Ch 28-DC Circuits! 1.) EMF & Terminal Voltage! 9.0 V 8.7 V 8.7 V V =! " Ir Terminal Open circuit internal! voltage voltage (emf) resistance" 2.)
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.) Resistors in series! What is the current flow in this circuit?! What is the voltage drop across each resistor?! 10! 30! 60! (Look at series movie in Richard s stuff ) 100 volts 4.)
Resistors in series! What is the current flow in this model?! The equivalent 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 equivalent = R 1 + R 2 + R 3 +... (for resistors in series) so we can write: 10! 30! 60! 100 volts 5.) Resistors in series! What is the current flow in this model?! The equivalent 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 equivalent = R 1 + R 2 + R 3 +... (for resistors in series) so we can write: 10! 30! 60! 100 volts What is the voltage drop across each resistor?! 6.)
Resistors in parallel! 7.) Resistors in parallel! What is the voltage drop over each resistor in this model?! What is the current through each resistor?! 10! 8! (Look at parallel movie in Richard s stuff ) 6! 24 volts 1 R equivalent = 1 R 1 + 1 R 2 + 1 R 3 +... (for resistors in parallel) 8.)
Example 1! Examine the circuit here.! 3! 6! 9! 18V a. What is the equivalent resistance?" R equiv =3! +6! +9! =18!! b. Find the current flowing throughout the circuit." I=V/R=18V/18! =1.0A! c. What is the potential at every point in the circuit." V 3 =IR=(1A)(3!)=3V; V 6 =6V; V 9 =9V! d. What is the Power delivered by the battery?" P=IV=(1.0A)(18V)=18W! e. What is the Power consumed by each resistor?" P 3 =IV=(1.0A)(3V)=3W; or use P 3 =I 2 R=(1A)(3!)=3W; or use P 3 =V 2 /R=(3V) 2 /3A=3W; P 6 =6W; P 9 =9W! 9.) Example 2! Examine the circuit here.! 24V 6! 4! 12! a. What is the equivalent resistance?" 1/R equiv =1/4! +1/6! +1/12!; R equiv =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 equiv = 24V/2! =12A; I 4 =V/R= (24V)/(4!)=6A; I 6 =4A; I 12 =2A! d. What is the Power delivered by the battery?" Total power P = (I) V = (12 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 12 ohm resistor, 96 watts for the 6 ohm resistor and 144 watts for the 6 ohm resistor for a total of (tada) 288 watts.! 10.)
Example 3! Examine the circuit here.! 6! 12! 36V 8! a. What is the equivalent 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?" 11.) Kirchhoff s Rules! 12.)
Kirchhoff s Rule 1! The Junction Rule - Total current into a node = total current out of a node! I 2 I 1 I 2 I 1 I 3 I 3 I 2 I 1 I 3 13.) Kirchhoff s Rule 2! The Loop Rule - The sum of the potential changes around a closed loop must equal zero.! 14.)
Using Kirchhoff s! 1. 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 equations using Kirchhoff s Junction Rule.(Make sure you don t duplicate any equations!)! 3. Write out a series of Loop equations using Kirchhoff s Loop Rule. (The total number of Junction and Loop equations = the number of unknown currents you re solving for.)! 4. Solve these simultaneous equations.! 15.) Example 4! Examine the circuit here.! 2 Ω a. What is the current flow, everywhere?" b. What are the voltages, everywhere?" 8 Ω 5 V 5 Ω c. What is the Power consumed/ produced, everywhere?" 15 Ω 10 V 16.)
Example 4! Examine the circuit here.! Apply Kirchoff s rules to circuit with currents (arbitrarily chosen here) to get:" 2 Ω I 1 5 V I 1 + I 3 = I 2 10!15I 3! 8I 2! 5! 5I 3 = 0 5 + 8I 2 + 2I 1 = 0 8 Ω I 2 5 Ω 15 Ω I 3 10 V 17.) Example 4! I 1! I 2 + I 3 = 0 0I 1! 8I 2! 20I 3 =!5 2I 1 + 8I 2 + 0I 3 =!5 1!1 1 0 0! 8! 20! 5 2 8 0! 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" " 1!1 1 0% $ ' $ 0! 8! 20! 5 ' # $ 2 8 0! 5& ' " 1 0 0!.833... % $ ' $ 0 1 0!.4166... ' # $ 0 0 1.4166... &' 18.)
Six Flags/Magic Mountain! When: Monday, April 5, 2010! 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 203.! 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 required.! 19.) Six Flags/Magic Mountain! 20.)
Six Flags/Magic Mountain! 21.) Six Flags/Magic Mountain! 22.)
Series & Parallel Activity! Examine the series and parallel circuits assembled in class... and don t touch the wires.! 23.) RC Circuits - Charge! V o!v capacitor!v resistor = 0 V o! q plate C! I instr = 0 V o! q plate C! dq dt R = 0 dq dt = V o R! q RC dq dt = V o R! q RC dq dt = " V oc RC! q % $ ' = V oc! q # RC& RC dq q!v o C =!1 RC dt q dq ( = q!v o C 0 t ( 0!1 RC dt q!t ln( q!v o C) 0 = RC ln( q!v oc!v o C ) =!t RC q(t) = CV o [1! e!t / RC ] q(t) = Q[1! e!t / RC ] 24.)
RC Circuits - Charging! The quantity! = 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 0.632 of its total value." q(t) = Q(1! e!t / RC ) 25.) RC Circuits - Discharging! What is the function for a discharging capacitor? What does! = RC represent for a discharging capacitor?" Q o q(t) = Q(e!t / RC ) t 26.)
RC Circuits - Current! q(t) = Q[1! e!t / RC ] I = dq dt I = d dt (Q[1! e!t / RC ]) I = Q( d dt 1! e!t / RC ) I = Q(!1 RC )(!e!t / RC ) I(t) = V o R e!t / RC!t / RC I(t) = I o e 27.) RC Circuits - Current! What is the function for current of a discharging capacitor?" I(t) =! Q RC e!t / RC 28.)
Example 5! An RC circuit has a 6V battery, a 200µF capacitor, and a 5000! 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." 29.) Galvanometers! A galvanometer is an ammeter that is designed to swing fulldeflection when.5 ma (.0005 amps) of current flows through it. If one is clever, a galvanometer can be used to build a voltmeter or larger current ammeter. We are about to be clever.! COM V! COM V! Ideal voltmeters have high resistance." Ideal ammeters have low resistance." 30.)
Ammeter:! Example: Create a 2 amp ammeter starting with the frame provided above (i.e., I ve provided the galvanometer you put in the circuit and determine any extraneous resistances). You may assume the galvanometer s resistance is 12 ohms. 31.) Example: Create a 2 amp ammeter. For the galvanometer to go full deflection--something you want it to do when 2 amps flow into it--you need to shunt some of the current away from the galvanometer. The current that does not flow into the galvanometer flows through the shunt resistor. In this case, that will be 2 -.0005 = 1.9995 amps. The two resistances are in parallel, so the voltage across each will be the same. As such, we can write: 2 A i g R g = i s R s R g =12! R s =?! 5x10 "4 A i s =1.9995 A ( ) 12 #! R $ 3x10 "3 # i g =5x10-4 A ( ) = 1.9995( R) Note: A short wire across the galvanometer s terminals will give us this resistance. Also, the equivalent resistance of a parallel combination of resistors is smaller than the smallest resistor in the combination. That means the equivalent resistance of this ammeter is less than three thousandths of an ohm. That is exactly what we expected. Ammeters have little net resistance to charge flow (hence current is not impacted by putting one directly into a circuit). 32.)
Voltmeter:! Example: Create a 2 volt voltmeter starting with the frame provided above (i.e., put in the circuit and determine any extraneous resistances). You may assume the galvanometer s resistance is 12 ohms. 33.) For the galvanometer to go full deflection--something you want it to do when 2 amps flow into it--you to have a half milliamp flow through it. Two volts across a 12 ohm galvanometer will produce a huge current, so you have to cut the current down some. You do that by putting a resistor in series with the galvanometer. Remembering that the current through resistors in series is the same, we can write: V = i g R g + i g R R g =12! R i g =5x10-4 A 2 volts ( ) 12 #! 2= 5x10 "4 A! R $ 4x10 3 # ( ) R ( ) + 5x10 "4 A ( ) Note: The equivalent resistance of a series combination of resistors is larger than the largest resistor in the combination. That means the equivalent resistance of this voltmeter is greater than four thousand ohm. That is exactly what we expected. Voltmeters have huge resistance to charge flow (hence little current passes through them when put in a circuit). 34.)
Mr. White s Ammeters! The galvanometer all by itself has some resistance (say 100 Ω), and deflects fully across the face of the meter at some current value (say 0.0005A). How can we alter this galvanometer so that it will be able to measure currents (deflect fully) up to 0.100A?! COM V! V full deflection = IR = (0.0005A)(100!) = 0.050V R needed = V I new = 0.05V 0.10A = 0.5! 1 100 + 1 = 1 R shunt 0.5 R shunt =.503! 35.) Mr. White s Voltmeters! COM V! The galvanometer all by itself has some resistance (say 100 Ω), and deflects fully across the face of the meter at some potential value (say 0.05 V). How can we alter this galvanometer so that it will be able to measure currents (deflect fully) up to 10 V?! V = IR = (0.0005 A)(100!) = 0.050 V R needed = V new 10V = I 0.0005A = 20,000! 100 + R = 20,000! R = 19,900! 36.)