Review on Coulomb’s Law and the electric field definitionExamplesSlide 3Slide 4Example – Charged DiskThe few examples that leads to Gauss’s lawChapter 24Electric Flux, perpendicular areaElectric Flux, with angle θElectric Flux, GeneralElectric Flux, Closed SurfaceFlux Through Closed Surface, cont.Flux Through Closed Surface, finalFlux Through a Cube, ExampleKarl Friedrich GaussGauss’s Law, IntroductionGauss’s Law from Coulomb’s LawGauss’s Law – GeneralGaussian Surface, ExampleGaussian Surface, Example 2Gauss’s Law – SummaryApplying Gauss’s LawConditions for a Gaussian SurfaceProblem type I: Field Due to a Spherically Symmetric Even Charge Distribution, including a point charge.Field inside the spherePlot the results (assume positive Q)Problem type II: Field at a Distance from a Straight Line of ChargeArguments for the flux calculationsNow apply Gauss Law to find the electric fieldProblem type III: Field Due to a Infinitely Large Plane of ChargeFind out the fluxSlide 32Other applications for Gauss Law: Electrostatic EquilibriumMore discussions about electrostatic equilibrium properties. Property 1: for a conductor, Fieldinside = 0Property 2: For a charged conductor, charge resides only on the surface, and the field inside the conductor is still zero.Property 3: Field’s Magnitude and Direction on the surfaceProperty 3: Field’s Magnitude and Direction, cont.Conducting Sphere and Shell ExampleSphere and Shell ExampleSphere and Shell Example, 3Slide 41Review on Coulomb’s Law and the electric field definitionCoulomb’s Law: the force between two point charges The electric field is defined asThe force a charge experiences in an electric filed121212rrF221022141rqqrqqketestqFE229C/mN 106987.8 ek22120mN/C 102854.8 EFqExamplesCalculate the electric field E at point P (0,0,z) generated by a ring of radius R, in the X-Y plane and its center at the origin of the coordinates. Total charge Q is evenly distributed on this ring. OXZYP(0,0,z)RZStep 1: formulas:testqFEandAPPrF2rdqqktesterdEAOZYP(0,0,z)RStep 2: known quantities: Q, R, Z.Step 3: Analyze to form the equation for the final solution:Example – Charged DiskThe ring has a radius R and a uniform charge density σChoose dq as a ring of radius rThe ring has a surface area 2π r drThe few examples that leads to Gauss’s lawElectric field of A point chargeAn infinitely long straight wire with evenly distributed chargeA wire loopA round diskAn infinitely large planeA solid sphere with evenly distributed chargeChapter 24Gauss’s LawElectric Flux, perpendicular areaElectric flux is the product of the magnitude of the electric field and the surface area, A, perpendicular to the field:ΦE = EACompare to a water flux in a tube:ΦW = –V1A1= V2A2This sign means water flows into the tubeElectric Flux, with angle θ The electric flux is proportional to the number of electric field lines penetrating some surfaceThe field lines may make some angle θ with the perpendicular to the surfaceThen ΦE = EA cosθ More precisely:And the electric field E has to be a constant all over the area A.AEcosEAEAEReview: direction of a surface = (outwards) normal to that surface.Electric Flux, GeneralIn the more general case, look at a small area elementIn general, this becomescosE i i i i iE AθDF = D = �DE Ar r0surfacelimiE i iAEE AdD �F = �DF = ���E Ar rThe surface integral means the integral must be evaluated over the surface in questionIn general, the value of the flux will depend both on the field pattern and on the surfaceThe unit of electric flux is N.m2/CElectric Flux, Closed SurfaceAssume a closed surfaceThe vectors point in different directionsAt each point, they are perpendicular to the surfaceBy convention, they point outwardiDArFlux Through Closed Surface, cont.At (1), the field lines are crossing the surface from the inside to the outside; θ < 90o, Φ is positiveAt (2), the field lines graze surface; θ = 90o, Φ = 0At (3), the field lines are crossing the surface from the outside to the inside;180o > θ > 90o, Φ is negativeFlux Through Closed Surface, finalThe net flux through the surface is proportional to the net number of lines leaving the surfaceThis net number of lines is the number of lines leaving the surface minus the number entering the surfaceIf En is the component of E perpendicular to the surface, then dAEdnEAEFlux Through a Cube, ExampleThe field lines pass through two surfaces perpendicularly and are parallel to the other four surfacesFor side 1, ΦE = -El 2For side 2, ΦE = El 2For the other sides, ΦE = 0Therefore, Φtotal = 0Karl Friedrich Gauss1777 – 1855Made contributions inElectromagnetismNumber theory like 1+2+3+…+100 = ?StatisticsNon-Euclidean geometryCometary orbital mechanicsA founder of the German Magnetic UnionStudies the Earth’s magnetic fieldGauss’s Law, IntroductionGauss’s law is an expression of the general relationship between the net electric flux through a closed surface and the charge enclosed by the surfaceThe closed surface is often called a gaussian surfaceGauss’s law is of fundamental importance in the study of electric fieldsGauss’s Law from Coulomb’s LawA positive point charge, q, is located at the center of a sphere of radius rAccording to Coulomb’s Law, the magnitude of the electric field everywhere on the surface of the sphere is The field lines are directed radially outward and are perpendicular to the surface at every point, soCombine these two equations, we have 24AE rEd AEEdAd AEdnE2rqkEe0222444qqkrrqkrEeeEGauss’s Law – GeneralThe net flux through any closed surface surrounding a charge q is given by q/εo and is independent of the shape of that surfaceThe net electric flux through a closed surface that surrounds no charge is zeroSince the electric field due to many charges is the vector sum of the electric fields produced by the individual charges, the flux through any closed surface can be expressed as 0AEqdE02121A)EE(AE...qqd...dEGaussian Surface, ExampleClosed surfaces
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