Phys2212_L04 - Physics 2212 Electricity and Magnetism...

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Physics 2212 Electricity and Magnetism Lecture 4 Computing Electric Fields
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10/06/09 Physics 2212 - Lecture 4 2 weak strong Review of Field Lines Q 2Q -Q Morecharge more field lines. Field lines stop on negativecharges. Field lines start on positivecharges. Field lines never cross. Field linespacing indicates field strength + + + Closed loops? Energy flow.
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10/06/09 Physics 2212 - Lecture 4 3 Recall : Computing Charged Object E-Fields Using Coulomb’s Law 1. Choosea coordinatesystem that will facilitateintegration. 2. Use any applicablesymmetries to set E-field components to zero or equal to each other. 3. Break up theobject into point-likeelements. 4. Write the Coulomb’s Law contribution to the E-field from a representativepoint-likeelement. 5. Integrate over theentireobject to get theE-field.
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10/06/09 Physics 2212 - Lecture 4 4 Recall: The E-Field of a Charged Ring A thin uniformly charged ring of radius R has a total charge Q . Find theelectric field on theaxis of thering .. thez-axis (perpendicular to thepage). Thelinear chargedensity of thering is λ = Q /(2 π R ). The system has cylindrical symmetry for rotations about theaxis, so along the z-axis E x =E y =0 and weneed only to find E z . Consider a small segment of thecircumferenceof thering of width dl = R d φ . Thecontribution to theelectric field on the z-axis is : 2 2 2 2 2 0 0 1 1 cos 4 4 z i dQ Rd z dE r R z R z λ φ θ πε πε = = + + ( 29 ( 29 ( 29 2 3/ 2 3/ 2 2 2 2 2 0 0 0 3/ 2 2 2 0 1 1 4 2 1 4 z Rz Rz E d R z R z Qz R z π λ λ φ πε ε πε = = + + = +
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10/06/09 Physics 2212 - Lecture 4 5 The on-axis E-Field of a Charged Ring (2) 3 0 1 ( ) 4 z Qz E z R R πε << = 2 0 1 ( ) 4 z Q E z R z πε = ( 29 3/2 2 2 0 1 4 z Qz E R z πε = + Near-field limit: linear Far-field limit: 1/r 2 (same as point charge)
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10/06/09 Physics 2212 - Lecture 4 6 Example: A Charged Ring A thin uniformly charged ring of radius R = 0.1 m has a total charge Q = 10 nC. Find theelectric field on theaxis of symmetry at z = 0.2 m from thecenter thering. ( 29 3/ 2 2 2 0 -8 9 2 2 2 2 3/ 2 3 1 4 (1.0 10 C)(0.2 m) (9.0 10 N m /C ) [(0.1 m) (0.2 m) ] 1.61 10 N/C z Qz E R z πε = + × = × + = ×
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10/06/09 Physics 2212 - Lecture 4 7 The E-Field of a Charged Disk A thin uniformly charged disk of radius R has a total charge Q . Find theelectric field E at points on thez axis perpendicular to the disk plane. Thechargearea-density of thedisk is η = Q/A = Q/( π R 2 ) .
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