no tagsLecture 011: Resistance to Electric CurrentSteveSekula, 28 February 2011 (created 24 February 2011)Current DensityWhat if electric current is occurring in a more complex medium (e.g. NOTin a wire)? What about through a cell membrane, which isn't simplydescribable as a single wire carrying ions into and out of the cell. In thatcase, we need a more general quantity: current density, the amount ofcurrent flowing per unit area of the path.Current density is by definition a vector, whose direction is given by thedirection of current flow at a given point and whose magnitude is thecurrent per unit area. Dividing out microscopic current equation by thearea of the conductor:We see the product of drift velocity direction and charge defines theoverall direction ( is always positive). So electron current densities pointopposite the velocity of electrons, as per our convention that current is inthe direction of positive charge motion.As an exercise, consider an ion pump in a cell membrane. What is an ionpump? It's a protein that is capable of allowing positive ions into or out ofthe cell itself. That protein is embedded in the cell membrane. We cananalyze the current and current density when a channel opens for 1ms andallows singly-ionized potassium ions through ( ). The radius ofthe channel, treating it like a little cylinder, is .What is the current? ANSWER: What is the current density? ANSWER: General Physics - E&M (PHY 1308) LectureNotesGeneral Physics - E&M (PHY 1308) LectureNotesJ qv : ~= nd~n 1:1 0 Â 14q e = +r :15mm = 0I Q=Át :8pA = Á = 1J =A :5 0 A=m = I = 2 Â 17 2General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...1 of 8 02/28/2011 07:56 PMConduction MechanismsThe mechanism by which conduction really occurs is beyond the scope ofthis class, but we can go a long way by using the following model (which isthe original model of conduction developed in the 1800s):charges are free to move in a conductor and respond to an electricfield by moving through the conductorconductors are not perfect, however, and the charges suffer collisionswith the atoms in the conductorFirst, electric fields in conductors? Didn't we use the fact that there are noconductors in electric fields to understand the capacitor problems?Remember, we're not in electrostatic equilibrium here. With the capacitor,we had a specific situation in which charge builds up across anuncrossable gap, and eventually current stops. Here, were are consideringa more general situation - when there is current, or when it doesn't stop. Inthat situation, we are NOT in electrostatic equilibrium and we CAN haveelectric fields in conductors.Collisions cause the charges to lose energy they had gained from theelectric field.These collisions provide an effective force that works againstthe electric field force:That means that there is a relationship between current density andelectric field that is not perfect, but somewhat diminished by the collisions.We can write this as: is a property of the material, and is called the conductivity of thematerial, and ranges between 0 and in magnitude.This is a microscopic version of something you might have heard of before:F total~= Felectric~+ Fcollisions~J E: ~= Û~Û 1 General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...2 of 8 02/28/2011 07:56 PM"Ohm's Law." Ohm's law relates the macroscopic current to the voltage. Wecan relate the microscopic and macroscopic laws.First of all, for most common conductors is independent of electric field(a constant). Such a materials is called "ohmic". Non-ohmic materials havea conductivity that DEPENDS on electric field. The microscopic version ofOhm's law is useful in biophysics, geophysics, astrophysics, and electricalengineering, studies that deal routinely with position-dependent electricfields. = constant: Ohmic: non-OhmicConductivity is a measure of how well charges respond to the electric field.A perfect conductor has , while a perfect insulator has . There isa related quantity, called resistivity:In terms of resistivity,JResistivity tells us how difficult it is for charge to move in a material. Thehigher the resistivity, the larger the electric field needed to provide thesame current density. The macroscopic property of electrical resistance isrelated to resistivity.The unit of resistivity is the . One is called an Ohm ( ), in honorof German Physicists Geog Ohm who explored the relationship betweencurrent and voltage. This the unit of resistivity is the , while units ofconductivity are .Resistivity and conductivity are some of the most variable properties weknow about.Û Û Û(E) ~Û = 1 Û = 0Ú =Û = 1~=ÚE~V =A Á m V=A Ê Ê Á m(Ê ) Á mÀ1General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...3 of 8 02/28/2011 07:56 PMMaterial Resistivity CopperGoldWaterHuman bloodSea waterSilicon 23.0GlassYou can use the current density relationship to electric field to determinethings like the electric field needed to drive current in your home orapartment. For instance, consider 15A of current moving through a copper wire. The electric field needed to drive this current is:Copper is really an excellent conductor, so it takes small electric fields todrive that current through your wiring. The value of copper in the currentglobal market is so great that during the peak of the recent housingforeclosure crisis, foreclosed homes were broken into and stripped ofcopper wiring, pipes, etc. Physics defines the technological value of thematerial, value defines the economic incentive, and that creates marketsand black markets. See? Physics is relevant. :-)The macroscopic version of Ohm's LawThe original version of Ohm's law, articulated by Ohm himself, is from themodern perspective the "macroscopic" version of this law. It depends noton the microscopic properties of the conductor and charge carriers, butrather on macroscopic properties such as the electric potential differenceand the overall flow of current. Ohm discovered that current the voltagecould be related linearly to one another in many materials, and this isexpressed as "Ohm's Law":Ê Á m1:68 0 Â 1À82:24 0 Â 1À82:6 0 Â 150:70 0:22 10 0 10À 1141:8mm jJ j Ej=Ú ! Ej jJ j 9mV=m ~= j~À j = Ú = ÚIA= 9General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...4 of 8 02/28/2011 07:56 PMIA given voltage
View Full Document
Unlocking...