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 Circuits and Currents Electric Current i Current Density J Drift Speed Resistance, Resistivity, Conductivity Ohm’s Law Power in Electric Circuits Examples Kirchhoff’s Rules applied to Circuits EMF’s - “Pumping” Charges Work, Energy, and EMF Simple Single Loop and Multi-Loop Circuits Summary

Current Definition: Net charge crossing a surface per unit time 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 junction i1 i2 i3 i1+i2=i3 Kirchhoff’s Rules: (summary) Junction Rule: S currents in = S currents out at any junction Voltage Rule: S DV’s = 0 for any closed path Current is the same across each cross-section of a wire + - i Current density J may vary [J] = current/area EMFs provide energy (electro-motive force) Resistances dissipate energy as heat Capacitances store energy in E field Inductances store energy in B field Energy 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

+ - 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? Microscopic level 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 vd conductivity resistivity

- + Drift speed: Do electrons in a current keep accelerating? 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. Drift speed in copper is 10-8 – 10-4 m/s. + - For E = 0 in conductor: no current, vD=0, J = 0, i = 0 For E not = 0 (battery voltage not 0): Note: Electrons drift rightward but vd and J are still leftward |q| = e = 1.6 x 10-19 C.

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 Resistance Definition: How much current flows through a device in response to a given 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 conducting wire. - 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. 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 DV Assume cylindrical resistors For insulators: r  infinity

Calculating resistance, given the 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

Resistivity Tables

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:

Current Through a Resistor 7-3: What is the current through the resistor in the following circuit, if V = 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 is doubled, which of the following might also have changed? The voltage across the resistor doubles. The resistance of the resistor doubles. The voltage in the wire between the battery and the resistor doubles. The voltage across the resistor drops by a factor of 2. The resistance of the resistor drops 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 (a special case) Definitions of resistance: but R could depend on applied V but r could depend on E Definition: OHMIC conductors and devices: Ratio of voltage drop to current is constant – it does not depend on applied voltage i.e., current is proportional to 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

Resistive Loads in Circuits Dissipate Power + - LOAD a b Apply voltage drop V across load Current flows through load which dissipates energy An EMF (e.g., a battery) does work, holding V and current i constant by expending potential energy As charge dq flows from a to b it loses P.E. = dU - potential is PE / unit charge - charge = current x time EXAMPLE: Space heater: Find rate of converting electrical energy to heat

EXAMPLE: EXAMPLE: EXAMPLE:

Circuit analysis with resistances and EMFs GENERAL ANALYSIS METHOD: Kirchhoff’S LAWS or RULES Junction Rule Charge conservation Loop Rule Energy conservation CIRCUIT ELEMENTS: PASSIVE: RESISTANCE, CAPACITANCE, INDUCTANCE ACTIVE: EMF’s (SOURCES OF POTENTIAL DIFFERENCE AND ENERGY) JUNCTIONS and BRANCHES i i1 i2 ….etc… RESISTANCE: POWER: i DV slope = 1/R OHM’s LAW: R is independent of DV or i

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 = Edq on charges (source of the energy in batteries is chemical) EMFs are “charge pumps” Unit: volts (V). Symbol: script E. Types of EMFs: batteries, electric generators, solar cells, fuel cells, etc. DC versus AC + - R E i CONVENTION: Current flows CW through circuit from + to – outside of EMF from – to + inside EMF Power dissipated by resistor: Power supplied by 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 also applied across load circuit Current & power not zero 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, 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 Generating Circuit Equations with the Kirchhoff Loop Rule Traversing a closed loop generates one equation. Multiple loops may be needed. The sum of voltage changes = zero around all closed loops in a multi-loop circuit) A branch is a series combination of devices between branches. Assume either current direction in each branch. Minus signs in the result may appear. Traverse each branch of the circuit with or against assumed current direction. Across resistances, voltage drop DV = - iR if following assumed current direction. Otherwise, voltage change is +iR. When crossing EMFs from – to +, DV = +E. Otherwise 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)

Equivalent resistance for resistors in series Junction Rule: The current through all of the resistances in series (a single branch) is identical. No information from Junction 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

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