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MIT 8 02T - Study Notes

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Class 13: OutlineExam 1 TopicsGeneral Exam SuggestionsFieldsPRS Questions:FieldsE Field and Potential: CreatingE Field and Potential: CreatingContinuous Sources: Charge DensityE Field and Potential: CreatingGauss’s Law:E Field and Potential: EffectsPRS Questions:Electric Fields and PotentialConductors in EquilibriumPRS Questions:ConductorsCapacitorsPRS Questions:CapacitorsDielectricsPRS Questions:DielectricsSAMPLE EXAM:The real exam has 5 concept, 3 analytical questionsQ: Point ChargesA: Point ChargesQ: Ring of ChargeA: Ring of ChargeQ: Spherical CapacitorA: Spherical CapacitorA: Spherical CapacitorQ: Find E from VA: Find E from VA: Find E from VQ: Parallel Plate CapacitorA: Parallel Plate CapacitorA: Parallel Plate CapacitorA: Parallel Plate Capacitor1P13-Class 13: OutlineHour 1:Concept Review / OverviewPRS Questions – possible exam questionsHour 2:Sample ExamEXAM Thursday: 7:30 – 9 pm2P13-Exam 1 Topics• Fields (visualizations)• Electric Field & Potential• Discrete Point Charges• Continuous Charge Distributions• Symmetric Distributions – Gauss’s Law• Conductors• Capacitance• Calculate for various geometries• Effects of dielectrics• Energy storage3P13-General Exam Suggestions• You should be able to complete every problem• If you are confused, ask• If it seems too hard, think some more• Look for hints in other problems• If you are doing math, you’re doing too much• Read directions completely (before & after)• Write down what you know before starting• Draw pictures, define (label) variables• Make sure that unknowns drop out of solution• Don’t forget units!4P13-FieldsGrass SeedsKnow how to readField LinesKnow how to draw• Field line density tells you field strength• Lines have tension (want to be straight)• Lines are repulsive (want to be far from other lines)• Lines begin and end on sources (charges) or ∞5P13-PRS Questions:Fields6P13-E Field and Potential: CreatingA point charge q creates a field and potential around it:2ˆ;eeqqkVkrr==ErGUse superposition for systems of charges;BBAAVVVV d=−∇ ∆ ≡ − =− ⋅∫EEsGGGThey are related:7P13-E Field and Potential: Creating2ˆ;eeqqkVkrr==ErGDiscrete set of point charges:Add up from each point charge2ˆ;eedq dqdk dVkrr==ErGContinuous charge distribution:Break charged object into small pieces, dq, and integrate8P13-Continuous Sources: Charge DensityCharge Densities:QAσ=QVρ=LQ=λdVdQρ=dLdQλ= dAdQσ=Don’t forget your geometry:2dA rdrπ=2cyldV rldrπ=24spheredV r drπ=dL dx=dL Rdθ=9P13-E Field and Potential: Creating2ˆ;eeqqkVkrr==ErGDiscrete set of point charges:Add up from each point charge2ˆ;eedq dqdk dVkrr==ErGContinuous charge distribution:Break charged object into small pieces, dq, and integrateSymmetric charged object:0S;inqdVε⋅= ∆≡−⋅∫∫ ∫EA E sGGGdUse Gauss’ law to get E everywhere, then integrate to get VGw10P13-0Sinqdε⋅=∫∫EAGGwGauss’s Law:GaussianPillboxSphericalSymmetryPlanar SymmetryCylindrical Symmetry11P13-E Field and Potential: Effectsq=FEGGIf you put a charged particle, q, in a field:To move a charged particle, q, in a field:WUqV=∆=∆12P13-PRS Questions:Electric Fields and Potential13P13-Conductors in EquilibriumConductors are equipotential objects:1) E = 0 inside2) Net charge inside is 03) E perpendicular to surface 4) Excess charge on surface5) Shielding – inside doesn’t “talk” to outside0εσ=E14P13-PRS Questions:ConductorsP13-CapacitorsTo calculate:1) Put on arbitrary ±Q 2) Calculate E3) Calculate ∆VQCV=∆CapacitanceIn Series & Parallel,parallel 1 2eqCCC=+series 1 2111eq,CCC=+Energy2332oEEudr drε==∫∫∫ ∫∫∫2221212VCVQCQU ∆=∆==16P13-PRS Questions:Capacitors17P13-DielectricsDielectrics locally weaken the electric field0;1EEκκ=≥0CCκ=Inserted into a capacitor: QCV=∆Hooked to a battery? Q increasesNot hooked up? V decreases18P13-PRS Questions:Dielectrics19P13-SAMPLE EXAM:The real exam has 5 concept, 3 analytical questions20P13-Q: Point ChargesA right isosceles triangle of side 2d has charges q, +2q and -q arranged on its vertices (see sketch). +q2q-qP2d2d(b) What is the potential at P, assuming V(∞)=0?(a) What is the electric field at point P, midway along the line connecting the +q and –q charges?(c) How much work to bring a charge -5Q from ∞ to P?21P13-A: Point ChargesAll charges a distance 2 from rd P=+q2q-qP2d2dr3kQr=∑ErGG()3342()()xkkqdEqdqdqdrr=++−−=33;xykkEQx E Qyrr→= =∑∑(a)()3()2 () 0ykEqdqdqdr=−+ +− =()22kQ k kqVqqqrr r==+−=∑V=(b)() ()()1055kqQWU QV QVPr−=∆ = − ∆ = − = W=(c)22P13-Q: Ring of ChargeA thin rod with a uniform charge per unit length λ is bent into the shape of a circle of radius Ra) Choose a coordinate system for the rod. Clearly indicate your choice of origin, and axes on the diagram above.b) Choose an infinitesimal charge element dq . Find an expression relating dq , λ, and your choice of length for dq .c) Find the vector components for the contribution of dq to the electric field along an axis perpendicular to the plane of the circle, a distance d above the plane of the circle. The axis passes through the center of the circle. Express the vector components in terms of your choice of unit vectorsd) What is the direction and magnitude of the electric field along the axis that passes through the center of the circle, perpendicular to the plane of the circle, and a distance d above the plane of the circle.e) What is the potential at that point, assuming V(∞)=0?23P13-A: Ring of Chargea) Origin & axes as pictureddq d Rdλλθ==Ab)dqθzxy3kdqdr=ErGGc)() ()22ˆˆˆcos sin ;RRdrRdθθ=− − + = +rijkGd) Horizontal components cancel, only find Ez3zzkdqEdE dr==∫∫230kdRdrπθλθ==∫32kd Rrλπ=e) Find the potential by same method:()kdqVd dVr==∫∫20kRdrπθλθ==∫2kRrλπ=d24P13-Q: Spherical CapacitorA conducting solid sphere of radius a, carrying a charge +Q is surrounded by a thin conducting spherical shell (inner radius b) with charge -Q .a) What is the direction and magnitude of the electric field E in the three regions below. Show how you obtain your expressions.1. r < a 2. a < r < b 3. r > bb) What is the electric potential V(r) in these same three regions. Take the electric potential to be zero at ∞.c) What is the electric potential difference between the outer shell and the inner cylinder, ∆V=V(b) - V(a)?d) What is the capacitance of this spherical capacitor?e) If a positive charge +2Q is


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MIT 8 02T - Study Notes

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