ANNOUNCEMENT Exam : Tuesday September 25, 208, 8 PM - 0 PM Location: Elliott Hall of Music (see seating chart) Covers all readings, lectures, homework from Chapters 2 through 23 Multiple choice (5-8 questions) Practice exams On the course website and on CHP Bring your student D card and your own one-page (two-sides) crib sheet Only a few equations will be given The equation sheet that will be given with the exam is posted on the course homepage Link on the right labeled Equationsheet t is your responsibility to create your own crib sheet Crib sheet 8.5 x ANNOUNCEMENT Can be handwritten, computer-generated, painted, etc. All calculators allowed Except web-enabled, internet or bluetooth connected Nothing that can communicate with other devices
9/24/8 Exam Seating Chart Resistors and Circuits http://www.wiseguysynth.com/larry/ kw200/200_construction.htm 2
This Class Resistors in series: Current through is same Voltage drop across is r i Resistors in parallel: Voltage drop across is same Current through is V/R i R effective = + +... R effective = + +... Solve Circuits V R 3 R 4 Resistors in Series Voltage drop across resistors in series Each has identical current V a V b = R V V = b c 2 Can replace with equivalent resistor Same total potential drop, same current V total =V ab +V bc = + = ( + ) = + 3
Another (ntuitive) Way Consider two cylindrical resistors with lengths L and L 2 L = ρ L A = ρ L 2 A V Put them together, end to end to make a longer one... L 2 = ρ L + L 2 A = + R = + Resistors in Parallel Voltage drop identical across resistors in parallel Current can vary across each V = = 2 = V = V Can replace with equivalent resistor Same potential drop, same total current total = V = V + V = + 4
= ρ L A = ρ L A 2 9/24/8 Another (ntuitive) Way Consider two cylindrical resistors of equal length L with cross-sectional areas A and A 2 V L A A 2 L Put them together, side by side to make one fatter one = A ρl ρl + A 2 ρl = ( A + A 2 ) = + = + Example Consider the ideal circuit shown: What is the relation between V a -V d and V a -V c? 2V a 50W 20W b 2 80W d c (a) (V a -V d ) > (V a -V c ) (b) (V a -V d ) = (V a -V c ) (c) (V a -V d ) < (V a -V c ) 5
Example Consider the ideal circuit shown: What is the relation between V a -V d and V a -V c? 2V a 50W 20W b 2 80W d c (a) (V a -V d ) > (V a -V c ) (b) (V a -V d ) = (V a -V c ) (c) (V a -V d ) < (V a -V c ) Assume cd is a perfect conductor Still an equipotential even though this example is not static => Points d and c are the same, electrically Kirchhoff s First Rule ( Loop Rule or Kirchhoff s Voltage Law ) The algebraic sum of the changes in potential in a complete traversal of any loop of circuit must be zero Move around circuit: Based on energy conservation KVL : V n = 0 e loop R R2 e 2 - + + - + e - - - e 2 = 0 A restatement that the potential difference is independent of path Applies to any circuit 6
Rules e R R2 e 2 - + + - + e - - - e 2 = 0 Loop direction is ARBTRARY Voltage gains enter equation with a + sign Voltage drops enter equation with a - sign Battery traverse - terminal to + terminal: V increases => +e + terminal to - terminal: V drops => -e Rules e R R2 e 2 - + + - + e - - - e 2 = 0 Resistor traverse Positive direction Voltage drops Enters the equation with a sign (-R) Negative direction Voltage rises Enters the equation with a + sign (+R) 7
Rules e R R2 e 2 No wrong set of paths (multi-loop circuits) Flip one path, just change every sign in that path s equation BUT - + + - + e - - - e 2 = 0 One or more of the currents in your solution may be NEGATVE Means actual current flow is opposite to the path you chose Where ONLY that one current is flowing Traverse loop from a to f in this case Loop Example b a e + - f R 4 loop V n = 0 c d + - R 3 ε 2 R 3 R 4 +ε = 0 e 2 e = ε ε 2 + + R 3 + R 4 f e < e 2, will be negative, i.e., will flow clockwise, opposite to path 8
nternal Resistance of an EMF Device Any real emf device has internal resistance e.g., a real battery Apply Kirchhoff s Loop Rule (clockwise) ε r R = 0 = ε R + r R V ab = ε r = ε R + r Kirchhoff s Second Rule (Junction Rule or Kirchhoff s Current Law ) The sum of the current entering any junction (or node ) must equal the sum of the currents leaving that junction 2 in = out Conservation of charge Branch currents Currents entering and leaving circuit nodes Each distinct branch must be assigned a current, i 9
How to Use Kirchhoff s Laws Analyze the circuit & identify all circuit nodes Use KCL ) = 2 + 3 e e 2 a + - b - + 2 c 3 dentify all independent loops R 3 Use KVL 2) e 3 R 3 = 0 3) e + e 2 2 = 0 4) = (3)-(2) = e 2 2 + 3 R 3 = 0 Only 2 are independent d How to Use Kirchhoff s Laws Solve for, 2, and 3 Find 2 and 3 in terms of From eqn. (2) ( ) / R 3 3 = ε + From eqn. (3) 2 = ( ε 2 ε ) / Solve for from eqn. () e e 2 a b + - - + 2 c R 3 = ε 2 ε ε R 3 ( + R 3 ) = d 3 ε 2 ε ε R 3 + + R 3 0
How to Use Kirchhoff s Laws Only works here since only 2 currents in Eqns. (2) & (3) e e 2 a + - b - + 2 c More general case 3 simultaneous eqns. Can use standard software R 3 3 d Example 25-6 2 equations in 2 unknowns 3 5 2 = 7 (2) 4 2 2 = 5 ( 5) 6 0 2 = 4 20 +0 2 = 25 =.5 A = + 2 Loop (abcdefa): 2 2 5 3( + 2 )+2 = 0 7 3 5 2 = 0 Loop 2 (bcdeb): 2 2 5 + 4 = 0 ( ) 5 2 = 7 3.5 2 = 0.5 A
Cramer s Rule c b x = c 2 b 2 a b = c b 2 c 2 b a b 2 a 2 b f a x + b y = c a 2 x + b 2 y = c 2 Then a 2 b 2 a c y = a 2 c 2 a b a 2 b 2 = a c 2 a 2 c a b 2 a 2 b Summary of Simple Circuits Resistors in series: Each resistor on same wire Current through each is same = + + R 3 +... Voltage drop across each is R i Equivalent resistance increases Resistors in parallel: Each resistor on different wire Voltage drop across is same = + + R 3 +... Current through is V /R i Equivalent resistance decreases 2
Series Summary of Resistor & Capacitor Combinations Resistors = n i= R i = C eq Capacitors n i= C i Parallel = n i= R i = + for n = 2 C eq = C C 2 C +C 2 for n = 2 C eq = n i= C i Problem Solving Tips When you are given a circuit, first carefully analyze circuit topology Find the nodes and distinct branches Pick Linearly ndependent subsets of each Assign branch currents Use Kirchhoff s First Rule for all independent loops in the circuit Sum of the voltages around these loops is zero Use Kirchhoff s Second Rule for all independent nodes in circuit 3
Ammeter & Voltmeter Ammeter, A, inserted into the circuit Voltmeter, V, across a circuit element Ammeter & Voltmeter Both disturb or perturb a circuit To minimize the perturbation: Resistance of A small compared to + + r deal ammeter has r = 0 Resistance of V large compared to deal voltmeter has infinite R 4
Galvanometer A meter that detects small currents passing through it Scales the reading proportional to R g = g g ~ 50 00 µa for full scale deflection R g ~ 00 Ω Make a.0 A Full-Scale Ammeter V g = R g g = P R P R P = R g g P = R g g g small parallel resistor (shunt) f R g = 00 W and g = 50 µa, then R P = 00 Ω 50 0 6 A A 50 0 6 A 00 Ω 50 0 6 A = 0.005 Ω A To change the scale, change R P 5
Voltmeter Assume R g = 00 W and g = 50 µa for full scale deflection (typical). Make a 00 V full scale deflection voltmeter. g (R S + R g ) =V R S + R g = 00V 50 0 6 A = 2 06 Ω 00 negligible Ohmmeter Battery in series with galvanometer and resistor Full-scale when a-b is shorted Reads resistance when resistor connected across a-b 6