain! Series & Parallel Activity! Examine the series and parallel circuits assembled in class... and don t touch the wires.! 2.) Six Flags/Magic Mountain! 23.) RC Circuits - Charge! d Vo =! dt R RC d " VoC % VoC! =$! '= dt # RC RC & RC d! = dt! VoC RC Vo! Vcapacitor! Vresistor = 22.) Vo! plate! Iinst R = C d Vo! plate! R = C dt d Vo =! dt R RC ( d =! VoC t! ( RC dt!t RC! VoC!t ln( )=!VoC RC ln(! VoC ) = (t) = CVo [! e!t / RC ] (t) = Q[! e!t / RC ] 24.)

7 RC Circuits - Charging! The uantity! = RC is called the time constant for the circuit, and (in this circuit) represents the amount of time (in seconds) for the charge to reach.632 of its total value." (t) = Q(! e!t / RC ) RC Circuits - Current! (t) = Q[! e!t / RC ] I = d dt I = d dt (Q[! e!t / RC ]) I = Q( d dt! e!t / RC ) I = Q(! RC )(!e!t / RC ) I(t) = V o R e!t / RC!t / RC I(t) = I o e 25.) 27.) RC Circuits - Discharging! What is the function for a discharging capacitor? What does! = RC represent for a discharging capacitor?" RC Circuits - Current! What is the function for current of a discharging capacitor?" I(t) =! Q RC e!t / RC Q o (t) = Q(e!t / RC ) t 26.) 28.)

8 Example 5! An RC circuit has a 6V battery, a 2µF capacitor, and a 5! resistor." a. Draw a picture of the circuit.! b. What is the flow of current through the circuit I o just after the switch is thrown?! c. What is the time constant for the circuit?! d. How much current is flowing in the circuit 5 seconds after the switch has been thrown?! e. Sketch a graph of the circuit s current vs. time, using at least 3 data points that you calculate." Ammeters! The galvanometer all by itself has some resistance (say Ω), and deflects fully across the face of the meter at some current value (say.a). How can we alter this galvanometer so that it will be able to measure currents (deflect fully) up to.a?! COM V! V full deflection = IR = (.A)(!) =.V R needed = V =.V I new.a =.! + = R shunt. R shunt =.! 29.) 3.) Galvanometers! A galvanometer is a type of meter that, in conjunction with appropriate resistors, will allow one to measure current (as an ammeter) and electric potential difference (as a voltmeter).! COM V! COM V! Ideal voltmeters have high resistance." Voltmeters! The galvanometer all by itself has some resistance (say Ω), and deflects fully across the face of the meter at some potential value (say.v). How can we alter this galvanometer so that it will be able to measure currents (deflect fully) up to V?! V = IR = (.A)(!) =.V R needed = V new = V I.A =,! + R =,! R = 99! COM V! Ideal ammeters have low resistance." 3.) 32.)