Electricity and Magnetism Lecture 07 Physics 121 Current

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Electricity and Magnetism Lecture 07 - Physics 121 Current, Resistance, DC Circuits: Y&F Chapter

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 “Pump” 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. Copyright R. Janow – FALL 2019 1

Definition of Current : Net rate of charge flow through some area Units: 1

Definition of Current : Net rate of charge flow through some area Units: 1 Ampere = 1 Coulomb per second Convention: flow is from + to – as if free charges are + Charge & current are conserved - charge does not pile up or vanish Current flowing through each cross-section of a wire is the same i + - At any Node (Junction) Kirchhoff’s Rules: (preview) Roles of circuit elements Current density J may vary [J] = current/area i 1 i 2 i 3 i 1+i 2=i 3 § Current (Node) Rule: S currents in = S currents out at any node § Voltage (Loop) Rule: S DV’s = 0 for any closed path § Voltage sources (EMFs, electro-motive forces) or current sources can supply energy to a circuit (or absorb it). § Resistances dissipate energy as heat. § Capacitances and Inductances store Copyright energy R. in Janow E or B fields. – FALL 2019

Junction Rule Example – Current Conservation 7 -1: What is the value of the

Junction Rule Example – Current Conservation 7 -1: What is the value of the current marked i ? A. B. C. D. E. 1 A. 2 A. 5 A. 7 A. Cannot determine from information given. 1 A 3 A 2 A 2 A 3 A 1 A 5 A 8 A i =7 A 6 A Copyright R. Janow – FALL 2019

Current density J: Current / Unit Area (Vector) i Small current density i +

Current density J: Current / Unit Area (Vector) i Small current density i + High current density A’ A - J=i/A (large) J’ = i / A’ (small) Conductor width varies …… but …. same current crosses larger or smaller surfaces. What varies is current density J. If density is uniform: units: Amperes/m 2 If density is non-uniform: What makes current flow in a conductor? E field. conductivity resistivity • E inside conductor not required to be zero…as • Conductor not in equilibrium if DV not zero Do electrons in a current keep accelerating? • Yes, for isolated charges in vacuum. • No, in a conducting solid, liquid, gas due to collisions • Recall: terminal velocity for falling object (collisions =>R. drift speed) Copyright Janow – FALL 2019

Collisions with ions & impurities, etc. cause resistance Charges move at constant drift speed

Collisions with ions & impurities, etc. cause resistance Charges move at constant drift speed v. D: E field in solid wire drives current. APPLIED FIELD = ZERO Charges in random motion left flow = right flow • • • + + - APPLIED FIELD NOT ZERO Accelerating charges collide frequently with fixed ions and flow with drift velocity Thermal motions (random motions) have speed Drift speed is tiny compared with thermal motions. Copper drift speed is 10 -8 – 10 -4 m/s. Drift Speed Formula: q = charge on each carrier v. D = s E qn Note: Electrons flow leftward in sketch but vd and J are still to the. R. right. Copyright Janow – FALL 2019

Increasing the Current 7 -2: When you increase the current density in a wire,

Increasing the Current 7 -2: When you increase the current density in a wire, what changes and what is constant? A. B. C. D. E. The density of charge carriers stays the same, and the drift speed increases. The drift speed stays the same, and the number of charge carriers increases. The charge carried by each charge carrier increases. The current density decreases. None of the above Copyright R. Janow – FALL 2019

Definition of Resistance : Ratio of current flowing through a conductor to the potential

Definition of Resistance : Ratio of current flowing through a conductor to the potential difference applied to it. E i DV L A i R depends on the material & geometry Note: C= Q/DV – inverse to R Apply voltage to a wire made of a good conductor. - very large current flows so R is small. Apply voltage to a poor conductor like carbon - tiny current flows so R is very large. Ohm’s Law means: R is independent of applied voltage V Circuit R Diagram Resistivity “r” : Property of a material itself Does not depend on size/geometry of a sample • The resistance of a device depends on resistivity r and also on shape. • For a given shape, different materials produce different currents for same DV • Assume cylindrical resistors For insulators: r infinity Copyright R. Janow – FALL 2019

Example: calculating resistance or resistivity resistance proportional to length inversely proportional to cross section

Example: calculating resistance or resistivity resistance proportional to length inversely proportional to cross section area 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, Use a table to identify material. Not Cu or Al, possibly an alloy Copyright R. Janow – FALL 2019

Current Through a Resistor 7 -3: What is the current through the resistor in

Current Through a Resistor 7 -3: What is the current through the resistor in the following circuit, if DV = 20 V and R = 100 W? A. B. C. D. E. 20 m. A. 5 m. A. 0. 2 A. 200 A. 5 A. V Circuit Diagram R Copyright R. Janow – FALL 2019

Current Through a Resistor 7 -4: The current in the preceding circuit is doubled.

Current Through a Resistor 7 -4: The current in the preceding circuit is doubled. Which of the following changes might have been the cause? A. B. C. D. E. The The The voltage across the resistor might have doubled. resistance of the resistor might have doubled. resistance and voltage might have both doubled. voltage across the resistor may have dropped by a factor of 2. resistance of the resistor may have dropped by a factor of 2. Note: there might be more than 1 right answer V Circuit Diagram R Copyright R. Janow – FALL 2019

Ohm’s Law and Ohmic materials (R could depend on applied V) Definitions of resistance:

Ohm’s Law and Ohmic materials (R could depend on applied V) Definitions of resistance: (r could depend on E) Definition of OHMIC conductors and devices: • Ratio of voltage drop to current is constant – NO dependance on applied voltage. • Voltage drop across a circuit element equals CONSTANT resistance times current. Resistivity does not depend on magnitude or direction of applied voltage Ohmic Materials e. g. , metals, carbon, … Non-Ohmic Materials e. g. , semiconductor diodes band gap constant slope = 1/R OHMIC CONDITION varying slope = 1/R is CONSTANT Copyright R. Janow – FALL 2019

Resistors Dissipate Power, Irreversibly i + L O A D V - 1 i

Resistors Dissipate Power, Irreversibly i + L O A D V - 1 i 2 • • Apply voltage drop V across load resistor Current flows through load which dissipates energy - irreversibly The EMF (battery) does work by expending potential energy, It maintains constant potential V and current i, As charge dq flows from 1 to 2 it loses P. E. = d. U - Potential V is PE / unit charge - Charge = current x time Copyright R. Janow – FALL 2019

EXAMPLE: Copyright R. Janow – FALL 2019

EXAMPLE: Copyright R. Janow – FALL 2019

Demonstration Source: Pearson Study Area - VTD Power dissipated = current 2 x resistance

Demonstration Source: Pearson Study Area - VTD Power dissipated = current 2 x resistance Chapter 25 Resistance in Copper and Nichrome https: //mediaplayer. pearsoncmg. com/assets/secs-vtd 36_irlosses Discussion: • • What made one paper sample ignite but not the other? What was different for the copper and Nichrome wire? Copyright R. Janow – FALL 2019

Circuit Analysis / Circuit Model i 1 Circuits consist of: ESSENTIAL NODES (junctions). .

Circuit Analysis / Circuit Model i 1 Circuits consist of: ESSENTIAL NODES (junctions). . . and … ESSENTIAL BRANCHES (elements connected in series, one current/branch) i i 2 i ANALYSIS METHOD: Kirchhoff’S LAWS or RULES Current (Junction) Rule: Charge conservation Loop (Voltage) Rule: Energy conservation RESISTANCE: POWER: OHM’s LAW: slope = 1/R i DV R is independent of DV or i Copyright R. Janow – FALL 2019

What EMFs do: • • • raise charges to higher potential energy, i. e.

What EMFs do: • • • raise charges to higher potential energy, i. e. move + charges from low to high potential maintain constant potential at terminals for any value of current do work d. W = Edq on charges (using chemical energy in batteries) • Unit: volts (V). • Types of EMFs: batteries, electric generators, solar cells, fuel cells, etc. • DC versus AC Symbol: script i + E - E. Power supplied by EMF: R i CONVENTION: Positive charges flow - from + to – outside of EMF - from – to + inside EMF Power dissipated by resistor: Copyright R. Janow – FALL 2019

Ideal EMF device • Zero internal battery resistance • Open switch: EMF = E

Ideal EMF device • Zero internal battery resistance • Open switch: EMF = E - zero current, zero power • Closed switch: EMF E is applied across load circuit - current & power not zero Real EMF device • Open switch: EMF still = E - r = internal EMF resistance in series, usually small • Closed switch: - V = E – ir across load, Pckt= i. V - Power dissipated in EMF Pemf = i(E-V) = i 2 r Multiple EMFs Assume EB > EA (ideal EMF’s) Which way does current i flow? • Apply kirchhoff Laws to find current • Answer: From EB to EA (CCW) • EB does + work, loses energy • EA is charged up, does negative work • R converts PE to heat • Load (motor or other) produces motion and/or heat R Copyright R. Janow – FALL 2019

Rules for generating circuit equations from the Kirchhoff Loop Rule: Multi-loop circuits. Assume unknown

Rules for generating circuit equations from the Kirchhoff Loop Rule: Multi-loop circuits. Assume unknown currents • Define: A branch (essential branch) is a series combination of circuit elements connecting to essential nodes at its endpoints. Each branch has exactly one current flowing in it through all components in series. • A loop may cross several branches. Traversing one closed loop generates one equation. The sum of voltage changes is zero around every closed loop in a multi-loop circuit. • Assume either current direction in each branch. Wrong choices imply minus signs. Imagine a test charge traversing branches, flowing with or against assumed current directions. • When crossing a resistance in the same direction assumed for current direction, add a voltage drop term DV = - i. R to the equation being built. Otherwise, add a voltage gain of DV = +i. R. • When crossing EMFs from – to +, write DV = +E, since the potential rises. • If traversing EMF from + to minus write DV= -E. The ‘dot’ product i. E determines whether power is actually supplied or dissipated. Copyright R. Janow – FALL 2019

EXAMPLE: Kirchhoff Loop Rule applied to a single loop circuit Follow circuit from a

EXAMPLE: Kirchhoff Loop Rule applied to a single loop circuit Follow circuit from a to b to a, same direction as i E E D Potential around the circuit = 0 Power in external ckt P = i. Vba = i(E – ir) P = i. E – i 2 r circuit battery dissipation drain dissipation Copyright R. Janow – FALL 2019

Example: Equivalent resistance for series resistors using Kirchhoff Rules Junction Rule: The current through

Example: Equivalent resistance for series resistors using Kirchhoff Rules Junction Rule: The current through all of the resistances in series (a single essential branch) is identical. No information from Junction/Current/Node Rule. Loop Rule: The sum of the potential changes around a closed loop path equals zero. Only one loop path exists: The equivalent circuit replaces the series resistors with a single equivalent resistance Req: same E, same i as above. 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 Copyright rule R. Janow – FALL 2019

Example: Equivalent resistance for resistors in parallel Loop Rule: The potential differences across each

Example: Equivalent resistance for resistors in parallel Loop Rule: The potential differences across each of the 4 parallel branches are the same. There are four unknown currents. Apply loop rule to 3 paths. i not in these equations 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 Copyright R. Janow – FALL 2019

Demonstration Chapter 26 Source: Pearson Study Area - VTD Bulbs Connected in Series and

Demonstration Chapter 26 Source: Pearson Study Area - VTD Bulbs Connected in Series and Parallel Circuits https: //mediaplayer. pearsoncmg. com/assets/secs-vtd 37_seriesparallel Discussion: • • Why did some of the bulbs look dimmer and others looked brighter? How do the equivalent resistances affect what you saw? Copyright R. Janow – FALL 2019

Resistors in series and parallel 7 -7: Four identical resistors are connected as shown

Resistors in series and parallel 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. c R R a R R Copyright R. Janow – FALL 2019

Capacitors in series and parallel 7 -8: Four identical capacitors are connected as shown

Capacitors in series and parallel 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. c C C a Copyright R. Janow – FALL 2019

EXAMPLE: Find i, V 1, V 2, V 3, P 1, P 2, P

EXAMPLE: Find i, V 1, V 2, V 3, P 1, P 2, P 3 R 1= 10 W + - i E =7 V R 2= 7 W R 3= 8 W EXAMPLE: Find currents and voltage drops i + - E =9 V i 1 i 2 R 1 R 2 i Copyright R. Janow – FALL 2019

EXAMPLE: MULTIPLE BATTERIES SINGLE LOOP i + - R 1= 10 W E 1

EXAMPLE: MULTIPLE BATTERIES SINGLE LOOP i + - R 1= 10 W E 1 = 8 V + - i E 2 = 3 V R 2= 15 W A battery (EMF) absorbs power (charges up) when I is opposite to E Copyright R. Janow – FALL 2019

EXAMPLE: Find the average current density J in a copper wire whose diameter is

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 Copyright R. Janow – FALL 2019

SUPPLEMENTARY MATERIAL Copyright R. Janow – FALL 2019

SUPPLEMENTARY MATERIAL Copyright R. Janow – FALL 2019

Resistivity depends on temperature: • Resistivity depends on temperature: Higher temperature greater thermal motion

Resistivity depends on temperature: • Resistivity depends on temperature: Higher temperature greater thermal motion more collisions higher resistance. 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 Simple linear model of resistivity: a = temperature coefficient Change the temperature from reference T 0 to T Coefficient a depends on the material Conductivity is the reciprocal of resistivity E i DV L Definition: A i Copyright R. Janow – FALL 2019

Resistivity Tables Copyright R. Janow – FALL 2019

Resistivity Tables Copyright R. Janow – FALL 2019

EXAMPLE: Calculate the current density Jions for ions in a gas Assume: • Doubly

EXAMPLE: Calculate the current density Jions for ions in a gas Assume: • Doubly charged positive ions • Density n = 2 x 108 ions/cm 3 • Ion drift speed v = 105 m/s d Find Jions – the current density for the ions only (forget Jelectrons) coul/ion ions/cm 3 m/s cm 3/m 3 A. /m 2 Copyright R. Janow – FALL 2019

Basic Circuit Elements and Symbols Basic circuit elements have two terminals Generic symbol: 2

Basic Circuit Elements and Symbols Basic circuit elements have two terminals Generic symbol: 2 1 Ideal basic circuit elements: Passive: resistance, capacitance, R C or inductance C L Active: voltage sources (EMF’s), current sources Ideal independent voltage source: • zero internal resistance • constant vs for any load Ideal independent current source: • constant is regardless of load + DC battery E DC vs + - AC vs is Dependent voltage or current sources: depend on a control voltage or current generated elsewhere within a circuit Copyright R. Janow – FALL 2019

Is a Component Supplying or Dissipating Power? Pabsorbed = dot product of Vrise with

Is a Component Supplying or Dissipating Power? Pabsorbed = dot product of Vrise with i (active sign convention) Voltage rise: + V - Circuit element absorbs power for V opposed to i + 1 - 2 Circuit element supplies power for V parallel to i + 1 - 2 V V i P = - Vi i P = + Vi Note: Some EE circuits texts use passive convention: treat Vdrop as positive. Copyright R. Janow – FALL 2019