1. Copyright R. Janow - Fall 2015 1 Electricity and Magnetism Lecture 07 - Physics 121 Current, Resistance, DC Circuits: Y&F Chapter 25 Sect. 1-5 Kirchhoff’s Laws: Y&F Chapter 26 Sect. 1 Currents and Charge Electric Current i Current Density J Drift Speed, Charge Carrier Collisions Resistance, Resistivity, Conductivity Ohm’s Law Power in Electric Circuits Examples Circuit Element Definitions Kirchhoff’s Rules EMF’s - “Pumping” Charges, Ideal and real EMFs Work, Energy, and EMF Simple Single Loop and Multi-Loop Circuits using Kirchhoff Rules Equivalent Series and Parallel Resistance formulas using Kirchhoff Rules.

2. Copyright R. Janow - Fall 2015 Definition of Current : Net flow rate of charge through some area Current is the same across each cross-section of a wire +- i Current density J may vary [J] = current/area Convention: flow is from + to – as if free charges are + Units: 1 Ampere = 1 Coulomb per second Charge / current is conserved - charge does not pile up or vanish At any Node (Junction) i1i1 i2i2 i3i3 i 1 +i 2 =i 3 Kirchhoff’s Rules: (summary)  Current (Node) Rule:  currents in =  currents out at any node  Voltage (Loop) Rule:   V’s = 0 for any closed path  Voltage sources (EMFs, electro-motive forces) or current sources can supply energy to a circuit  Resistances dissipate energy as heat  Capacitances and Inductances store energy in E or B fields Energy or Power in a circuit:

3. Copyright R. Janow - Fall 2015 Junction Rule Example – Current Conservation 7-1: What is the value of the current marked i ? A.1 A. B.2 A. C.5 A. D.7 A. E.Cannot determine from information given. 5 A 2 A 3 A 1 A 6 A i 3 A 8 A 1 A = 7 A

4. Copyright R. Janow - Fall 2015 Current density J: Current / Unit Area (Vector) What makes current flow? Microscopic level conductivity Same current crosses larger or smaller Surfaces, current density J varies + - i A A’ J’ = i / A’ (small) J = i / A (large) High current density in this region Small current density in this region i units: Amperes/m 2 For uniform density: For non-uniform density: Drift speed: Do electrons in a current keep accelerating? Yes, for isolated charge in vacuum. No, in a conducting solid, liquid, gas resistivity

5. Copyright R. Janow - Fall 2015 COLLISIONS with ions, impurities, etc. cause resistance Charges move at constant drift speed v D : Thermal motions (random motions) have speed Drift speed is tiny compared with thermal motions. Drift speed in copper is 10  8 – 10 -4 m/s. E field in solid wire drives current. APPLIED FIELD = 0 Charges in random motion flow left = flow right ++++ ---- APPLIED FIELD NOT ZERO Moving charges collide with fixed ions and flow with drift velocity v d + - Note: Electrons drift rightward but v d and J are still leftward |q| = e = 1.6 x 10 -19 C. For E = 0 in conductor: no current, v D =0, J = 0, i = 0 For E not = 0 (battery voltage not 0):

6. Copyright R. Janow - Fall 2015 EXAMPLE: Calculate the current density J ions for ions in a gas Assume: Doubly charged positive ions Density n = 2 x 10 8 ions/cm 3 Ion drift speed v d = 10 5 m/s Find J ions – the current density for the ions only (forget J electrons ) coul/ion ions/cm 3 m/scm 3 /m 3 A./m 2

7. Copyright R. Janow - Fall 2015 Increasing the Current 7-2: When you increase the current in a wire, what changes and what is constant? A.The density of charge carriers stays the same, and the drift speed increases. B.The drift speed stays the same, and the number of charge carriers increases. C.The charge carried by each charge carrier increases. D.The current density decreases. E.None of the above

8. Copyright R. Janow - Fall 2015 Definition of Resistance : Amount of current flowing through a conductor per unit of applied potential difference. R depends on the material & geometry Note: C= Q/ D V – inverse to R i i DVDV E L A Apply voltage to a wire made of a good conductor. - Very large current flows so R is small. Apply voltage to a poorly conducting material like carbon - Tiny current flows so R is very large. R Circuit Diagram V Resistivity “ r ” : Property of a material itself (as is dielectric constant). Does not depend on dimensions The resistance of a device depends on resistivity r and also depends on shape For a given shape, different materials produce different currents for same D V Assume cylindrical resistors For insulators: r  infinity

9. Copyright R. Janow - Fall 2015 Example: calculating resistance or resistivity resistance proportional to length inversely proportional to cross section area resistivity EXAMPLE: Find R for a 10 m long iron wire, 1 mm in diameter Find the potential difference across R if i = 10 A. (Amperes) EXAMPLE: Find resistivity of a wire with R = 50 m W, diameter d = 1 mm, length L = 2 m Use a table to identify material. Not Cu or Al, possibly an alloy

10. Copyright R. Janow - Fall 2015 Resistivity depends on temperature: Resistivity depends on temperature: Higher temperature  greater thermal motion  more collisions  higher resistance. Conductivity is the reciprocal of resistivity i i DVDV E L A Definition: SOME SAMPLE RESISTIVITY VALUES r in W. m @ 20 o C. 1.72 x 10 -8 copper 9.7 x 10 -8 iron 2.30 x 10 +3 pure silicon Reference Temperature Change the temperature from reference T 0 to T Coefficient  depends on the material Simple model of resistivity:  = temperature coefficient

11. Copyright R. Janow - Fall 2015 Resistivity Tables

12. Copyright R. Janow - Fall 2015 Current Through a Resistor 7-3: What is the current through the resistor in the following circuit, if V = 20 V and R = 100 ? A.20 mA. B.5 mA. C.0.2 A. D.200 A. E.5 A. R V Circuit Diagram

13. Copyright R. Janow - Fall 2015 Current Through a Resistor 7-4: If the current is doubled, which of the following might also have changed? A.The voltage across the resistor might have doubled. B.The resistance of the resistor might have doubled. C.The resistance and voltage might have both doubled. D.The voltage across the resistor may have dropped by a factor of 2. E.The resistance of the resistor may have dropped by a factor of 2. R V Circuit Diagram Note: there might be more than 1 right answer

14. Copyright R. Janow - Fall 2015 Ohm’s Law and Ohmic materials (a special case) Definitions of resistance: (R could depend on applied V) (  could depend on E) Definition: For OHMIC conductors and devices: Ratio of voltage drop to current is constant – NO dependance on applied voltage. i.e., current through circuit element equals CONSTANT resistance times applied V Resistivity does not depend on magnitude or direction of applied voltage Ohmic Materials e.g., metals, carbon,… Non-Ohmic Materials e.g., semiconductor diodes constant slope = 1/R varying slope = 1/R band gap OHMIC CONDITION is CONSTANT

15. Copyright R. Janow - Fall 2015 Resistive Loads in Circuits Dissipate Power i V + i - LOADLOAD 1 2 Apply voltage drop V across basic circuit element (load) Current flows through load which dissipates energy An EMF (e.g., a battery) does work, holding potential V and current i constant by expending potential energy As charge dq flows from 1 to 2 it loses P.E. = dU - potential is PE / unit charge - charge = current x time element absorbs power P=-Vi 1 2 i V + - 1 2 i V - + element supplies power P= +Vi 1 2 i V + - 1 2 i V - + Voltage rise: + - V Active sign convention: P absorbed = dot product of V rise with i Note: Some EE circuits texts use passive convention: model voltage drops V drop as positive.

16. Copyright R. Janow - Fall 2015 EXAMPLE:

17. Copyright R. Janow - Fall 2015 Basic circuit elements have two terminals Basic Circuit Elements and Symbols 1 2 Ideal basic circuit elements: Generic symbol: Active: voltage sources ( EMF’s ), current sources L C C or R Passive: resistance, capacitance, inductance isis Ideal independent voltage source: zero internal resistance constant v s for any load Ideal independent current source: constant i s regardless of load vsvs + - E + - vsvs DC battery AC Dependent voltage or current sources: depend on a control voltage or current generated elsewhere within a circuit

18. Copyright R. Janow - Fall 2015 Circuit analysis with resistances and EMFs ANALYSIS METHOD: Kirchhoff’S LAWS or RULES Current (Junction) Rule: Charge conservation Loop (Voltage) Rule: Energy conservation i VV slope = 1/R RESISTANCE: POWER: OHM’s LAW: R is independent of  V or i CIRCUIT ELEMENTS ARE CONNECTED TOGETHER AT NODES i i1i1 i2i2 CIRCUITS CONSIST OF JUNCTIONS (ESSENTIAL NODES)... and … ESSENTIAL BRANCHES (elements connected in series,one current/branch) i

19. Copyright R. Janow - Fall 2015 EMFs “pump” charges to higher energy EMFs (electromotive force) such as batteries supply energy: –move charges from low to high potential (potential energy). –maintain constant potential at terminals –do work dW = E dq on charges (source of the energy in batteries is chemical) –EMFs are “charge pumps” Unit: volts (V). Variable name: script E. Types of EMFs: batteries, electric generators, solar cells, fuel cells, etc. DC versus AC + - R E i i CONVENTION: Current flows CW through circuit from + to – outside of EMF from – to + inside EMF Power is dissipated by resistor: Power supplied by EMF:

20. Copyright R. Janow - Fall 2015 Zero internal battery resistance Open switch: EMF = E no current, zero power Closed switch: EMF E is also applied across load circuit Current & power not zero Ideal EMF device R Multiple EMFs Assume E B > E A (ideal EMF’s) Which way does current i flow? Apply kirchhoff Laws to find current Answer: From E B to E A E B does work, loses energy E A is charged up R converts PE to heat Load (motor, other) produces motion and/or heat Real EMF device Open switch: EMF still = E r = internal EMF resistance in series, usually small ~ 1 W Closed switch: V = E – ir across load, P ckt = iV Power dissipated in EMF P emf = i( E -V) = i 2 r

21. Copyright R. Janow - Fall 2015 The Kirchhoff Loop Rule Generates Circuit Equations The sum of voltage changes = zero around all closed loops in a multi-loop circuit) Traversing one closed loop generates one equation. Multiple loops may be needed. A branch is a series combination of circuit elements between essential nodes. Assume either current direction in each branch. Minus signs may appear in the result. Traverse each branch of a loop with or against assumed current direction. Across resistances, write voltage drop DV = - iR if following assumed current direction. Otherwise, write voltage change = +iR. When crossing EMFs from – to +, write DV = + E. Otherwise write DV= - E Dot product i. E determines whether power is actually supplied or dissipated Follow circuit from a to b to a, same direction as i Potential around the circuit EXAMPLE: Single loop (circuit with battery (internal resistance r) E E P = i E – i 2 r Power in External Ckt circuit dissipation battery drain battery dissipation P = iV = i( E – ir)

22. Copyright R. Janow - Fall 2015 The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above Loop Rule: The sum of the potential differences around a closed loop equals zero. Only one loop path exists: Example: Equivalent resistance for resistors in series Junction Rule: The current through all of the resistances in series (a single essential branch) is identical. No information from Junction/Current/Node Rule CONCLUSION: The equivalent resistance for a series combination is the sum of the individual resistances and is always greater than any one of them. inverse of series capacitance rule

23. Copyright R. Janow - Fall 2015 Example: Equivalent resistance for resistors in parallel Loop Rule: The potential differences across each of the 4 parallel branches are the same. Four unknown currents. Apply loop rule to 3 paths. Junction Rule: The sum of the currents flowing in equals the sum of the currents flowing out. Combine equations for all the upper junctions at “a” (same at “b”). The equivalent circuit replaces the series resistors with a single equivalent resistance: same E, same i as above. CONCLUSION: The reciprocal of the equivalent resistance for a parallel combination is the sum of the individual reciprocal resistances and is always smaller than any one of them. inverse of parallel capacitance rule i not in these equations

24. Copyright R. Janow - Fall 2015 7-7: Four identical resistors are connected as shown in the figure. Find the equivalent resistance between points a and c. A. 4 R. B. 3 R. C. 2.5 R. D. 0.4 R. E.Cannot determine from information given. Resistors in series and parallel R R R R c a

25. Copyright R. Janow - Fall 2015 7-8: Four identical capacitors are connected as shown in figure. Find the equivalent capacitance between points a and c. A. 4 C. B. 3 C. C. 2.5 C. D. 0.4 C. E. Cannot determine from information given. Capacitors in series and parallel C CC C c a

26. Copyright R. Janow - Fall 2015 EXAMPLE: + - R 1 = 10  R 2 = 7  R 3 = 8  E = 7 V i + - E = 9 V R1R1 R2R2 i1i1 i2i2 i i Find i, V 1, V 2, V 3, P 1, P 2, P 3 Find currents and voltage drops

27. Copyright R. Janow - Fall 2015 + - + - R 1 = 10  R 2 = 15  E 1 = 8 V E 2 = 3 V i i EXAMPLE: MULTIPLE BATTERIES SINGLE LOOP A battery (EMF) absorbs power (charges up) when I is opposite to E

28. Copyright R. Janow - Fall 2015 EXAMPLE: Find the average current density J in a copper wire whose diameter is 1 mm carrying current of i = 1 ma. Suppose diameter is 2 mm instead. Find J’: Current i is unchanged Calculate the drift velocity for the 1 mm wire as above? About 3 m/year !! So why do electrical signals on wires seem to travel at the speed of light (300,000 km/s)? Calculating n for copper: One conduction electron per atom