no tagsLecture 003: Electric Field and Simple Distributions of ChargeSteveSekula, 29 January 2011 (created 24 January 2011)Goals of this lectureDiscuss in detail Coulomb's Lawtwo point chargesDiscuss principle of superpositionmore than 2 point chargesDiscuss the concept of the electric field, the FOUNDATIONAL conceptof electricity and magnetismallows us to not worry about what charge is being acted upon andthus envision complex geometriesDiscuss the electric dipoledipole force and field, torque, etc.Discuss how to handle very large distributions of chargecalculus!Electric ForceAs indicated by demonstrations from last lecture, electric charge is able toexert a force. We tend not to notice this force most of the time because theelectrons and protons in our bodies, and in the work around us, are largelypaired up and thus electrically neutral (zero electric charge) on a humanscale.As the soda can and the candle flame demonstrations show, electric chargeand force go hand-in-hand at both the microscopic (ions in the candleflame) and macroscopic (soda can attracted by charged plastic) levels.Many observations and measurements of the relationship between:General Physics - E&M (PHY 1308) LectureNotesGeneral Physics - E&M (PHY 1308) LectureNotesGeneral Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...1 of 7 01/29/2011 11:19 AMThe magnitude of the charges involvedthe distance between the charges (it's direction AND magnitude)the sign of the chargesthe force exerted between chargeshave been carefully studied. The result is mathematical statement that hasbeen upheld by thousands upon thousands of repeated experiments carriedout over hundreds of years - a LAW. This law is known as "Coulomb's Law":where is the force VECTOR (magnitude and direction) that charge 1exerts on charge 2. is a constant, determined from repeatedexperimentation, whose value is:Let's draw a picture of this and illustrate the pieces of this formula. Itcombines two key areas of mathematics: standard algebra and vectoralgebra. The picture will help us to parse the meaning of this formula,considering two cases: a pair of like-signed charged, and a pair of opposite-signed charges.When solving a problem it's good to develop a strategy for attacking thinksstep-by-step with Coulomb's Law.Strategies for dealing with problems involving theforce between two particlesRemember: Coulomb's Law reminds us that force is a VECTOR, and thusCoulomb's Law provides us both with the MAGNITUDE and theDIRECTION of the force exerted by charge 1 on charge 2Interpret the problem to determine what you need to figure out: First,make sure that you're dealing with the electric force alone, and that no1.F r 12~=r2k Á q1Á q2^F 12~k k :0 0 N =C = 9 Â 19Á m2 2General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...2 of 7 01/29/2011 11:19 AMother forces also need to be considered. Identify the charge or chargeson which you want to calculate the force. Next, identify the charge orcharges producing the force - the SOURCE charges. In the aboveexample, we implicitly chose charge 1 and the source and wanted toknow the force it exerted on charge 2.Develop a set of information for attacking the problem: Begin with aDRAWING that shows the charges. If you're given coordinates for thecharges, place the charges on coordinates; if not, define a suitablecoordinate system that makes the problem easier. Determine the unitvectors in Coulomb's Law ( ). If two charges lie along the samecoordinate axis, then the unit vector will be one of , , . When thecharges don't lie simply on a single coordinate axis, find the unit vectorby writing the vector and determining the unit vector as follows:Please note that for simplicity, we will in the future write2.Evaluate the electric force: Apply Coulomb's Law, using the pieces youassembled during the Interpretation phase of the problem3.Assess the outcome: Think about your answer to see if it makes sense.Check the little things: does you force point in the direction youexpect, based on the charges involved? Compare to an order-of-magnitude estimate of the answer: ignoring the specific values of thenumbers involved and multiple just by the appropriate powers of ten,does your answer get within a factor of about 10 of your fully computedanswer?1.Problem Solving: Coulomb's LawTo attack a problem involving Coulomb's Law, you need to keep a fewdefinitions in mind: is the force that charge 1 exerts on charge 2 is the charge of the source charge (and is a signed quantity) and r ^i ^j ^k ^r 12~ r ^r : ^ Ñr12~jr j12~jr j : 12~ = r12F 12~q 1q 2General Physics - E&M (PHY 1308) - Lecture Notes file:///home/sekula/Dropbox/Documents/Notebook...3 of 7 01/29/2011 11:19 AMis the charge of the target charge (the one on which you are trying todetermine the force)the unit vector always points from the source charge to the targetchargedouble-check any results using what you know about charges:like charges REPELunlike charges ATTRACTSee lecture slides for a demonstration calculationPoint Charges and the Principle of SuperpositionWhen dealing with more than one pair of charges, you need a strategy forcomputing the force on a charge, , given a number of other charges (where runs from 1 to N and labels each of the remaining charges).Because force is a vector, to find the total force on you add the forces(vectors) exerted on by the charges . The force that exerts on isunaffected by the force exerts on - this allows us to superpose theindividual forces to find the total force. This is not obvious, but its realityhas been upheld by experiments and observations of nature. Nature didn'thave to be this simple, but it is.Why do you need this? Coulomb's Law applies to point charges - chargedobjects whose size is negligible. However, the real world is populated bycharge distributions - a collection of many charges spread out overspace. For instance:molecules are an example of distributions of charges - protons andelectrons - and those distributions matter when you are thinking abouthow different molecules interact with one another (and, since they aresimilarly sized, you cannot neglect their dimensions).your heart contains a charge distribution, which accumulates duringsystole (contraction of the heart) and causes heart muscle tissue tocontract and pump bloodTherefore, we are often confronted with situations where we need to dealwith a
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