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.

Definition of Current : Net rate of charge flow through some 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 Current is the same across each cross-section of a wire + - i Current density J may vary [J] = current/area At any Node (Junction) i1 i2 i3 i1+i2=i3 Kirchhoff’s Rules: (summary) 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 energy in E or B fields Energy or Power in a circuit:

Junction Rule Example – Current Conservation 7-1: What is the value of the current marked i ? 1 A. 2 A. 5 A. 7 A. Cannot determine from information given. 5 A 2 A 3 A 1 A 6 A i 3 A 1 A = 7 A 8 A

+ - Current density J: Current / Unit Area (Vector) 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 units: Amperes/m2 For uniform density: For non-uniform density: What makes current flow in a conductor? conductivity resistivity Yes, for isolated charges in vacuum. No, in a conducting solid, liquid, gas Recall: terminal velocity for falling object (collisions, drift speed) Do electrons in a current keep accelerating?

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

A./m2 Assume: Doubly charged positive ions EXAMPLE: Calculate the current density Jions for ions in a gas Assume: Doubly charged positive ions Density n = 2 x 108 ions/cm3 Ion drift speed vd = 105 m/s Find Jions – the current density for the ions only (forget Jelectrons) coul/ion ions/cm3 m/s cm3/m3 A./m2

Increasing the Current 7-2: When you increase the current in a wire, what changes and what is constant? 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

Resistivity “r” : Property of a material itself Definition of Resistance : Ratio of current flowing through a conductor to applied potential difference. i DV E L A R depends on the material & geometry Note: C= Q/DV – inverse to R R Circuit Diagram V 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: in addition R is independent of applied voltage Resistivity “r” : Property of a material itself 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 DV Assume cylindrical resistors For insulators: r  infinity

Example: calculating resistance or resistivity 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 mW, diameter d = 1 mm, length L = 2 m Use a table to identify material. Not Cu or Al, possibly an alloy

Change the temperature from reference T0 to T Resistivity depends on temperature: Resistivity depends on temperature: Higher temperature  greater thermal motion  more collisions  higher resistance. r in W.m @ 20o C. 1.72 x 10-8 copper 9.7 x 10-8 iron 2.30 x 10+3 pure silicon Reference Temperature SOME SAMPLE RESISTIVITY VALUES Change the temperature from reference T0 to T Coefficient a depends on the material Simple model of resistivity: a = temperature coefficient Conductivity is the reciprocal of resistivity i DV E L A Definition:

Resistivity Tables

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? 20 mA. 5 mA. 0.2 A. 200 A. 5 A. R V Circuit Diagram

Current Through a Resistor 7-4: If the current in the circuit is doubled, which of the following might be responsible? The voltage across the resistor might have doubled. The resistance of the resistor might have doubled. The resistance and voltage might have both doubled. The voltage across the resistor may have dropped by a factor of 2. 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

Definitions of resistance: Ohm’s Law and Ohmic materials Definitions of resistance: (R could depend on applied V) (r could depend on E) Definition: For OHMIC conductors and devices: Ratio of voltage drop to current is constant – NO dependance on applied voltage. 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

A Resistor Dissipating Power V + - LOAD 1 2 Apply voltage drop V across load resistor Current flows through load which dissipates energy The EMF (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 V is PE / unit charge - Charge = current x time element absorbs power for V opposed to i element supplies power for V parallel to i 1 2 i V + - Voltage rise: Pabsorbed = dot product of Vrise with i (active sign convention) Note: Some EE circuits texts use passive convention: treat Vdrop as positive. P = - Vi P = + Vi

EXAMPLE: EXAMPLE: EXAMPLE:

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

Circuit analysis with resistances and EMFs CIRCUITS CONSIST OF JUNCTIONS (ESSENTIAL NODES) ... and … ESSENTIAL BRANCHES (elements connected in series,one current/branch) ANALYSIS METHOD: Kirchhoff’S LAWS or RULES Current (Junction) Rule: Charge conservation Loop (Voltage) Rule: Energy conservation i DV slope = 1/R RESISTANCE: POWER: OHM’s LAW: R is independent of DV or i

EMFs “pump” charges to higher energy EMFs (independent voltage sources): move charges from low to high potential (raise potential energy). maintain constant potential at terminals for any value of current do work dW = Edq on charges (source of the energy in batteries is chemical) Unit: volts (V). Variable name: script E. Types of EMFs: batteries, electric generators, solar cells, fuel cells, etc. DC versus AC Power dissipated by resistor: Power supplied by EMF: + - R E i CONVENTION: Positive charges flow - from + to – outside of EMF - from – to + inside EMF

Ideal EMF device Real EMF device Multiple EMFs Zero internal battery resistance Open switch: EMF = E no 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= iV Power dissipated in EMF Pemf = i(E-V) = i2r 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 EB does + work, loses energy EA is charged up R converts PE to heat Load (motor, other) produces motion and/or heat

Potential around the circuit How to generate circuit equations from the Kirchhoff Loop Rule The sum of voltage changes = zero around every closed loop in a multi-loop circuit) Traverse one closed loop to generate each equation. 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 within 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 P = iE – i2r Power in External Ckt circuit dissipation battery drain P = iV = i(E – ir)

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 Loop Rule: The sum of the potential differences around a closed loop equals zero. Only one loop path exists: The equivalent circuit replaces the series resistors with a single equivalent resistance: 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 rule

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. 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

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. 4 R. 3 R. 2.5 R. 0.4 R. Cannot determine from information given. c R R R R a

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. 4 C. 3 C. 2.5 C. 0.4 C. Cannot determine from information given. c C C C C a

+ - + - EXAMPLE: Find i, V1, V2, V3, P1, P2, P3 EXAMPLE: R1= 10 W R2= 7 W R3= 8 W E = 7 V i EXAMPLE: Find currents and voltage drops i + - E = 9 V R1 R2 i1 i2

EXAMPLE: MULTIPLE BATTERIES SINGLE LOOP + - R1= 10 W R2= 15 W E1 = 8 V E2 = 3 V i A battery (EMF) absorbs power (charges up) when I is opposite to E

So why do electrical signals on wires seem 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